Wikiversity enwikiversity https://en.wikiversity.org/wiki/Wikiversity:Main_Page MediaWiki 1.39.0-wmf.25 first-letter Media Special Talk User User talk Wikiversity Wikiversity talk File File talk MediaWiki MediaWiki talk Template Template talk Help Help talk Category Category talk School School talk Portal Portal talk Topic Topic talk Collection Collection talk Draft Draft talk TimedText TimedText talk Module Module talk Gadget Gadget talk Gadget definition Gadget definition talk 4 28 2416471 2415595 2022-08-19T11:49:52Z Qgil-WMF 289584 /* Invitation to join the Movement Strategy Forum */ new section wikitext text/x-wiki {{Wikiversity:Colloquium/Header}} <!-- MESSAGES GO BELOW --> == TemplateScripts = Templates + JavaScript == Hi! I'd like to propose enabling [[c:Help:TemplateScripts|TemplateScripts]] on the English Wikiversity. It's not a MediaWiki extension, but a few lines of JavaScript added to [[MediaWiki:Common.js]] that basically allow to run JavaScript from templates, '''as long as the code is on the MediaWiki namespace and with the "TemplateScript-" prefix''', which requires an authorized user and community consensus to get there. The system is enabled on the Spanish Wikipedia where it's used for easy signing of polls and projects (see blue button [[:es:Wikiproyecto:Veganismo/participantes|here]]), for navigating [[Template:Excerpt#Excerpt trees|excerpt trees]] (see box with tree icon [[:es:Discusión:Ciencia|here]]), for injecting interactive widgets on some articles ([[:es:Hormiga de Langton|here]] and [[:es:Juego de la vida|here]]) and more recently for creating interactive forms that inject content into other pages (see template [[:es:Plantilla:Formulario|here]], soon to be used on admin boards). My immediate goal on Wikiversity is to use it to develop a tool to make [[Wikidebate|wikidebates]] more friendly. However I believe some of the existing scripts, particularly the ones for creating forms and signing pages, can be very useful on Wikiversity overall, as well as in some specific projects like [[Automata theory]] and [[Conway's Game of Life]]. So what do you think? [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|talk]]) 21:12, 29 June 2022 (UTC) :Hmm well, there's no support, but no objections either. If no one objects in a week or two, I may implement this since I got the necessary permission (and years of involvement in Wikiversity and other Wikimedia projects, I dare say). Cheers! [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 19:23, 1 August 2022 (UTC) ::Well, after a prudent amount of time with no objections, I just enabled TemplateScripts. See [[Template:Game of Life]] and [[Conway's Game of Life#Sandbox]] for a first example of a template using this feature. Cheers! [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 21:38, 13 August 2022 (UTC) :::As promised, I developed [[MediaWiki:TemplateScript-Wikidebate.js]] to make [[wikidebates]] more friendly, see for example the [add objection] buttons in [[Should cannabis be legal?]] [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 21:32, 15 August 2022 (UTC) ::::Thanks for doing this and sharing this example {{ping|Sophivorus}} -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:36, 16 August 2022 (UTC) ==[[Template:Notice]]== The rendering of this template now gives the message: Page Module:Message box/ambox.css has no content. This message shows on some pages that use the template e.g., the [[Template:Notice|template page itself]] and pages such as [[Motivation and emotion/Lectures/Introduction]] but not when tested on other pages e.g., [[User:Jtneill/sandbox]] or below {{Notice|I've checked the history of this and related templates, but couldn't see any recent changes}}. Anyone know what's happened and how to fix? {{ping|Dave Braunschweig}} Sincerely, James -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 08:36, 16 August 2022 (UTC) :@[[User:Jtneill|Jtneill]] Hi! Not sure what happened but I just created an empty [[Module:Message box/ambox.css]] to at least get rid of the error message. [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 12:27, 16 August 2022 (UTC) :I think it's related to a recent Wikimedia update about how CSS pages are managed. I just noticed [[Module:Message box/fmbox.css]] causing similar messages and created that one. If anyone notices similar messages, let us know or create an empty page to (temporarily?) resolve the issue. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 15:31, 16 August 2022 (UTC) == Invitation to join the Movement Strategy Forum == The [https://forum.movement-strategy.org/ Movement Strategy Forum] (MS Forum) is a multilingual collaborative space for all conversations about Movement Strategy implementation. We are inviting all Movement participants to collaborate on the MS Forum. The goal of the forum is to build community collaboration using an inclusive multilingual platform. The [[metawiki:Special:MyLanguage/Movement Strategy|Movement Strategy]] is a collaborative effort to imagine and build the future of the Wikimedia Movement. Anyone can contribute to the Movement Strategy, from a comment to a full-time project. Join this forum with your Wikimedia account, engage in conversations, and ask questions in your language. The Movement Strategy and Governance team launched the proposal for this MS Forum in May. After a 2-month review period, we have just published the [[metawiki:Movement Strategy/Forum/Community Review Report|Community Review Report]]. It includes a summary of the discussions, metrics, and information about the next steps. We look forward to seeing you at the MS Forum! [[User:Qgil-WMF|Qgil-WMF]] ([[User talk:Qgil-WMF|discuss]] • [[Special:Contributions/Qgil-WMF|contribs]]) 11:49, 19 August 2022 (UTC) gibaf1fwjd1x1h7lm9d6tfar7otfh66 4 5947 2416348 2415992 2022-08-18T23:06:04Z Dave Braunschweig 426084 Reverted edits by [[Special:Contributions/23.184.144.52|23.184.144.52]] ([[User_talk:23.184.144.52|talk]]) to last version by [[User:Dave Braunschweig|Dave Braunschweig]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki {{About Wikiversity}} Wikiversity is a project of the [[Foundation:Home|Wikimedia Foundation]]. To contact the Wikimedia Foundation, please refer to their [[Foundation:Contact_us|Contact]] webpage. == Contacting users on Wikiversity == There are a number of ways you can contact users on Wikiversity. You may * Post to a particular community member's [[Wikiversity:Talk page|talk page]]; * Post to a general interest page with a larger audience, such as the [[Wikiversity:Colloquium|Colloquium]] * [[Wikiversity:Chat|Chat]] with Wikiversity users; * Contact the [[Wikiversity:Support staff|support staff]] amongst the Wikiversity community; or * Write to the mailing list: [http://mail.wikimedia.org/mailman/listinfo/wikiversity-l Subscribe] to ''Wikiversity-l'' and browse the [http://mail.wikimedia.org/pipermail/wikiversity-l/ archives]. [[Wikiversity:Main Page|Return to the Wikiversity main page]] [[Category:Wikiversity]] [[de:Wikiversity:Kontakt]] [[el:Βοήθεια:Επικοινωνία]] [[pt:Wikiversidade:Contato]] neu6vmfiqcbrk23v8qsiaa31omn2jt7 0 32273 2416176 2413379 2022-08-18T16:22:36Z Nsebhastian 2948486 /* Semicolons */ wikitext text/x-wiki <noinclude>{{Header}}</noinclude> ==Basics== *JavaScript is a popular web-page scripting language, and is supported by almost every [[Browse|browser]].<ref>http://en.wikipedia.org/wiki/Comparison_of_web_browsers#JavaScript_support</ref><ref>Flanagan, David JavaScript:The Definitive Guide. O'Reilly Media, 2006 , p. 1.</ref> *It adds interactivity to [[w:Internet technologies|web-technology]] pages.<ref>Flanagan, David JavaScript:The Definitive Guide. O'Reilly Media, 2006 , p. 236.</ref> *You can use editing tools on your device or within supporting software. *JavaScript can work with the user's (client) resources or server resources. *It is easy to learn the basics. ===Step One: HTML & Scripts=== {{Activity | Open up your text editor of choice and create a basic page (a basic note pad works great). Now I am going to teach you how to write on your webpage using JavaScript. Type the following into your text editor: | <syntaxhighlight lang="html4strict"> <html> <head><title>Hello World!</title></head> <body> <script type="text/javascript"></syntaxhighlight> <syntaxhighlight lang="javascript">document.write('Hello World!');</syntaxhighlight><syntaxhighlight lang="html4strict"> </script> </body> </html> </syntaxhighlight> As you can see, our JavaScript instruction (<code>document.write('Hello World!');</code>) is added directly into your HTML by adding the <code><script></code> tag. Now save the text file with an HTML file extension. Use the "save as" function for your text editor and change the filename from something like "<code>New Text Document.txt</code>" to "<code>Hello World.html</code>". Use the .html in place of the .txt, .doc, .rtf, etc... Open your newly saved file in a browser. If all went well "Hello World!" will display in your browser as HTML content. If not, perhaps set your computers folder options to show filename extensions. Javascript and 'filename changes' are not always available (by default). }} Video Resources: [http://thenewboston.org/watch.php?cat=10&number=1 Javascript 1 ] To start off with, the <code><script type="text/javascript"></code> tag tells the browser that whatever comes between that tag and the coming <code></script></code> tag is script, and the <code>type="text/javascript"</code> tells it that it is JavaScript. Similarly, to use [[w:VBScript | VBScript]], you might use <code>type="text/vbscript"</code> instead of <code>"text/javascript"</code>. For the purposes of this course, though, you should only ever need to use the JavaScript tag. It is also important to note that the <code><script></code> tag is case sensitive. What this means is if you use lowercase letters in the tag, all the letters in the tag must be lowercase, and uppercase if you use any uppercase. For example, if you wrote <ScRiPt TyPe="TeXt/JaVaScRiPt"> Not only would you waste a lot of time typing, but you would also confuse your browser. You are probably wondering when I'm going to explain the rest of what you did. Well, <code>document.write()</code> is the JavaScript standard for writing text to the browser window. The 'document' clause refers to the HTML webpage (termed a document) as a whole; what follows ('<code>.write()</code>') is a command for the document object to carry out. In this case, you told the page to write the classic "<code>Hello World</code>" to the screen. ====Comments==== <syntaxhighlight lang="javascript">//(Brian Crowder) A 'Hello source-reader' contribution.</syntaxhighlight> . ===Step Two: Variables=== One of the things JavaScript is used for is the manipulation of data. In JavaScript, pieces of data can be stored either in variables or arrays. Variables are declared using the ''var'' keyword: var numValue; var textValue; var binaryValue; and can be given values when they are created: var numValue = 3; var textValue = "This is text."; var binaryValue = true; Multiple variables can even be declared at the same time: var numValue = 3, textValue = "This is text.", binaryValue = true; Variables in JavaScript are weakly typed, meaning that the types of individual variables are not determined by the programmer. Unlike many languages, JavaScript only provides a generic ''var'' rather than separate types for integers, floating point numbers, characters, and strings. For example, in the statement var x = 0; it is fairly obvious to any individual reading the code that <code>x</code> is a number. However, if the statement x = x + "This is text that is probably not a number"; were to appear later in the code, JavaScript would change the variable to a text variable. This is one of the reasons that JavaScript is referred to as weakly typed; at any particular point in time, it is impossible to tell exactly what type a particular variable is. ===Step Three: Operators=== ===== Arithmetic Operators ===== {| border=1 cellspacing=0 cellpadding=5 | '''Operator''' | '''Description''' | '''Example''' | '''Result''' |- | + | Addition | x = 2 y = 2 x + y | 4 |- | - | Subtraction | x = 5 y = 2 x - y | 3 |- | * | Multiplication | x = 5 y = 4 x * y | 20 |- | / | Division | 15 / 5 5 / 2 | 3 2.5 |- |% | Modulus (division remainder) | 5 % 2 10 % 4 10 % 2 | 1 2 0 |- | ++ | Increment | x = 5 x++ | 6 |- | -- ''(two adjacent [[w:Hyphen|hyphens]])'' | Decrement | x = 5 x-- | 4 |} ===== Assignment Operators ===== {| border=1 cellspacing=0 cellpadding=5 | '''Operator''' | '''Example''' | '''Is The Same As''' |- | = | x = y | x = y |- | += | x += y | x = x + y |- | -= | x -= y | x = x - y |- | *= | x *= y | x = x * y |- | /= | x /= y | x = x / y |- | %= | x %= y | x = x % y |} ===== Comparison Operators ===== {| border=1 cellspacing=0 cellpadding=5 | '''Operator''' | '''Description''' | '''Example''' |- | == | is equal to | 5 == 8 returns false |- | === | is equal to (checks for both value and type) | x = 5 y = "5" x == y returns true x === y returns false |- | != | is not equal | 5 != 8 returns true |- | > | is greater than | 5 > 8 returns false |- | < | is less than | 5 < 8 returns true |- | >= | is greater than or equal to | 5 >= 8 returns false |- | <= | is less than or equal to | 5 <= 8 returns true |} ===== Logical Operators ===== {| border=1 cellspacing=0 cellpadding=5 | '''Operator''' | '''Description''' | '''Example''' |- | && | and | x = 6 y = 3 (x < 10 && y > 1) returns true |- | <nowiki>||</nowiki><!-- Fixed --> | or | x = 6 y = 3 (x == 5 || y == 5) returns false |- | ! | not | x = 6 y = 3 <nowiki>!(x == y) returns true</nowiki> <!-- Fixed. --> |} ===== String Operator ===== The string operator is used to [[w:Concatenate|concatenate]] two strings together. For example: <blockquote><code> var x = "Hello " + "world!" </code></blockquote> {{Activity | Repeat our first activity with a variable and operator as shown below: | <syntaxhighlight lang="html4strict"> <!DOCTYPE html> <html> <head><title>Hello World, Again!</title></head> <body> <script type="text/javascript"></syntaxhighlight> <syntaxhighlight lang="javascript"> var x = "Hello " + "world!" document.write(x);</syntaxhighlight> <syntaxhighlight lang="html4strict"> </script> </body> </html> </syntaxhighlight> The above example results in the variable x equaling "Hello world!". *Save the document as helloAgain.html and open it in a browser. }} ===== Conditional Operator ===== JavaScript also contains a conditional operator that assigns a value to a variable based on some condition. The syntax would be:HTML <pre> variablename = ( condition ) ? value1 : value2 </pre> ''value1'' is the value of the expression if "''condition''" is fulfilled. ''value2'' is its value in the case "''condition''" is not fulfilled. For example: <blockquote><code> var x = "Hello " + "world!" </code></blockquote> {{Activity | Repeat our activity with a conditional operator that assigns a value to a variable as shown below: | <syntaxhighlight lang="html4strict"> <!DOCTYPE html> <html> <head><title>Hello World, Again!</title></head> <body> <script type="text/javascript"></syntaxhighlight> <syntaxhighlight lang="javascript"> var x = "hello"; var towrite = ( x == "hello")? "Hello world, again!" : "Look at the state of that!" document.write(towrite); //The example results in "Hello world, again!" because x is "hello" and condition "Hello world, again!" is logically true. </syntaxhighlight> <syntaxhighlight lang="html4strict"> </script> <p> Save the document as helloToo.html and open it in a browser. Change the line var x = "hello" to var x = "punk" and use CRTL + S to save. Run it in your browser. </p> </body> </html> </syntaxhighlight> }} == Semicolons == JavaScript require semicolons to be inserted to end statements and expressions.<ref>{{Cite web|url=https://sebhastian.com/the-proper-guide-to-semicolon-in-javascript/|title=The Proper Guide to Semicolon in JavaScript|last=Sebhastian|first=Nathan|website=Nathan Sebhastian|language=en|access-date=2022-08-18}}</ref> When you forget to add semicolon between commands, JavaScript adds it automatically so that it knows where to start the next command. The rules of JavaScript automatic semicolon insertion are as follows: # When the next line starts with a closing braces <code>}</code> , closing the current block # When there is a <code>return</code> statement without a semicolon # When there is a <code>break</code> statement without a semicolon # When there is a <code>throw</code> statement without a semicolon # When there is a <code>continue</code> statement without a semicolon # When the final statement of the code is reached (The last line on the JavaScript file) ==References== {{reflist}} <noinclude> {{CourseCat}} {{Footer}} </noinclude> pd26ma48d6qhakbhnxvfw32crp9xy6m 0 39681 2416466 2307305 2022-08-19T09:33:05Z 103.158.43.18 /* Elliptic, Hyperbolic, and Parabolic PDEs */ wikitext text/x-wiki Partial differential equations (PDEs) are the most common method by which we model physical problems in engineering. Finite element methods are one of many ways of solving PDEs. This handout reviews the basics of PDEs and discusses some of the classes of PDEs in brief. The contents are based on '' Partial Differential Equations in Mechanics'' volumes 1 and 2 by A.P.S. Selvadurai and ''Nonlinear Finite Elements of Continua and Structures'' by T. Belytschko, W.K. Liu, and B. Moran. == Definition of a PDE == A PDE is a relationship between an unknown function of several variables and its partial derivatives. Let <math>u(x_1, x_2, x_3, t)</math> be an unknown function. The '' independent'' variables are <math>x_1</math>, <math>x_2</math>, <math>x_3</math>, and <math>t</math>. We usually write :<math> u = u(x_1, x_2, x_3, t) </math> and say that <math>u</math> is the '' dependent'' variable. Partial derivatives are denoted by expressions such as :<math> u_{,1} = \frac{\partial u}{\partial x_1} ~;~~ u_{,2} = \frac{\partial u}{\partial x_2} ~;~~ u_{,11} = \frac{\partial^2 u}{\partial x_1\partial x_1} \equiv \frac{\partial^2 u}{\partial x_1^2} ~;~~ u_{,12} = \frac{\partial^2 u}{\partial x_1\partial x_2}~. </math> Some examples of partial differential equations are :<math>\begin{align} u_{,t} = u_{,1} + u_{,2} &\Leftrightarrow \frac{\partial u}{\partial t} = \frac{\partial u}{\partial x_1} + \frac{\partial u}{\partial x_2} \\ \nabla^2 u = 0 \Leftrightarrow u_{,11} + u_{,22} + u_{,33} = 0 &\Leftrightarrow \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2} = 0 \\ u_{,1111} = u_{,22} + u &\Leftrightarrow \frac{\partial^4 u}{\partial x_1^4} = \frac{\partial^2 u}{\partial x_2^2} + u ~. \end{align}</math> An example of a system of partial differential equations is :<math> \boldsymbol{\nabla} (\boldsymbol{\nabla} \bullet \mathbf{u}) + \nabla^2 \mathbf{u} + \mathbf{f} = \mathbf{0} \Leftrightarrow u_{k,ki} + u_{i,jj} + f_i = 0 </math> In expanded form this system of equations is :<math>\begin{align} \frac{\partial^2 u_1}{\partial x_1^2} + \frac{\partial^2 u_2}{\partial x_2\partial x_1} + \frac{\partial^2 u_3}{\partial x_3\partial x_1} + \frac{\partial^2 u_1}{\partial x_1^2} + \frac{\partial^2 u_1}{\partial x_2^2} + \frac{\partial^2 u_1}{\partial x_3^2} + f_1 & = 0 \\ \frac{\partial^2 u_1}{\partial x_1\partial x_2} + \frac{\partial^2 u_2}{\partial x_2^2} + \frac{\partial^2 u_3}{\partial x_3\partial x_2} + \frac{\partial^2 u_2}{\partial x_1^2} + \frac{\partial^2 u_2}{\partial x_2^2} + \frac{\partial^2 u_2}{\partial x_3^2} + f_2 & = 0 \\ \frac{\partial^2 u_1}{\partial x_1\partial x_3} + \frac{\partial^2 u_2}{\partial x_2\partial x_3} + \frac{\partial^2 u_3}{\partial x_3^2} + \frac{\partial^2 u_3}{\partial x_1^2} + \frac{\partial^2 u_3}{\partial x_2^2} + \frac{\partial^2 u_3}{\partial x_3^2} + f_3 & = 0 \end{align}</math> It is often more convenient to write PDEs in vector notation or index notation. == Order of a PDE == The order of a PDE is determined by the highest derivative in the equation. For example, :<math>\begin{align} \frac{\partial u}{\partial t} - \frac{\partial u}{\partial x} & = 0 ~~~~\text{is a first-order PDE.}\\ \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} + \frac{\partial^2 u}{\partial x_3^2} & = 0 ~~~~\text{is a second-order PDE.}\\ \frac{\partial^4 u}{\partial x_1^4} + \frac{\partial^2 u}{\partial x_2^2} - u & = 0 ~~~~\text{is a fourth-order PDE.}\\ \left(\frac{\partial u}{\partial x_1}\right)^3 + \frac{\partial u}{\partial x_2} + u^4 & = 0 ~~~~\text{is a first-order PDE.} \end{align}</math> == Linear and nonlinear PDEs == A '' linear'' PDE is one that is of first degree in all of its field variables and partial derivatives. For example, :<math>\begin{align} \frac{\partial u}{\partial x_1} + \frac{\partial u}{\partial x_2} & = 0 ~~~\text{is linear}~.\\ \frac{\partial u}{\partial x_1} + \left(\frac{\partial u}{\partial x_2}\right)^2 & = 0 ~~~\text{is nonlinear}~.\\ \frac{\partial u}{\partial x_1} + \frac{\partial u}{\partial x_2} + u^2 & = 0 ~~~\text{is nonlinear}~.\\ \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2} & = x_1 ~~~\text{is linear}~.\\ \frac{\partial^2 u}{\partial x_1^2} + u\frac{\partial^2 u}{\partial x_2^2} & = 0 ~~~\text{is quasilinear}~. \end{align}</math> The above equations can also be written in operator notation as :<math>\begin{align} D(u) = 0 & ~~\text{where}~~ D := \frac{\partial }{\partial x_1} + \frac{\partial }{\partial x_2}~. \\ D(u) = 0 & ~~\text{where}~~ D := \frac{\partial }{\partial x_1} + \left(\frac{\partial }{\partial x_2}\right)^2~.\\ D(u) = 0 & ~~\text{where}~~ D := \frac{\partial }{\partial x_1} + \frac{\partial }{\partial x_2} + u^2~.\\ D(u) = x_1 & ~~\text{where}~~ D := \frac{\partial^2 }{\partial x_1^2} + \frac{\partial^2 }{\partial x_2^2}~.\\ D(u) = 0 & ~~\text{where}~~ D := \frac{\partial^2 }{\partial x_1^2} + u\frac{\partial^2 }{\partial x_2^2}~. \end{align}</math> == Homogeneous PDEs == Let <math>L</math> be a linear operator. Then a linear partial differential equation can be written in the form :<math> L(u) = f(x_1,x_2,x_3,t)~. </math> If <math>f(x_1,x_2,x_3,t) = 0</math>, the PDE is called ''homogeneous''. For example, :<math>\begin{align} \frac{\partial u}{\partial t} + \frac{\partial u}{\partial x_1} + \frac{\partial u}{\partial x_2} + \frac{\partial u}{\partial x_3} & = 0 ~~~\text{is homogeneous}~.\\ \frac{\partial u}{\partial t} + \frac{\partial u}{\partial x_1} + \frac{\partial u}{\partial x_2} + \frac{\partial u}{\partial x_3} & = x_1 + x_2 ~~~\text{is nonhomogeneous}~.\\ \end{align}</math> == Elliptic, Hyperbolic, and Parabolic PDEs == We usually come across three-types of second-order PDEs in mechanics. These are classified as '' elliptic'', '' hyperbolic'', and '' parabolic''. The equations of elasticity (without inertial terms) are '' elliptic PDEs''. '' Hyperbolic PDEs'' describe wave propagation phenomena. The heat conduction equation is an example of a '' parabolic PDE''. Each type of PDE has certain characteristics that help determine if a particular finite element approach is appropriate to the problem being described by the PDE. Interestingly, just knowing the type of PDE can give us insight into how smooth the solution is, how fast information propagates, and the effect of initial and boundary conditions. * In hyperbolic PDEs, the smoothness of the solution depends on the smoothness of the initial and boundary conditions. For instance, if there is a jump in the data at the start or at the boundaries, then the jump will propagate as a shock in the solution. If, in addition, the PDE is '' nonlinear'', then shocks may develop even though the initial conditions and the boundary conditions are smooth. In a system modeled with a hyperbolic PDE, information travels at a finite speed referred to as the ''wavespeed''. Information is not transmitted until the wave arrives. * In contrast, the solutions of elliptic PDEs are always smooth, even if the initial and boundary conditions are rough (though there may be singularities at sharp corners). In addition, boundary data at any point affect the solution at all points in the domain. * Parabolic PDEs are usually time dependent and represent the diffusion-like processes. Solutions are smooth in space but may possess singularities. However, information travels at infinite speed in a parabolic system. Suppose we have a second-order PDE of the form :<math> a(x_1,x_2) \frac{\partial^2 u}{\partial x_1^2} + b(x_1,x_2) \frac{\partial^2 u}{\partial x_1\partial x_2} + c(x_1,x_2) \frac{\partial^2 u}{\partial x_2^2} + d(x_1,x_2) \frac{\partial u}{\partial x_1} + e(x_1,x_2) \frac{\partial u}{\partial x_2} + f(x_1,x_2) u = g(x_1,x_2) </math> Then, the PDE is called '' elliptic'' if :<math> { b^2 - 4ac < 0 } </math> An example is :<math> \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_1\partial x_2} + \frac{\partial^2 u}{\partial x_2^2} = x_1 \frac{\partial u}{\partial x_1} </math> The PDE is called '' hyperbolic'' if :<math> { b^2 - 4ac > 0 } </math> An example is :<math> \frac{\partial^2 u}{\partial x_1^2} + 3\frac{\partial^2 u}{\partial x_1\partial x_2} + \frac{\partial^2 u}{\partial x_2^2} = x_1 \frac{\partial u}{\partial x_1} </math> The PDE is called '' parabolic'' if :<math> { b^2 - 4ac = 0 } </math> An example is :<math> \frac{\partial^2 u}{\partial x_1^2} + 2\frac{\partial^2 u}{\partial x_1\partial x_2} + \frac{\partial^2 u}{\partial x_2^2} = x_1 \frac{\partial u}{\partial x_1} </math> == Solutions to Common PDEs == Partial differential equation appear in several areas of physics and engineering. A firm grasp of how to solve [[:Ordinary differential equations|ordinary differential equations]] is required to solve PDEs. In particular, solutions to the Sturm-Liouville problems should be familiar to anyone attempting to solve PDEs. *A [[/Laplace Equation/|tutorial on how to solve the Laplace equation]] *A [[/Poisson Equation/|tutorial on how to solve the Poisson equation]] *A [[/Separation of variables method/|tutorial on how to apply the method of separation of variables]] [[Category:Finite element analysis]] [[Category:Computational solid mechanics]] [[Category:Mathematics]] [[Category:Nonlinear finite elements]] == Application of PDEs in Physics and Engineering == There are many applications of partial differential equations in physics and engineering. Here are some examples: * The diffusion equation ** The [[Heat equation|heat equation]] * The wave equation * [[Maxwell's equations|Maxwell's equations]] == Resources == * [[w:Numerical partial differential equations]] * [http://www.youtube.com/view_play_list?p=EC88901EBADDD980 MIT 18.03 Differential Equations, Spring 2006] [[fr:Équation différentielle]] The Heat conduction equation of 2-D is elliptic in space and parabolic in time. [[Category:Partial differential equations| ]] [[Category:Mathematics]] dtvpci7mc5khi6ek1n9p20hsjlf9ibh 0 54718 2416166 2415878 2022-08-18T14:40:02Z Huz and Buz 2928717 /* Chapter 25 */ wikitext text/x-wiki {{RightTOC}} <[[Bible, English, King James, According to the documentary hypothesis]] According to the [[w:documentary hypothesis|documentary hypothesis]], [[w:Numbers|Numbers]] is composed from a number of originally independant sources joined by a [[w:redaction|redactor]]. There follows the text of Numbers in the [[w:King James Version|King James Version]], with sources highlighted according to the documentary hypothesis *The [[w:Priestly source|"Priestly source"]] is highlighted in {{font|color=#888800|olive yellow}} <small>{{font|color=#0000FF|([[Bible, English, King James, Documentary Hypothesis, Priestly source|view in isolation]])}}</small> *The [[w:Jahwist|"Jahwist source"]] is highlighted in {{font|color=#000088|navy blue}} <small>{{font|color=#0000FF|([[Bible, English, King James, Documentary Hypothesis, JE source, Jahwist source|view in isolation]])}}</small> *The [[w:Elohist|"Elohist source"]] is highlighted in {{font|color=#008888|teal blueish grey}} <small>{{font|color=#0000FF|([[Bible, English, King James, Documentary Hypothesis, JE source, Elohist source|view in isolation]])}}</small> *The [[w:Torah redactor|"Additions by the redactor"]] are highlighted in {{font|color=#880000|maroon red}} ==Chapter 1== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses in the wilderness of Sinai, in the tabernacle of the congregation, on the first day of the second month, in the second year after they were come out of the land of Egypt, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Take ye the sum of all the congregation of the children of Israel, after their families, by the house of their fathers, with the number of their names, every male by their polls;}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| From twenty years old and upward, all that are able to go forth to war in Israel: thou and Aaron shall number them by their armies.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And with you there shall be a man of every tribe; every one head of the house of his fathers.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And these are the names of the men that shall stand with you: of the tribe of Reuben; Elizur the son of Shedeur.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Of Simeon; Shelumiel the son of Zurishaddai.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Of Judah; Nahshon the son of Amminadab.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| Of Issachar; Nethaneel the son of Zuar.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| Of Zebulun; Eliab the son of Helon.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| Of the children of Joseph: of Ephraim; Elishama the son of Ammihud: of Manasseh; Gamaliel the son of Pedahzur.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| Of Benjamin; Abidan the son of Gideoni.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| Of Dan; Ahiezer the son of Ammishaddai.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Of Asher; Pagiel the son of Ocran.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Of Gad; Eliasaph the son of Deuel.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| Of Naphtali; Ahira the son of Enan.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| These were the renowned of the congregation, princes of the tribes of their fathers, heads of thousands in Israel.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And Moses and Aaron took these men which are expressed by their names:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And they assembled all the congregation together on the first day of the second month, and they declared their pedigrees after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, by their polls.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| As the LORD commanded Moses, so he numbered them in the wilderness of Sinai.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And the children of Reuben, Israel's eldest son, by their generations, after their families, by the house of their fathers, according to the number of the names, by their polls, every male from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Reuben, were forty and six thousand and five hundred.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Of the children of Simeon, by their generations, after their families, by the house of their fathers, those that were numbered of them, according to the number of the names, by their polls, every male from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Simeon, were fifty and nine thousand and three hundred.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| Of the children of Gad, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Gad, were forty and five thousand six hundred and fifty.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| Of the children of Judah, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Judah, were threescore and fourteen thousand and six hundred. }} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| Of the children of Issachar, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Issachar, were fifty and four thousand and four hundred.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| Of the children of Zebulun, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Zebulun, were fifty and seven thousand and four hundred.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| Of the children of Joseph, namely, of the children of Ephraim, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Ephraim, were forty thousand and five hundred.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| Of the children of Manasseh, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Manasseh, were thirty and two thousand and two hundred.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| Of the children of Benjamin, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Benjamin, were thirty and five thousand and four hundred.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| Of the children of Dan, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Dan, were threescore and two thousand and seven hundred.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| Of the children of Asher, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Asher, were forty and one thousand and five hundred.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| Of the children of Naphtali, throughout their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Naphtali, were fifty and three thousand and four hundred.}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| These are those that were numbered, which Moses and Aaron numbered, and the princes of Israel, being twelve men: each one was for the house of his fathers.}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| So were all those that were numbered of the children of Israel, by the house of their fathers, from twenty years old and upward, all that were able to go forth to war in Israel; }} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| Even all they that were numbered were six hundred thousand and three thousand and five hundred and fifty.}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| But the Levites after the tribe of their fathers were not numbered among them.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| For the LORD had spoken unto Moses, saying,}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| Only thou shalt not number the tribe of Levi, neither take the sum of them among the children of Israel:}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| But thou shalt appoint the Levites over the tabernacle of testimony, and over all the vessels thereof, and over all things that belong to it: they shall bear the tabernacle, and all the vessels thereof; and they shall minister unto it, and shall encamp round about the tabernacle.}} <small>{{font|color=#0000FF|51}}</small>{{font|color=#888800| And when the tabernacle setteth forward, the Levites shall take it down: and when the tabernacle is to be pitched, the Levites shall set it up: and the stranger that cometh nigh shall be put to death.}} <small>{{font|color=#0000FF|52}}</small>{{font|color=#888800| And the children of Israel shall pitch their tents, every man by his own camp, and every man by his own standard, throughout their hosts.}} <small>{{font|color=#0000FF|53}}</small>{{font|color=#888800| But the Levites shall pitch round about the tabernacle of testimony, that there be no wrath upon the congregation of the children of Israel: and the Levites shall keep the charge of the tabernacle of testimony.}} <small>{{font|color=#0000FF|54}}</small>{{font|color=#888800| And the children of Israel did according to all that the LORD commanded Moses, so did they.}} ==Chapter 2== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Every man of the children of Israel shall pitch by his own standard, with the ensign of their father's house: far off about the tabernacle of the congregation shall they pitch.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And on the east side toward the rising of the sun shall they of the standard of the camp of Judah pitch throughout their armies: and Nahshon the son of Amminadab shall be captain of the children of Judah.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were threescore and fourteen thousand and six hundred.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And those that do pitch next unto him shall be the tribe of Issachar: and Nethaneel the son of Zuar shall be captain of the children of Issachar.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And his host, and those that were numbered thereof, were fifty and four thousand and four hundred.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Then the tribe of Zebulun: and Eliab the son of Helon shall be captain of the children of Zebulun.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And his host, and those that were numbered thereof, were fifty and seven thousand and four hundred.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| All that were numbered in the camp of Judah were an hundred thousand and fourscore thousand and six thousand and four hundred, throughout their armies. These shall first set forth.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| On the south side shall be the standard of the camp of Reuben according to their armies: and the captain of the children of Reuben shall be Elizur the son of Shedeur.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And his host, and those that were numbered thereof, were forty and six thousand and five hundred.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And those which pitch by him shall be the tribe of Simeon: and the captain of the children of Simeon shall be Shelumiel the son of Zurishaddai.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were fifty and nine thousand and three hundred.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Then the tribe of Gad: and the captain of the sons of Gad shall be Eliasaph the son of Reuel.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were forty and five thousand and six hundred and fifty.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| All that were numbered in the camp of Reuben were an hundred thousand and fifty and one thousand and four hundred and fifty, throughout their armies. And they shall set forth in the second rank.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| Then the tabernacle of the congregation shall set forward with the camp of the Levites in the midst of the camp: as they encamp, so shall they set forward, every man in his place by their standards.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| On the west side shall be the standard of the camp of Ephraim according to their armies: and the captain of the sons of Ephraim shall be Elishama the son of Ammihud.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were forty thousand and five hundred.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And by him shall be the tribe of Manasseh: and the captain of the children of Manasseh shall be Gamaliel the son of Pedahzur.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were thirty and two thousand and two hundred.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Then the tribe of Benjamin: and the captain of the sons of Benjamin shall be Abidan the son of Gideoni.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were thirty and five thousand and four hundred.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| All that were numbered of the camp of Ephraim were an hundred thousand and eight thousand and an hundred, throughout their armies. And they shall go forward in the third rank.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| The standard of the camp of Dan shall be on the north side by their armies: and the captain of the children of Dan shall be Ahiezer the son of Ammishaddai.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were threescore and two thousand and seven hundred.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And those that encamp by him shall be the tribe of Asher: and the captain of the children of Asher shall be Pagiel the son of Ocran.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were forty and one thousand and five hundred.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Then the tribe of Naphtali: and the captain of the children of Naphtali shall be Ahira the son of Enan.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were fifty and three thousand and four hundred.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| All they that were numbered in the camp of Dan were an hundred thousand and fifty and seven thousand and six hundred. They shall go hindmost with their standards.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| These are those which were numbered of the children of Israel by the house of their fathers: all those that were numbered of the camps throughout their hosts were six hundred thousand and three thousand and five hundred and fifty.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| But the Levites were not numbered among the children of Israel; as the LORD commanded Moses.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| And the children of Israel did according to all that the LORD commanded Moses: so they pitched by their standards, and so they set forward, every one after their families, according to the house of their fathers.}} ==Chapter 3== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| These also are the generations of Aaron and Moses in the day that the LORD spake with Moses in mount Sinai.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And these are the names of the sons of Aaron; Nadab the firstborn, and Abihu, Eleazar, and Ithamar.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| These are the names of the sons of Aaron, the priests which were anointed, whom he consecrated to minister in the priest's office.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And Nadab and Abihu died before the LORD, when they offered strange fire before the LORD, in the wilderness of Sinai, and they had no children: and Eleazar and Ithamar ministered in the priest's office in the sight of Aaron their father.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Bring the tribe of Levi near, and present them before Aaron the priest, that they may minister unto him.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And they shall keep his charge, and the charge of the whole congregation before the tabernacle of the congregation, to do the service of the tabernacle.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And they shall keep all the instruments of the tabernacle of the congregation, and the charge of the children of Israel, to do the service of the tabernacle.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And thou shalt give the Levites unto Aaron and to his sons: they are wholly given unto him out of the children of Israel.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And thou shalt appoint Aaron and his sons, and they shall wait on their priest's office: and the stranger that cometh nigh shall be put to death.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying, }} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And I, behold, I have taken the Levites from among the children of Israel instead of all the firstborn that openeth the matrix among the children of Israel: therefore the Levites shall be mine;}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Because all the firstborn are mine; for on the day that I smote all the firstborn in the land of Egypt I hallowed unto me all the firstborn in Israel, both man and beast: mine shall they be: I am the LORD.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And the LORD spake unto Moses in the wilderness of Sinai, saying,}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| Number the children of Levi after the house of their fathers, by their families: every male from a month old and upward shalt thou number them.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And Moses numbered them according to the word of the LORD, as he was commanded.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And these were the sons of Levi by their names; Gershon, and Kohath, and Merari.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And these are the names of the sons of Gershon by their families; Libni, and Shimei.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And the sons of Kohath by their families; Amram, and Izehar, Hebron, and Uzziel.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And the sons of Merari by their families; Mahli, and Mushi. These are the families of the Levites according to the house of their fathers.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Of Gershon was the family of the Libnites, and the family of the Shimites: these are the families of the Gershonites.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Those that were numbered of them, according to the number of all the males, from a month old and upward, even those that were numbered of them were seven thousand and five hundred.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| The families of the Gershonites shall pitch behind the tabernacle westward.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| And the chief of the house of the father of the Gershonites shall be Eliasaph the son of Lael.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the charge of the sons of Gershon in the tabernacle of the congregation shall be the tabernacle, and the tent, the covering thereof, and the hanging for the door of the tabernacle of the congregation,}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And the hangings of the court, and the curtain for the door of the court, which is by the tabernacle, and by the altar round about, and the cords of it for all the service thereof.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And of Kohath was the family of the Amramites, and the family of the Izeharites, and the family of the Hebronites, and the family of the Uzzielites: these are the families of the Kohathites.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| In the number of all the males, from a month old and upward, were eight thousand and six hundred, keeping the charge of the sanctuary.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| The families of the sons of Kohath shall pitch on the side of the tabernacle southward.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| And the chief of the house of the father of the families of the Kohathites shall be Elizaphan the son of Uzziel.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| And their charge shall be the ark, and the table, and the candlestick, and the altars, and the vessels of the sanctuary wherewith they minister, and the hanging, and all the service thereof.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And Eleazar the son of Aaron the priest shall be chief over the chief of the Levites, and have the oversight of them that keep the charge of the sanctuary.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| Of Merari was the family of the Mahlites, and the family of the Mushites: these are the families of Merari.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| And those that were numbered of them, according to the number of all the males, from a month old and upward, were six thousand and two hundred.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| And the chief of the house of the father of the families of Merari was Zuriel the son of Abihail: these shall pitch on the side of the tabernacle northward.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And under the custody and charge of the sons of Merari shall be the boards of the tabernacle, and the bars thereof, and the pillars thereof, and the sockets thereof, and all the vessels thereof, and all that serveth thereto,}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| And the pillars of the court round about, and their sockets, and their pins, and their cords.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| But those that encamp before the tabernacle toward the east, even before the tabernacle of the congregation eastward, shall be Moses, and Aaron and his sons, keeping the charge of the sanctuary for the charge of the children of Israel; and the stranger that cometh nigh shall be put to death.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| All that were numbered of the Levites, which Moses and Aaron numbered at the commandment of the LORD, throughout their families, all the males from a month old and upward, were twenty and two thousand.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| And the LORD said unto Moses, Number all the firstborn of the males of the children of Israel from a month old and upward, and take the number of their names.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| And thou shalt take the Levites for me (I am the LORD) instead of all the firstborn among the children of Israel; and the cattle of the Levites instead of all the firstlings among the cattle of the children of Israel.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| And Moses numbered, as the LORD commanded him, all the firstborn among the children of Israel.}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| And all the firstborn males by the number of names, from a month old and upward, of those that were numbered of them, were twenty and two thousand two hundred and threescore and thirteen.}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying, }} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| Take the Levites instead of all the firstborn among the children of Israel, and the cattle of the Levites instead of their cattle; and the Levites shall be mine: I am the LORD.}} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| And for those that are to be redeemed of the two hundred and threescore and thirteen of the firstborn of the children of Israel, which are more than the Levites;}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| Thou shalt even take five shekels apiece by the poll, after the shekel of the sanctuary shalt thou take them: (the shekel is twenty gerahs:)}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| And thou shalt give the money, wherewith the odd number of them is to be redeemed, unto Aaron and to his sons.}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| And Moses took the redemption money of them that were over and above them that were redeemed by the Levites:}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| Of the firstborn of the children of Israel took he the money; a thousand three hundred and threescore and five shekels, after the shekel of the sanctuary:}} <small>{{font|color=#0000FF|51}}</small>{{font|color=#888800| And Moses gave the money of them that were redeemed unto Aaron and to his sons, according to the word of the LORD, as the LORD commanded Moses.}} ==Chapter 4== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying, }} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Take the sum of the sons of Kohath from among the sons of Levi, after their families, by the house of their fathers,}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| From thirty years old and upward even until fifty years old, all that enter into the host, to do the work in the tabernacle of the congregation.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| This shall be the service of the sons of Kohath in the tabernacle of the congregation, about the most holy things:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And when the camp setteth forward, Aaron shall come, and his sons, and they shall take down the covering vail, and cover the ark of testimony with it:}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And shall put thereon the covering of badgers' skins, and shall spread over it a cloth wholly of blue, and shall put in the staves thereof.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And upon the table of shewbread they shall spread a cloth of blue, and put thereon the dishes, and the spoons, and the bowls, and covers to cover withal: and the continual bread shall be thereon:}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And they shall spread upon them a cloth of scarlet, and cover the same with a covering of badgers' skins, and shall put in the staves thereof.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And they shall take a cloth of blue, and cover the candlestick of the light, and his lamps, and his tongs, and his snuffdishes, and all the oil vessels thereof, wherewith they minister unto it:}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And they shall put it and all the vessels thereof within a covering of badgers' skins, and shall put it upon a bar.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And upon the golden altar they shall spread a cloth of blue, and cover it with a covering of badgers' skins, and shall put to the staves thereof:}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And they shall take all the instruments of ministry, wherewith they minister in the sanctuary, and put them in a cloth of blue, and cover them with a covering of badgers' skins, and shall put them on a bar:}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And they shall take away the ashes from the altar, and spread a purple cloth thereon:}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And they shall put upon it all the vessels thereof, wherewith they minister about it, even the censers, the fleshhooks, and the shovels, and the basons, all the vessels of the altar; and they shall spread upon it a covering of badgers' skins, and put to the staves of it.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And when Aaron and his sons have made an end of covering the sanctuary, and all the vessels of the sanctuary, as the camp is to set forward; after that, the sons of Kohath shall come to bear it: but they shall not touch any holy thing, lest they die. These things are the burden of the sons of Kohath in the tabernacle of the congregation.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And to the office of Eleazar the son of Aaron the priest pertaineth the oil for the light, and the sweet incense, and the daily meat offering, and the anointing oil, and the oversight of all the tabernacle, and of all that therein is, in the sanctuary, and in the vessels thereof.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And the LORD spake unto Moses and unto Aaron saying,}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| Cut ye not off the tribe of the families of the Kohathites from among the Levites:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| But thus do unto them, that they may live, and not die, when they approach unto the most holy things: Aaron and his sons shall go in, and appoint them every one to his service and to his burden:}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| But they shall not go in to see when the holy things are covered, lest they die.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Take also the sum of the sons of Gershon, throughout the houses of their fathers, by their families;}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| From thirty years old and upward until fifty years old shalt thou number them; all that enter in to perform the service, to do the work in the tabernacle of the congregation.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| This is the service of the families of the Gershonites, to serve, and for burdens:}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And they shall bear the curtains of the tabernacle, and the tabernacle of the congregation, his covering, and the covering of the badgers' skins that is above upon it, and the hanging for the door of the tabernacle of the congregation,}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And the hangings of the court, and the hanging for the door of the gate of the court, which is by the tabernacle and by the altar round about, and their cords, and all the instruments of their service, and all that is made for them: so shall they serve.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| At the appointment of Aaron and his sons shall be all the service of the sons of the Gershonites, in all their burdens, and in all their service: and ye shall appoint unto them in charge all their burdens.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| This is the service of the families of the sons of Gershon in the tabernacle of the congregation: and their charge shall be under the hand of Ithamar the son of Aaron the priest.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| As for the sons of Merari, thou shalt number them after their families, by the house of their fathers;}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| From thirty years old and upward even unto fifty years old shalt thou number them, every one that entereth into the service, to do the work of the tabernacle of the congregation.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| And this is the charge of their burden, according to all their service in the tabernacle of the congregation; the boards of the tabernacle, and the bars thereof, and the pillars thereof, and sockets thereof,}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And the pillars of the court round about, and their sockets, and their pins, and their cords, with all their instruments, and with all their service: and by name ye shall reckon the instruments of the charge of their burden.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| This is the service of the families of the sons of Merari, according to all their service, in the tabernacle of the congregation, under the hand of Ithamar the son of Aaron the priest.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| And Moses and Aaron and the chief of the congregation numbered the sons of the Kohathites after their families, and after the house of their fathers,}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| From thirty years old and upward even unto fifty years old, every one that entereth into the service, for the work in the tabernacle of the congregation:}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And those that were numbered of them by their families were two thousand seven hundred and fifty.}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| These were they that were numbered of the families of the Kohathites, all that might do service in the tabernacle of the congregation, which Moses and Aaron did number according to the commandment of the LORD by the hand of Moses.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| And those that were numbered of the sons of Gershon, throughout their families, and by the house of their fathers,}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| From thirty years old and upward even unto fifty years old, every one that entereth into the service, for the work in the tabernacle of the congregation,}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| Even those that were numbered of them, throughout their families, by the house of their fathers, were two thousand and six hundred and thirty.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| These are they that were numbered of the families of the sons of Gershon, of all that might do service in the tabernacle of the congregation, whom Moses and Aaron did number according to the commandment of the LORD.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| And those that were numbered of the families of the sons of Merari, throughout their families, by the house of their fathers,}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| From thirty years old and upward even unto fifty years old, every one that entereth into the service, for the work in the tabernacle of the congregation,}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| Even those that were numbered of them after their families, were three thousand and two hundred.}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| These be those that were numbered of the families of the sons of Merari, whom Moses and Aaron numbered according to the word of the LORD by the hand of Moses.}} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| All those that were numbered of the Levites, whom Moses and Aaron and the chief of Israel numbered, after their families, and after the house of their fathers,}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| From thirty years old and upward even unto fifty years old, every one that came to do the service of the ministry, and the service of the burden in the tabernacle of the congregation.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| Even those that were numbered of them, were eight thousand and five hundred and fourscore,}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| According to the commandment of the LORD they were numbered by the hand of Moses, every one according to his service, and according to his burden: thus were they numbered of him, as the LORD commanded Moses.}} ==Chapter 5== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Command the children of Israel, that they put out of the camp every leper, and every one that hath an issue, and whosoever is defiled by the dead:}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| Both male and female shall ye put out, without the camp shall ye put them; that they defile not their camps, in the midst whereof I dwell.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And the children of Israel did so, and put them out without the camp: as the LORD spake unto Moses, so did the children of Israel.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying, }} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Speak unto the children of Israel, When a man or woman shall commit any sin that men commit, to do a trespass against the LORD, and that person be guilty;}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Then they shall confess their sin which they have done: and he shall recompense his trespass with the principal thereof, and add unto it the fifth part thereof, and give it unto him against whom he hath trespassed.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| But if the man have no kinsman to recompense the trespass unto, let the trespass be recompensed unto the LORD, even to the priest; beside the ram of the atonement, whereby an atonement shall be made for him.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And every offering of all the holy things of the children of Israel, which they bring unto the priest, shall be his.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And every man's hallowed things shall be his: whatsoever any man giveth the priest, it shall be his.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| Speak unto the children of Israel, and say unto them, If any man's wife go aside, and commit a trespass against him,}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And a man lie with her carnally, and it be hid from the eyes of her husband, and be kept close, and she be defiled, and there be no witness against her, neither she be taken with the manner;}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And the spirit of jealousy come upon him, and he be jealous of his wife, and she be defiled: or if the spirit of jealousy come upon him, and he be jealous of his wife, and she be not defiled:}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| Then shall the man bring his wife unto the priest, and he shall bring her offering for her, the tenth part of an ephah of barley meal; he shall pour no oil upon it, nor put frankincense thereon; for it is an offering of jealousy, an offering of memorial, bringing iniquity to remembrance.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And the priest shall bring her near, and set her before the LORD:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And the priest shall take holy water in an earthen vessel; and of the dust that is in the floor of the tabernacle the priest shall take, and put it into the water:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And the priest shall set the woman before the LORD, and uncover the woman's head, and put the offering of memorial in her hands, which is the jealousy offering: and the priest shall have in his hand the bitter water that causeth the curse:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And the priest shall charge her by an oath, and say unto the woman, If no man have lain with thee, and if thou hast not gone aside to uncleanness with another instead of thy husband, be thou free from this bitter water that causeth the curse:}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| But if thou hast gone aside to another instead of thy husband, and if thou be defiled, and some man have lain with thee beside thine husband:}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Then the priest shall charge the woman with an oath of cursing, and the priest shall say unto the woman, The LORD make thee a curse and an oath among thy people, when the LORD doth make thy thigh to rot, and thy belly to swell;}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And this water that causeth the curse shall go into thy bowels, to make thy belly to swell, and thy thigh to rot: And the woman shall say, Amen, amen.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And the priest shall write these curses in a book, and he shall blot them out with the bitter water:}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| And he shall cause the woman to drink the bitter water that causeth the curse: and the water that causeth the curse shall enter into her, and become bitter.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| Then the priest shall take the jealousy offering out of the woman's hand, and shall wave the offering before the LORD, and offer it upon the altar:}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And the priest shall take an handful of the offering, even the memorial thereof, and burn it upon the altar, and afterward shall cause the woman to drink the water.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And when he hath made her to drink the water, then it shall come to pass, that, if she be defiled, and have done trespass against her husband, that the water that causeth the curse shall enter into her, and become bitter, and her belly shall swell, and her thigh shall rot: and the woman shall be a curse among her people.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| And if the woman be not defiled, but be clean; then she shall be free, and shall conceive seed.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| This is the law of jealousies, when a wife goeth aside to another instead of her husband, and is defiled;}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| Or when the spirit of jealousy cometh upon him, and he be jealous over his wife, and shall set the woman before the LORD, and the priest shall execute upon her all this law.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| Then shall the man be guiltless from iniquity, and this woman shall bear her iniquity.}} ==Chapter 6== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Speak unto the children of Israel, and say unto them, When either man or woman shall separate themselves to vow a vow of a Nazarite, to separate themselves unto the LORD:}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| He shall separate himself from wine and strong drink, and shall drink no vinegar of wine, or vinegar of strong drink, neither shall he drink any liquor of grapes, nor eat moist grapes, or dried.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| All the days of his separation shall he eat nothing that is made of the vine tree, from the kernels even to the husk.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| All the days of the vow of his separation there shall no razor come upon his head: until the days be fulfilled, in the which he separateth himself unto the LORD, he shall be holy, and shall let the locks of the hair of his head grow.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| All the days that he separateth himself unto the LORD he shall come at no dead body.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| He shall not make himself unclean for his father, or for his mother, for his brother, or for his sister, when they die: because the consecration of his God is upon his head.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| All the days of his separation he is holy unto the LORD.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And if any man die very suddenly by him, and he hath defiled the head of his consecration; then he shall shave his head in the day of his cleansing, on the seventh day shall he shave it.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And on the eighth day he shall bring two turtles, or two young pigeons, to the priest, to the door of the tabernacle of the congregation:}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And the priest shall offer the one for a sin offering, and the other for a burnt offering, and make an atonement for him, for that he sinned by the dead, and shall hallow his head that same day.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And he shall consecrate unto the LORD the days of his separation, and shall bring a lamb of the first year for a trespass offering: but the days that were before shall be lost, because his separation was defiled.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And this is the law of the Nazarite, when the days of his separation are fulfilled: he shall be brought unto the door of the tabernacle of the congregation:}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And he shall offer his offering unto the LORD, one he lamb of the first year without blemish for a burnt offering, and one ewe lamb of the first year without blemish for a sin offering, and one ram without blemish for peace offerings,}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And a basket of unleavened bread, cakes of fine flour mingled with oil, and wafers of unleavened bread anointed with oil, and their meat offering, and their drink offerings.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And the priest shall bring them before the LORD, and shall offer his sin offering, and his burnt offering:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And he shall offer the ram for a sacrifice of peace offerings unto the LORD, with the basket of unleavened bread: the priest shall offer also his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And the Nazarite shall shave the head of his separation at the door of the tabernacle of the congregation, and shall take the hair of the head of his separation, and put it in the fire which is under the sacrifice of the peace offerings.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And the priest shall take the sodden shoulder of the ram, and one unleavened cake out of the basket, and one unleavened wafer, and shall put them upon the hands of the Nazarite, after the hair of his separation is shaven:}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And the priest shall wave them for a wave offering before the LORD: this is holy for the priest, with the wave breast and heave shoulder: and after that the Nazarite may drink wine.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| This is the law of the Nazarite who hath vowed, and of his offering unto the LORD for his separation, beside that that his hand shall get: according to the vow which he vowed, so he must do after the law of his separation.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| Speak unto Aaron and unto his sons, saying, On this wise ye shall bless the children of Israel, saying unto them,}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| The LORD bless thee, and keep thee:}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| The LORD make his face shine upon thee, and be gracious unto thee:}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| The LORD lift up his countenance upon thee, and give thee peace.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And they shall put my name upon the children of Israel, and I will bless them.}} ==Chapter 7== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And it came to pass on the day that Moses had fully set up the tabernacle, and had anointed it, and sanctified it, and all the instruments thereof, both the altar and all the vessels thereof, and had anointed them, and sanctified them;}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| That the princes of Israel, heads of the house of their fathers, who were the princes of the tribes, and were over them that were numbered, offered:}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And they brought their offering before the LORD, six covered wagons, and twelve oxen; a wagon for two of the princes, and for each one an ox: and they brought them before the tabernacle.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| Take it of them, that they may be to do the service of the tabernacle of the congregation; and thou shalt give them unto the Levites, to every man according to his service.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And Moses took the wagons and the oxen, and gave them unto the Levites.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Two wagons and four oxen he gave unto the sons of Gershon, according to their service:}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And four wagons and eight oxen he gave unto the sons of Merari, according unto their service, under the hand of Ithamar the son of Aaron the priest.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| But unto the sons of Kohath he gave none: because the service of the sanctuary belonging unto them was that they should bear upon their shoulders.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And the princes offered for dedicating of the altar in the day that it was anointed, even the princes offered their offering before the altar.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And the LORD said unto Moses, They shall offer their offering, each prince on his day, for the dedicating of the altar.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And he that offered his offering the first day was Nahshon the son of Amminadab, of the tribe of Judah:}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And his offering was one silver charger, the weight thereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them were full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| One spoon of ten shekels of gold, full of incense:}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Nahshon the son of Amminadab.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| On the second day Nethaneel the son of Zuar, prince of Issachar, did offer:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| He offered for his offering one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| One spoon of gold of ten shekels, full of incense:}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Nethaneel the son of Zuar.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| On the third day Eliab the son of Helon, prince of the children of Zebulun, did offer:}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Eliab the son of Helon.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| On the fourth day Elizur the son of Shedeur, prince of the children of Reuben, did offer:}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| His offering was one silver charger of the weight of an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Elizur the son of Shedeur.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| On the fifth day Shelumiel the son of Zurishaddai, prince of the children of Simeon, did offer:}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Shelumiel the son of Zurishaddai.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| On the sixth day Eliasaph the son of Deuel, prince of the children of Gad, offered:}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| His offering was one silver charger of the weight of an hundred and thirty shekels, a silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Eliasaph the son of Deuel.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| On the seventh day Elishama the son of Ammihud, prince of the children of Ephraim, offered:}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|51}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|52}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|53}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Elishama the son of Ammihud.}} <small>{{font|color=#0000FF|54}}</small>{{font|color=#888800| On the eighth day offered Gamaliel the son of Pedahzur, prince of the children of Manasseh:}} <small>{{font|color=#0000FF|55}}</small>{{font|color=#888800| His offering was one silver charger of the weight of an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|56}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|57}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|58}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|59}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Gamaliel the son of Pedahzur.}} <small>{{font|color=#0000FF|60}}</small>{{font|color=#888800| On the ninth day Abidan the son of Gideoni, prince of the children of Benjamin, offered:}} <small>{{font|color=#0000FF|61}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|62}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|63}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|64}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|65}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Abidan the son of Gideoni.}} <small>{{font|color=#0000FF|66}}</small>{{font|color=#888800| On the tenth day Ahiezer the son of Ammishaddai, prince of the children of Dan, offered:}} <small>{{font|color=#0000FF|67}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|68}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|69}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|70}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|71}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Ahiezer the son of Ammishaddai.}} <small>{{font|color=#0000FF|72}}</small>{{font|color=#888800| On the eleventh day Pagiel the son of Ocran, prince of the children of Asher, offered:}} <small>{{font|color=#0000FF|73}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|74}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|75}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|76}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|77}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Pagiel the son of Ocran.}} <small>{{font|color=#0000FF|78}}</small>{{font|color=#888800| On the twelfth day Ahira the son of Enan, prince of the children of Naphtali, offered:}} <small>{{font|color=#0000FF|79}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|80}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|81}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|82}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|83}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Ahira the son of Enan.}} <small>{{font|color=#0000FF|84}}</small>{{font|color=#888800| This was the dedication of the altar, in the day when it was anointed, by the princes of Israel: twelve chargers of silver, twelve silver bowls, twelve spoons of gold:}} <small>{{font|color=#0000FF|85}}</small>{{font|color=#888800| Each charger of silver weighing an hundred and thirty shekels, each bowl seventy: all the silver vessels weighed two thousand and four hundred shekels, after the shekel of the sanctuary:}} <small>{{font|color=#0000FF|86}}</small>{{font|color=#888800| The golden spoons were twelve, full of incense, weighing ten shekels apiece, after the shekel of the sanctuary: all the gold of the spoons was an hundred and twenty shekels.}} <small>{{font|color=#0000FF|87}}</small>{{font|color=#888800| All the oxen for the burnt offering were twelve bullocks, the rams twelve, the lambs of the first year twelve, with their meat offering: and the kids of the goats for sin offering twelve.}} <small>{{font|color=#0000FF|88}}</small>{{font|color=#888800| And all the oxen for the sacrifice of the peace offerings were twenty and four bullocks, the rams sixty, the he goats sixty, the lambs of the first year sixty. This was the dedication of the altar, after that it was anointed.}} <small>{{font|color=#0000FF|89}}</small>{{font|color=#888800| And when Moses was gone into the tabernacle of the congregation to speak with him, then he heard the voice of one speaking unto him from off the mercy seat that was upon the ark of testimony, from between the two cherubims: and he spake unto him.}} ==Chapter 8== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Speak unto Aaron and say unto him, When thou lightest the lamps, the seven lamps shall give light over against the candlestick.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And Aaron did so; he lighted the lamps thereof over against the candlestick, as the LORD commanded Moses.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And this work of the candlestick was of beaten gold, unto the shaft thereof, unto the flowers thereof, was beaten work: according unto the pattern which the LORD had shewed Moses, so he made the candlestick.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Take the Levites from among the children of Israel, and cleanse them.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And thus shalt thou do unto them, to cleanse them: Sprinkle water of purifying upon them, and let them shave all their flesh, and let them wash their clothes, and so make themselves clean.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| Then let them take a young bullock with his meat offering, even fine flour mingled with oil, and another young bullock shalt thou take for a sin offering.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And thou shalt bring the Levites before the tabernacle of the congregation: and thou shalt gather the whole assembly of the children of Israel together:}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And thou shalt bring the Levites before the LORD: and the children of Israel shall put their hands upon the Levites:}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And Aaron shall offer the Levites before the LORD for an offering of the children of Israel, that they may execute the service of the LORD.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And the Levites shall lay their hands upon the heads of the bullocks: and thou shalt offer the one for a sin offering, and the other for a burnt offering, unto the LORD, to make an atonement for the Levites.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And thou shalt set the Levites before Aaron, and before his sons, and offer them for an offering unto the LORD.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Thus shalt thou separate the Levites from among the children of Israel: and the Levites shall be mine.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And after that shall the Levites go in to do the service of the tabernacle of the congregation: and thou shalt cleanse them, and offer them for an offering.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| For they are wholly given unto me from among the children of Israel; instead of such as open every womb, even instead of the firstborn of all the children of Israel, have I taken them unto me.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| For all the firstborn of the children of Israel are mine, both man and beast: on the day that I smote every firstborn in the land of Egypt I sanctified them for myself.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And I have taken the Levites for all the firstborn of the children of Israel.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And I have given the Levites as a gift to Aaron and to his sons from among the children of Israel, to do the service of the children of Israel in the tabernacle of the congregation, and to make an atonement for the children of Israel: that there be no plague among the children of Israel, when the children of Israel come nigh unto the sanctuary.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And Moses, and Aaron, and all the congregation of the children of Israel, did to the Levites according unto all that the LORD commanded Moses concerning the Levites, so did the children of Israel unto them.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And the Levites were purified, and they washed their clothes; and Aaron offered them as an offering before the LORD; and Aaron made an atonement for them to cleanse them.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And after that went the Levites in to do their service in the tabernacle of the congregation before Aaron, and before his sons: as the LORD had commanded Moses concerning the Levites, so did they unto them.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| This is it that belongeth unto the Levites: from twenty and five years old and upward they shall go in to wait upon the service of the tabernacle of the congregation:}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And from the age of fifty years they shall cease waiting upon the service thereof, and shall serve no more:}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| But shall minister with their brethren in the tabernacle of the congregation, to keep the charge, and shall do no service. Thus shalt thou do unto the Levites touching their charge.}} ==Chapter 9== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses in the wilderness of Sinai, in the first month of the second year after they were come out of the land of Egypt, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Let the children of Israel also keep the passover at his appointed season.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| In the fourteenth day of this month, at even, ye shall keep it in his appointed season: according to all the rites of it, and according to all the ceremonies thereof, shall ye keep it.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And Moses spake unto the children of Israel, that they should keep the passover.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And they kept the passover on the fourteenth day of the first month at even in the wilderness of Sinai: according to all that the LORD commanded Moses, so did the children of Israel.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And there were certain men, who were defiled by the dead body of a man, that they could not keep the passover on that day: and they came before Moses and before Aaron on that day:}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And those men said unto him, We are defiled by the dead body of a man: wherefore are we kept back, that we may not offer an offering of the LORD in his appointed season among the children of Israel?}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And Moses said unto them, Stand still, and I will hear what the LORD will command concerning you.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| Speak unto the children of Israel, saying, If any man of you or of your posterity shall be unclean by reason of a dead body, or be in a journey afar off, yet he shall keep the passover unto the LORD.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| The fourteenth day of the second month at even they shall keep it, and eat it with unleavened bread and bitter herbs.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| They shall leave none of it unto the morning, nor break any bone of it: according to all the ordinances of the passover they shall keep it.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| But the man that is clean, and is not in a journey, and forbeareth to keep the passover, even the same soul shall be cut off from among his people: because he brought not the offering of the LORD in his appointed season, that man shall bear his sin.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And if a stranger shall sojourn among you, and will keep the passover unto the LORD; according to the ordinance of the passover, and according to the manner thereof, so shall he do: ye shall have one ordinance, both for the stranger, and for him that was born in the land.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And on the day that the tabernacle was reared up the cloud covered the tabernacle, namely, the tent of the testimony: and at even there was upon the tabernacle as it were the appearance of fire, until the morning.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| So it was alway: the cloud covered it by day, and the appearance of fire by night.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And when the cloud was taken up from the tabernacle, then after that the children of Israel journeyed: and in the place where the cloud abode, there the children of Israel pitched their tents.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| At the commandment of the LORD the children of Israel journeyed, and at the commandment of the LORD they pitched: as long as the cloud abode upon the tabernacle they rested in their tents.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And when the cloud tarried long upon the tabernacle many days, then the children of Israel kept the charge of the LORD, and journeyed not.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And so it was, when the cloud was a few days upon the tabernacle; according to the commandment of the LORD they abode in their tents, and according to the commandment of the LORD they journeyed.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And so it was, when the cloud abode from even unto the morning, and that the cloud was taken up in the morning, then they journeyed: whether it was by day or by night that the cloud was taken up, they journeyed.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Or whether it were two days, or a month, or a year, that the cloud tarried upon the tabernacle, remaining thereon, the children of Israel abode in their tents, and journeyed not: but when it was taken up, they journeyed.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| At the commandment of the LORD they rested in the tents, and at the commandment of the LORD they journeyed: they kept the charge of the LORD, at the commandment of the LORD by the hand of Moses.}} ==Chapter 10== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Make thee two trumpets of silver; of a whole piece shalt thou make them: that thou mayest use them for the calling of the assembly, and for the journeying of the camps.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And when they shall blow with them, all the assembly shall assemble themselves to thee at the door of the tabernacle of the congregation.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And if they blow but with one trumpet, then the princes, which are heads of the thousands of Israel, shall gather themselves unto thee.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| When ye blow an alarm, then the camps that lie on the east parts shall go forward.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| When ye blow an alarm the second time, then the camps that lie on the south side shall take their journey: they shall blow an alarm for their journeys.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| But when the congregation is to be gathered together, ye shall blow, but ye shall not sound an alarm.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And the sons of Aaron, the priests, shall blow with the trumpets; and they shall be to you for an ordinance for ever throughout your generations.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And if ye go to war in your land against the enemy that oppresseth you, then ye shall blow an alarm with the trumpets; and ye shall be remembered before the LORD your God, and ye shall be saved from your enemies.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| Also in the day of your gladness, and in your solemn days, and in the beginnings of your months, ye shall blow with the trumpets over your burnt offerings, and over the sacrifices of your peace offerings; that they may be to you for a memorial before your God: I am the LORD your God.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And it came to pass on the twentieth day of the second month, in the second year, that the cloud was taken up from off the tabernacle of the testimony.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And the children of Israel took their journeys out of the wilderness of Sinai; and the cloud rested in the wilderness of Paran.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And they first took their journey according to the commandment of the LORD by the hand of Moses.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| In the first place went the standard of the camp of the children of Judah according to their armies: and over his host was Nahshon the son of Amminadab.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And over the host of the tribe of the children of Issachar was Nethaneel the son of Zuar.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And over the host of the tribe of the children of Zebulun was Eliab the son of Helon.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And the tabernacle was taken down; and the sons of Gershon and the sons of Merari set forward, bearing the tabernacle.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And the standard of the camp of Reuben set forward according to their armies: and over his host was Elizur the son of Shedeur.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And over the host of the tribe of the children of Simeon was Shelumiel the son of Zurishaddai.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And over the host of the tribe of the children of Gad was Eliasaph the son of Deuel.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And the Kohathites set forward, bearing the sanctuary: and the other did set up the tabernacle against they came.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And the standard of the camp of the children of Ephraim set forward according to their armies: and over his host was Elishama the son of Ammihud.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And over the host of the tribe of the children of Manasseh was Gamaliel the son of Pedahzur.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| And over the host of the tribe of the children of Benjamin was Abidan the son of Gideoni.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the standard of the camp of the children of Dan set forward, which was the rereward of all the camps throughout their hosts: and over his host was Ahiezer the son of Ammishaddai.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And over the host of the tribe of the children of Asher was Pagiel the son of Ocran.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And over the host of the tribe of the children of Naphtali was Ahira the son of Enan.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| Thus were the journeyings of the children of Israel according to their armies, when they set forward.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#000088| And Moses said unto Hobab, the son of Raguel the Midianite, Moses' father in law, We are journeying unto the place of which the LORD said, I will give it you: come thou with us, and we will do thee good: for the LORD hath spoken good concerning Israel.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#000088| And he said unto him, I will not go; but I will depart to mine own land, and to my kindred.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#000088| And he said, Leave us not, I pray thee; forasmuch as thou knowest how we are to encamp in the wilderness, and thou mayest be to us instead of eyes.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#000088| And it shall be, if thou go with us, yea, it shall be, that what goodness the LORD shall do unto us, the same will we do unto thee.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#000088| And they departed from the mount of the LORD three days' journey: and the ark of the covenant of the LORD went before them in the three days' journey, to search out a resting place for them.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#000088| And the cloud of the LORD was upon them by day, when they went out of the camp.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#000088| And it came to pass, when the ark set forward, that Moses said, Rise up, LORD, and let thine enemies be scattered; and let them that hate thee flee before thee.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#000088| And when it rested, he said, Return, O LORD, unto the many thousands of Israel.}} ==Chapter 11== <small>{{font|color=#0000FF|1}}</small>{{font|color=#000088| And when the people complained, it displeased the LORD: and the LORD heard it; and his anger was kindled; and the fire of the LORD burnt among them, and consumed them that were in the uttermost parts of the camp.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#000088| And the people cried unto Moses; and when Moses prayed unto the LORD, the fire was quenched.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#000088| And he called the name of the place Taberah: because the fire of the LORD burnt among them.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#000088| And the mixt multitude that was among them fell a lusting: and the children of Israel also wept again, and said, Who shall give us flesh to eat?}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#000088| We remember the fish, which we did eat in Egypt freely; the cucumbers, and the melons, and the leeks, and the onions, and the garlick:}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#000088| But now our soul is dried away: there is nothing at all, beside this manna, before our eyes.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#880000| And the manna was as coriander seed, and the colour thereof as the colour of bdellium.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#880000| And the people went about, and gathered it, and ground it in mills, or beat it in a mortar, and baked it in pans, and made cakes of it: and the taste of it was as the taste of fresh oil.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#880000| And when the dew fell upon the camp in the night, the manna fell upon it.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#000088| Then Moses heard the people weep throughout their families, every man in the door of his tent: and the anger of the LORD was kindled greatly; Moses also was displeased.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#008888| And Moses said unto the LORD, Wherefore hast thou afflicted thy servant? and wherefore have I not found favour in thy sight, that thou layest the burden of all this people upon me?}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#008888| Have I conceived all this people? have I begotten them, that thou shouldest say unto me, Carry them in thy bosom, as a nursing father beareth the sucking child, unto the land which thou swarest unto their fathers?}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#000088| Whence should I have flesh to give unto all this people? for they weep unto me, saying, Give us flesh, that we may eat.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#008888| I am not able to bear all this people alone, because it is too heavy for me.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#008888| And if thou deal thus with me, kill me, I pray thee, out of hand, if I have found favour in thy sight; and let me not see my wretchedness.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#008888| And the LORD said unto Moses, Gather unto me seventy men of the elders of Israel, whom thou knowest to be the elders of the people, and officers over them; and bring them unto the tabernacle of the congregation, that they may stand there with thee.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#008888| And I will come down and talk with thee there: and I will take of the spirit which is upon thee, and will put it upon them; and they shall bear the burden of the people with thee, that thou bear it not thyself alone.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#000088| And say thou unto the people, Sanctify yourselves against to morrow, and ye shall eat flesh: for ye have wept in the ears of the LORD, saying, Who shall give us flesh to eat? for it was well with us in Egypt: therefore the LORD will give you flesh, and ye shall eat.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#000088| Ye shall not eat one day, nor two days, nor five days, neither ten days, nor twenty days;}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#000088| But even a whole month, until it come out at your nostrils, and it be loathsome unto you: because that ye have despised the LORD which is among you, and have wept before him, saying, Why came we forth out of Egypt?}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#000088| And Moses said, The people, among whom I am, are six hundred thousand footmen; and thou hast said, I will give them flesh, that they may eat a whole month.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#000088| Shall the flocks and the herds be slain for them, to suffice them? or shall all the fish of the sea be gathered together for them, to suffice them?}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#000088| And the LORD said unto Moses, Is the LORD's hand waxed short? thou shalt see now whether my word shall come to pass unto thee or not.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#000088| And Moses went out, and told the people the words of the LORD, }} {{font|color=#880000|and}} {{font|color=#008888|gathered the seventy men of the elders of the people, and set them round about the tabernacle.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#008888| And the LORD came down in a cloud, and spake unto him, and took of the spirit that was upon him, and gave it unto the seventy elders: and it came to pass, that, when the spirit rested upon them, they prophesied, and did not cease.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#008888| But there remained two of the men in the camp, the name of the one was Eldad, and the name of the other Medad: and the spirit rested upon them; and they were of them that were written, but went not out unto the tabernacle: and they prophesied in the camp.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#008888| And there ran a young man, and told Moses, and said, Eldad and Medad do prophesy in the camp.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#008888| And Joshua the son of Nun, the servant of Moses, one of his young men, answered and said, My lord Moses, forbid them.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#008888| And Moses said unto him, Enviest thou for my sake? would God that all the LORD's people were prophets, and that the LORD would put his spirit upon them!}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#008888| And Moses gat him into the camp, he and the elders of Israel.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#000088| And there went forth a wind from the LORD, and brought quails from the sea, and let them fall by the camp, as it were a day's journey on this side, and as it were a day's journey on the other side, round about the camp, and as it were two cubits high upon the face of the earth.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#000088| And the people stood up all that day, and all that night, and all the next day, and they gathered the quails: he that gathered least gathered ten homers: and they spread them all abroad for themselves round about the camp.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#000088| And while the flesh was yet between their teeth, ere it was chewed, the wrath of the LORD was kindled against the people, and the LORD smote the people with a very great plague.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#000088| And he called the name of that place Kibrothhattaavah: because there they buried the people that lusted.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#880000| And the people journeyed from Kibrothhattaavah unto Hazeroth; and abode at Hazeroth.}} ==Chapter 12== <small>{{font|color=#0000FF|1}}</small>{{font|color=#008888| And Miriam and Aaron spake against Moses}} {{font|color=#880000| because of the Ethiopian woman whom he had married: for he had married an Ethiopian woman.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#008888| And they said, Hath the LORD indeed spoken only by Moses? hath he not spoken also by us? And the LORD heard it.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#008888| (Now the man Moses was very meek, above all the men which were upon the face of the earth.)}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#008888| And the LORD spake suddenly unto Moses, and unto Aaron, and unto Miriam, Come out ye three unto the tabernacle of the congregation. And they three came out.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#008888| And the LORD came down in the pillar of the cloud, and stood in the door of the tabernacle, and called Aaron and Miriam: and they both came forth.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#008888| And he said, Hear now my words: If there be a prophet among you, I the LORD will make myself known unto him in a vision, and will speak unto him in a dream.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#008888| My servant Moses is not so, who is faithful in all mine house. }} <small>{{font|color=#0000FF|8}}</small>{{font|color=#008888| With him will I speak mouth to mouth, even apparently, and not in dark speeches; and the similitude of the LORD shall he behold: wherefore then were ye not afraid to speak against my servant Moses?}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#008888| And the anger of the LORD was kindled against them; and he departed.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#008888| And the cloud departed from off the tabernacle; and, behold, Miriam became leprous, white as snow: and Aaron looked upon Miriam, and, behold, she was leprous.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#008888| And Aaron said unto Moses, Alas, my lord, I beseech thee, lay not the sin upon us, wherein we have done foolishly, and wherein we have sinned.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#008888| Let her not be as one dead, of whom the flesh is half consumed when he cometh out of his mother's womb.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#008888| And Moses cried unto the LORD, saying, Heal her now, O God, I beseech thee.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#008888| And the LORD said unto Moses, If her father had but spit in her face, should she not be ashamed seven days? let her be shut out from the camp seven days, and after that let her be received in again.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#008888| And Miriam was shut out from the camp seven days: and the people journeyed not till Miriam was brought in again.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#008888| And afterward the people removed from Hazeroth, and pitched in the wilderness of Paran.}} ==Chapter 13== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Send thou men, that they may search the land of Canaan, which I give unto the children of Israel: of every tribe of their fathers shall ye send a man, every one a ruler among them.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And Moses by the commandment of the LORD sent them from the wilderness of Paran: all those men were heads of the children of Israel.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And these were their names: of the tribe of Reuben, Shammua the son of Zaccur.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| Of the tribe of Simeon, Shaphat the son of Hori.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Of the tribe of Judah, Caleb the son of Jephunneh.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Of the tribe of Issachar, Igal the son of Joseph.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| Of the tribe of Ephraim, Oshea the son of Nun.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| Of the tribe of Benjamin, Palti the son of Raphu.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| Of the tribe of Zebulun, Gaddiel the son of Sodi.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| Of the tribe of Joseph, namely, of the tribe of Manasseh, Gaddi the son of Susi.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| Of the tribe of Dan, Ammiel the son of Gemalli.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Of the tribe of Asher, Sethur the son of Michael.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Of the tribe of Naphtali, Nahbi the son of Vophsi.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| Of the tribe of Gad, Geuel the son of Machi.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| These are the names of the men which Moses sent to spy out the land. And Moses called Oshea the son of Nun Jehoshua.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And Moses sent them to spy out the land of Canaan, }} {{font|color=#000088|and said unto them, Get you up this way southward, and go up into the mountain:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#000088|And see the land, what it is, and the people that dwelleth therein, whether they be strong or weak, few or many;}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#000088| And what the land is that they dwell in, whether it be good or bad; and what cities they be that they dwell in, whether in tents, or in strong holds;}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#000088| And what the land is, whether it be fat or lean, whether there be wood therein, or not. And be ye of good courage, and bring of the fruit of the land. Now the time was the time of the firstripe grapes.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800|So they went up, and searched the land from the wilderness of Zin unto Rehob, as men come to Hamath.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#000088| And they ascended by the south, and came unto Hebron; where Ahiman, Sheshai, and Talmai, the children of Anak, were. (Now Hebron was built seven years before Zoan in Egypt.)}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#000088| And they came unto the brook of Eshcol, and cut down from thence a branch with one cluster of grapes, and they bare it between two upon a staff; and they brought of the pomegranates, and of the figs.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#000088| The place was called the brook Eshcol, because of the cluster of grapes which the children of Israel cut down from thence.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And they returned from searching of the land after forty days.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And they went and came to Moses, and to Aaron, and to all the congregation of the children of Israel, unto the wilderness of Paran,}} {{font|color=#000088|to Kadesh; and brought back word unto them, and unto all the congregation, and shewed them the fruit of the land.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#000088| And they told him, and said, We came unto the land whither thou sentest us, and surely it floweth with milk and honey; and this is the fruit of it.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#000088| Nevertheless the people be strong that dwell in the land, and the cities are walled, and very great: and moreover we saw the children of Anak there.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#000088| The Amalekites dwell in the land of the south: and the Hittites, and the Jebusites, and the Amorites, dwell in the mountains: and the Canaanites dwell by the sea, and by the coast of Jordan.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#000088| And Caleb stilled the people before Moses, and said, Let us go up at once, and possess it; for we are well able to overcome it.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#000088| But the men that went up with him said, We be not able to go up against the people; for they are stronger than we.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And they brought up an evil report of the land which they had searched unto the children of Israel, saying, The land, through which we have gone to search it, is a land that eateth up the inhabitants thereof; and all the people that we saw in it are men of a great stature.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#000088| And there we saw the giants, the sons of Anak, which come of the giants: and we were in our own sight as grasshoppers, and so we were in their sight.}} ==Chapter 14== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And all the congregation lifted up their voice, and cried;}} {{font|color=#000088| and the people wept that night.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And all the children of Israel murmured against Moses and against Aaron: and the whole congregation said unto them, Would God that we had died in the land of Egypt! or would God we had died in this wilderness!}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And wherefore hath the LORD brought us unto this land, to fall by the sword, that our wives and our children should be a prey? were it not better for us to return into Egypt?}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And they said one to another, Let us make a captain, and let us return into Egypt.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| Then Moses and Aaron fell on their faces before all the assembly of the congregation of the children of Israel.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And Joshua the son of Nun, and Caleb the son of Jephunneh, which were of them that searched the land, rent their clothes:}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800|And they spake unto all the company of the children of Israel, saying, The land, which we passed through to search it, is an exceeding good land.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| If the LORD delight in us, then he will bring us into this land, and give it us; a land which floweth with milk and honey.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| Only rebel not ye against the LORD, neither fear ye the people of the land; for they are bread for us: their defence is departed from them, and the LORD is with us: fear them not.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| But all the congregation bade stone them with stones. And the glory of the LORD appeared in the tabernacle of the congregation before all the children of Israel.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#000088| And the LORD said unto Moses, How long will this people provoke me? and how long will it be ere they believe me, for all the signs which I have shewed among them?}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#000088| I will smite them with the pestilence, and disinherit them, and will make of thee a greater nation and mightier than they.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#000088| And Moses said unto the LORD, Then the Egyptians shall hear it, (for thou broughtest up this people in thy might from among them;)}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#000088| And they will tell it to the inhabitants of this land: for they have heard that thou LORD art among this people, that thou LORD art seen face to face, and that thy cloud standeth over them, and that thou goest before them, by day time in a pillar of a cloud, and in a pillar of fire by night.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#000088| Now if thou shalt kill all this people as one man, then the nations which have heard the fame of thee will speak, saying,}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#000088| Because the LORD was not able to bring this people into the land which he sware unto them, therefore he hath slain them in the wilderness.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#000088| And now, I beseech thee, let the power of my Lord be great, according as thou hast spoken, saying,}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#000088| The LORD is longsuffering, and of great mercy, forgiving iniquity and transgression, and by no means clearing the guilty, visiting the iniquity of the fathers upon the children unto the third and fourth generation.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#000088| Pardon, I beseech thee, the iniquity of this people according unto the greatness of thy mercy, and as thou hast forgiven this people, from Egypt even until now.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#000088| And the LORD said, I have pardoned according to thy word: }} <small>{{font|color=#0000FF|21}}</small>{{font|color=#000088| But as truly as I live, all the earth shall be filled with the glory of the LORD.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#000088| Because all those men which have seen my glory, and my miracles, which I did in Egypt and in the wilderness, and have tempted me now these ten times, and have not hearkened to my voice;}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#000088| Surely they shall not see the land which I sware unto their fathers, neither shall any of them that provoked me see it:}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#000088| But my servant Caleb, because he had another spirit with him, and hath followed me fully, him will I bring into the land whereinto he went; and his seed shall possess it.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#000088| (Now the Amalekites and the Canaanites dwelt in the valley.) Tomorrow turn you, and get you into the wilderness by the way of the Red sea.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying, }} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| How long shall I bear with this evil congregation, which murmur against me? I have heard the murmurings of the children of Israel, which they murmur against me.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| Say unto them, As truly as I live, saith the LORD, as ye have spoken in mine ears, so will I do to you:}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Your carcases shall fall in this wilderness; and all that were numbered of you, according to your whole number, from twenty years old and upward which have murmured against me.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| Doubtless ye shall not come into the land, concerning which I sware to make you dwell therein, save Caleb the son of Jephunneh, and Joshua the son of Nun.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| But your little ones, which ye said should be a prey, them will I bring in, and they shall know the land which ye have despised.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| But as for you, your carcases, they shall fall in this wilderness.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| And your children shall wander in the wilderness forty years, and bear your whoredoms, until your carcases be wasted in the wilderness.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| After the number of the days in which ye searched the land, even forty days, each day for a year, shall ye bear your iniquities, even forty years, and ye shall know my breach of promise.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| I the LORD have said, I will surely do it unto all this evil congregation, that are gathered together against me: in this wilderness they shall be consumed, and there they shall die.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And the men, which Moses sent to search the land, who returned, and made all the congregation to murmur against him, by bringing up a slander upon the land,}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| Even those men that did bring up the evil report upon the land, died by the plague before the LORD.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| But Joshua the son of Nun, and Caleb the son of Jephunneh, which were of the men that went to search the land, lived still.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#000088| And Moses told these sayings unto all the children of Israel: and the people mourned greatly.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#000088| And they rose up early in the morning, and gat them up into the top of the mountain, saying, Lo, we be here, and will go up unto the place which the LORD hath promised: for we have sinned.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#000088| And Moses said, Wherefore now do ye transgress the commandment of the LORD? but it shall not prosper.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#000088| Go not up, for the LORD is not among you; that ye be not smitten before your enemies.}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#000088| For the Amalekites and the Canaanites are there before you, and ye shall fall by the sword: because ye are turned away from the LORD, therefore the LORD will not be with you.}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#000088| But they presumed to go up unto the hill top: nevertheless the ark of the covenant of the LORD, and Moses, departed not out of the camp.}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#000088| Then the Amalekites came down, and the Canaanites which dwelt in that hill, and smote them, and discomfited them, even unto Hormah.}} ==Chapter 15== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Speak unto the children of Israel, and say unto them, When ye be come into the land of your habitations, which I give unto you,}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And will make an offering by fire unto the LORD, a burnt offering, or a sacrifice in performing a vow, or in a freewill offering, or in your solemn feasts, to make a sweet savour unto the LORD, of the herd or of the flock:}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| Then shall he that offereth his offering unto the LORD bring a meat offering of a tenth deal of flour mingled with the fourth part of an hin of oil.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And the fourth part of an hin of wine for a drink offering shalt thou prepare with the burnt offering or sacrifice, for one lamb.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Or for a ram, thou shalt prepare for a meat offering two tenth deals of flour mingled with the third part of an hin of oil.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And for a drink offering thou shalt offer the third part of an hin of wine, for a sweet savour unto the LORD.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And when thou preparest a bullock for a burnt offering, or for a sacrifice in performing a vow, or peace offerings unto the LORD:}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| Then shall he bring with a bullock a meat offering of three tenth deals of flour mingled with half an hin of oil.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And thou shalt bring for a drink offering half an hin of wine, for an offering made by fire, of a sweet savour unto the LORD.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| Thus shall it be done for one bullock, or for one ram, or for a lamb, or a kid.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| According to the number that ye shall prepare, so shall ye do to every one according to their number.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| All that are born of the country shall do these things after this manner, in offering an offering made by fire, of a sweet savour unto the LORD.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And if a stranger sojourn with you, or whosoever be among you in your generations, and will offer an offering made by fire, of a sweet savour unto the LORD; as ye do, so he shall do.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| One ordinance shall be both for you of the congregation, and also for the stranger that sojourneth with you, an ordinance for ever in your generations: as ye are, so shall the stranger be before the LORD.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| One law and one manner shall be for you, and for the stranger that sojourneth with you.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| Speak unto the children of Israel, and say unto them, When ye come into the land whither I bring you,}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| Then it shall be, that, when ye eat of the bread of the land, ye shall offer up an heave offering unto the LORD.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| Ye shall offer up a cake of the first of your dough for an heave offering: as ye do the heave offering of the threshingfloor, so shall ye heave it.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Of the first of your dough ye shall give unto the LORD an heave offering in your generations.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And if ye have erred, and not observed all these commandments, which the LORD hath spoken unto Moses,}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| Even all that the LORD hath commanded you by the hand of Moses, from the day that the LORD commanded Moses, and henceforward among your generations;}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| Then it shall be, if ought be committed by ignorance without the knowledge of the congregation, that all the congregation shall offer one young bullock for a burnt offering, for a sweet savour unto the LORD, with his meat offering, and his drink offering, according to the manner, and one kid of the goats for a sin offering.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the priest shall make an atonement for all the congregation of the children of Israel, and it shall be forgiven them; for it is ignorance: and they shall bring their offering, a sacrifice made by fire unto the LORD, and their sin offering before the LORD, for their ignorance:}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And it shall be forgiven all the congregation of the children of Israel, and the stranger that sojourneth among them; seeing all the people were in ignorance.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And if any soul sin through ignorance, then he shall bring a she goat of the first year for a sin offering.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| And the priest shall make an atonement for the soul that sinneth ignorantly, when he sinneth by ignorance before the LORD, to make an atonement for him; and it shall be forgiven him.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Ye shall have one law for him that sinneth through ignorance, both for him that is born among the children of Israel, and for the stranger that sojourneth among them.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| But the soul that doeth ought presumptuously, whether he be born in the land, or a stranger, the same reproacheth the LORD; and that soul shall be cut off from among his people.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| Because he hath despised the word of the LORD, and hath broken his commandment, that soul shall utterly be cut off; his iniquity shall be upon him.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And while the children of Israel were in the wilderness, they found a man that gathered sticks upon the sabbath day.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| And they that found him gathering sticks brought him unto Moses and Aaron, and unto all the congregation.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| And they put him in ward, because it was not declared what should be done to him.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| And the LORD said unto Moses, The man shall be surely put to death: all the congregation shall stone him with stones without the camp.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And all the congregation brought him without the camp, and stoned him with stones, and he died; as the LORD commanded Moses.}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| Speak unto the children of Israel, and bid them that they make them fringes in the borders of their garments throughout their generations, and that they put upon the fringe of the borders a ribband of blue:}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| And it shall be unto you for a fringe, that ye may look upon it, and remember all the commandments of the LORD, and do them; and that ye seek not after your own heart and your own eyes, after which ye use to go a whoring:}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| That ye may remember, and do all my commandments, and be holy unto your God.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| I am the LORD your God, which brought you out of the land of Egypt, to be your God: I am the LORD your God.}} ==Chapter 16== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| Now Korah, the son of Izhar, the son of Kohath, the son of Levi,}} {{font|color=#008888| and Dathan and Abiram, the sons of Eliab,}} {{font|color=#880000|and On, the son of Peleth,}} {{font|color=#008888| son}}{{font|color=#888800|s}} {{font|color=#008888|of Reuben, took men:}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And they rose up before Moses, with certain of the children of Israel, two hundred and fifty princes of the assembly, famous in the congregation, men of renown:}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And they gathered themselves together against Moses and against Aaron, and said unto them, Ye take too much upon you, seeing all the congregation are holy, every one of them, and the LORD is among them: wherefore then lift ye up yourselves above the congregation of the LORD?}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And when Moses heard it, he fell upon his face:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And he spake unto Korah and unto all his company, saying, Even to morrow the LORD will shew who are his, and who is holy; and will cause him to come near unto him: even him whom he hath chosen will he cause to come near unto him.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| This do; Take you censers, Korah, and all his company;}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And put fire therein, and put incense in them before the LORD to morrow: and it shall be that the man whom the LORD doth choose, he shall be holy: ye take too much upon you, ye sons of Levi.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And Moses said unto Korah, Hear, I pray you, ye sons of Levi:}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| Seemeth it but a small thing unto you, that the God of Israel hath separated you from the congregation of Israel, to bring you near to himself to do the service of the tabernacle of the LORD, and to stand before the congregation to minister unto them?}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And he hath brought thee near to him, and all thy brethren the sons of Levi with thee: and seek ye the priesthood also?}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| For which cause both thou and all thy company are gathered together against the LORD: and what is Aaron, that ye murmur against him?}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#008888| And Moses sent to call Dathan and Abiram, the sons of Eliab: which said, We will not come up:}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#008888| Is it a small thing that thou hast brought us up out of a land that floweth with milk and honey, to kill us in the wilderness, except thou make thyself altogether a prince over us?}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#008888| Moreover thou hast not brought us into a land that floweth with milk and honey, or given us inheritance of fields and vineyards: wilt thou put out the eyes of these men? we will not come up.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#008888| And Moses was very wroth, and said unto the LORD, Respect not thou their offering: I have not taken one ass from them, neither have I hurt one of them.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And Moses said unto Korah, Be thou and all thy company before the LORD, thou, and they, and Aaron, to morrow:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And take every man his censer, and put incense in them, and bring ye before the LORD every man his censer, two hundred and fifty censers; thou also, and Aaron, each of you his censer.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And they took every man his censer, and put fire in them, and laid incense thereon, and stood in the door of the tabernacle of the congregation with Moses and Aaron.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And Korah gathered all the congregation against them unto the door of the tabernacle of the congregation: and the glory of the LORD appeared unto all the congregation.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying, }} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Separate yourselves from among this congregation, that I may consume them in a moment.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And they fell upon their faces, and said, O God, the God of the spirits of all flesh, shall one man sin, and wilt thou be wroth with all the congregation?}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| Speak unto the congregation, saying, Get you up from about the tabernacle}} {{font|color=#880000|of Korah, Dathan, and Abiram.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#008888| And Moses rose up and went unto Dathan and Abiram; and the elders of Israel followed him.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And he spake unto the congregation,}} {{font|color=#008888| saying, Depart, I pray you, from the tents of these wicked men, and touch nothing of their's, lest ye be consumed in all their sins.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| So they gat up from the tabernacle }} {{font|color=#880000| of Korah, Dathan, and Abiram,}} {{font|color=#888800| on every side:}} {{font|color=#008888| and Dathan and Abiram came out, and stood in the door of their tents, and their wives, and their sons, and their little children.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#008888| And Moses said, Hereby ye shall know that the LORD hath sent me to do all these works; for I have not done them of mine own mind.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#008888| If these men die the common death of all men, or if they be visited after the visitation of all men; then the LORD hath not sent me.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#008888| But if the LORD make a new thing, and the earth open her mouth, and swallow them up, with all that appertain unto them, and they go down quick into the pit; then ye shall understand that these men have provoked the LORD.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#008888| And it came to pass, as he had made an end of speaking all these words, that the ground clave asunder that was under them:}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#008888| And the earth opened her mouth, and swallowed them up, and their houses,}} {{font|color=#880000| and all the men that appertained unto Korah,}} {{font|color=#008888| and all their goods.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#008888| They, and all that appertained to them, went down alive into the pit, and the earth closed upon them: and they perished from among the congregation.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#008888| And all Israel that were round about them fled at the cry of them: for they said, Lest the earth swallow us up also.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| And there came out a fire from the LORD, and consumed the two hundred and fifty men that offered incense.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying, }} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| Speak unto Eleazar the son of Aaron the priest, that he take up the censers out of the burning, and scatter thou the fire yonder; for they are hallowed.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| The censers of these sinners against their own souls, let them make them broad plates for a covering of the altar: for they offered them before the LORD, therefore they are hallowed: and they shall be a sign unto the children of Israel.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| And Eleazar the priest took the brasen censers, wherewith they that were burnt had offered; and they were made broad plates for a covering of the altar:}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| To be a memorial unto the children of Israel, that no stranger, which is not of the seed of Aaron, come near to offer incense before the LORD; that he be not as Korah, and as his company: as the LORD said to him by the hand of Moses.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| But on the morrow all the congregation of the children of Israel murmured against Moses and against Aaron, saying, Ye have killed the people of the LORD.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| And it came to pass, when the congregation was gathered against Moses and against Aaron, that they looked toward the tabernacle of the congregation: and, behold, the cloud covered it, and the glory of the LORD appeared.}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| And Moses and Aaron came before the tabernacle of the congregation.}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| Get you up from among this congregation, that I may consume them as in a moment. And they fell upon their faces. }} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| And Moses said unto Aaron, Take a censer, and put fire therein from off the altar, and put on incense, and go quickly unto the congregation, and make an atonement for them: for there is wrath gone out from the LORD; the plague is begun.}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| And Aaron took as Moses commanded, and ran into the midst of the congregation; and, behold, the plague was begun among the people: and he put on incense, and made an atonement for the people.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| And he stood between the dead and the living; and the plague was stayed.}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| Now they that died in the plague were fourteen thousand and seven hundred, beside them that died about the matter of Korah.}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| And Aaron returned unto Moses unto the door of the tabernacle of the congregation: and the plague was stayed.}} ==Chapter 17== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Speak unto the children of Israel, and take of every one of them a rod according to the house of their fathers, of all their princes according to the house of their fathers twelve rods: write thou every man's name upon his rod.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And thou shalt write Aaron's name upon the rod of Levi: for one rod shall be for the head of the house of their fathers.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And thou shalt lay them up in the tabernacle of the congregation before the testimony, where I will meet with you.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And it shall come to pass, that the man's rod, whom I shall choose, shall blossom: and I will make to cease from me the murmurings of the children of Israel, whereby they murmur against you.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And Moses spake unto the children of Israel, and every one of their princes gave him a rod apiece, for each prince one, according to their fathers' houses, even twelve rods: and the rod of Aaron was among their rods.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And Moses laid up the rods before the LORD in the tabernacle of witness.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And it came to pass, that on the morrow Moses went into the tabernacle of witness; and, behold, the rod of Aaron for the house of Levi was budded, and brought forth buds, and bloomed blossoms, and yielded almonds.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And Moses brought out all the rods from before the LORD unto all the children of Israel: and they looked, and took every man his rod.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And the LORD said unto Moses, Bring Aaron's rod again before the testimony, to be kept for a token against the rebels; and thou shalt quite take away their murmurings from me, that they die not.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And Moses did so: as the LORD commanded him, so did he. }} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And the children of Israel spake unto Moses, saying, Behold, we die, we perish, we all perish.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Whosoever cometh any thing near unto the tabernacle of the LORD shall die: shall we be consumed with dying?}} ==Chapter 18== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD said unto Aaron, Thou and thy sons and thy father's house with thee shall bear the iniquity of the sanctuary: and thou and thy sons with thee shall bear the iniquity of your priesthood.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And thy brethren also of the tribe of Levi, the tribe of thy father, bring thou with thee, that they may be joined unto thee, and minister unto thee: but thou and thy sons with thee shall minister before the tabernacle of witness.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And they shall keep thy charge, and the charge of all the tabernacle: only they shall not come nigh the vessels of the sanctuary and the altar, that neither they, nor ye also, die.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And they shall be joined unto thee, and keep the charge of the tabernacle of the congregation, for all the service of the tabernacle: and a stranger shall not come nigh unto you.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And ye shall keep the charge of the sanctuary, and the charge of the altar: that there be no wrath any more upon the children of Israel.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And I, behold, I have taken your brethren the Levites from among the children of Israel: to you they are given as a gift for the LORD, to do the service of the tabernacle of the congregation.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Therefore thou and thy sons with thee shall keep your priest's office for everything of the altar, and within the vail; and ye shall serve: I have given your priest's office unto you as a service of gift: and the stranger that cometh nigh shall be put to death.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And the LORD spake unto Aaron, Behold, I also have given thee the charge of mine heave offerings of all the hallowed things of the children of Israel; unto thee have I given them by reason of the anointing, and to thy sons, by an ordinance for ever.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| This shall be thine of the most holy things, reserved from the fire: every oblation of their's, every meat offering of their's, and every sin offering of their's, and every trespass offering of their's which they shall render unto me, shall be most holy for thee and for thy sons.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| In the most holy place shalt thou eat it; every male shall eat it: it shall be holy unto thee.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And this is thine; the heave offering of their gift, with all the wave offerings of the children of Israel: I have given them unto thee, and to thy sons and to thy daughters with thee, by a statute for ever: every one that is clean in thy house shall eat of it.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| All the best of the oil, and all the best of the wine, and of the wheat, the firstfruits of them which they shall offer unto the LORD, them have I given thee.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And whatsoever is first ripe in the land, which they shall bring unto the LORD, shall be thine; every one that is clean in thine house shall eat of it.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Every thing devoted in Israel shall be thine.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| Every thing that openeth the matrix in all flesh, which they bring unto the LORD, whether it be of men or beasts, shall be thine: nevertheless the firstborn of man shalt thou surely redeem, and the firstling of unclean beasts shalt thou redeem.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And those that are to be redeemed from a month old shalt thou redeem, according to thine estimation, for the money of five shekels, after the shekel of the sanctuary, which is twenty gerahs.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| But the firstling of a cow, or the firstling of a sheep, or the firstling of a goat, thou shalt not redeem; they are holy: thou shalt sprinkle their blood upon the altar, and shalt burn their fat for an offering made by fire, for a sweet savour unto the LORD.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And the flesh of them shall be thine, as the wave breast and as the right shoulder are thine.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| All the heave offerings of the holy things, which the children of Israel offer unto the LORD, have I given thee, and thy sons and thy daughters with thee, by a statute for ever: it is a covenant of salt for ever before the LORD unto thee and to thy seed with thee.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And the LORD spake unto Aaron, Thou shalt have no inheritance in their land, neither shalt thou have any part among them: I am thy part and thine inheritance among the children of Israel.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And, behold, I have given the children of Levi all the tenth in Israel for an inheritance, for their service which they serve, even the service of the tabernacle of the congregation.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Neither must the children of Israel henceforth come nigh the tabernacle of the congregation, lest they bear sin, and die.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| But the Levites shall do the service of the tabernacle of the congregation, and they shall bear their iniquity: it shall be a statute for ever throughout your generations, that among the children of Israel they have no inheritance.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| But the tithes of the children of Israel, which they offer as an heave offering unto the LORD, I have given to the Levites to inherit: therefore I have said unto them, Among the children of Israel they shall have no inheritance.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| Thus speak unto the Levites, and say unto them, When ye take of the children of Israel the tithes which I have given you from them for your inheritance, then ye shall offer up an heave offering of it for the LORD, even a tenth part of the tithe.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And this your heave offering shall be reckoned unto you, as though it were the corn of the threshingfloor, and as the fulness of the winepress.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| Thus ye also shall offer an heave offering unto the LORD of all your tithes, which ye receive of the children of Israel; and ye shall give thereof the LORD's heave offering to Aaron the priest.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Out of all your gifts ye shall offer every heave offering of the LORD, of all the best thereof, even the hallowed part thereof out of it.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| Therefore thou shalt say unto them, When ye have heaved the best thereof from it, then it shall be counted unto the Levites as the increase of the threshingfloor, and as the increase of the winepress.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| And ye shall eat it in every place, ye and your households: for it is your reward for your service in the tabernacle of the congregation.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And ye shall bear no sin by reason of it, when ye have heaved from it the best of it: neither shall ye pollute the holy things of the children of Israel, lest ye die.}} ==Chapter 19== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying, }} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| This is the ordinance of the law which the LORD hath commanded, saying, Speak unto the children of Israel, that they bring thee a red heifer without spot, wherein is no blemish, and upon which never came yoke:}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And ye shall give her unto Eleazar the priest, that he may bring her forth without the camp, and one shall slay her before his face:}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And Eleazar the priest shall take of her blood with his finger, and sprinkle of her blood directly before the tabernacle of the congregation seven times:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And one shall burn the heifer in his sight; her skin, and her flesh, and her blood, with her dung, shall he burn:}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And the priest shall take cedar wood, and hyssop, and scarlet, and cast it into the midst of the burning of the heifer.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Then the priest shall wash his clothes, and he shall bathe his flesh in water, and afterward he shall come into the camp, and the priest shall be unclean until the even.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And he that burneth her shall wash his clothes in water, and bathe his flesh in water, and shall be unclean until the even.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And a man that is clean shall gather up the ashes of the heifer, and lay them up without the camp in a clean place, and it shall be kept for the congregation of the children of Israel for a water of separation: it is a purification for sin.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And he that gathereth the ashes of the heifer shall wash his clothes, and be unclean until the even: and it shall be unto the children of Israel, and unto the stranger that sojourneth among them, for a statute for ever.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| He that toucheth the dead body of any man shall be unclean seven days.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| He shall purify himself with it on the third day, and on the seventh day he shall be clean: but if he purify not himself the third day, then the seventh day he shall not be clean.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Whosoever toucheth the dead body of any man that is dead, and purifieth not himself, defileth the tabernacle of the LORD; and that soul shall be cut off from Israel: because the water of separation was not sprinkled upon him, he shall be unclean; his uncleanness is yet upon him.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| This is the law, when a man dieth in a tent: all that come into the tent, and all that is in the tent, shall be unclean seven days.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And every open vessel, which hath no covering bound upon it, is unclean.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And whosoever toucheth one that is slain with a sword in the open fields, or a dead body, or a bone of a man, or a grave, shall be unclean seven days.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And for an unclean person they shall take of the ashes of the burnt heifer of purification for sin, and running water shall be put thereto in a vessel:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And a clean person shall take hyssop, and dip it in the water, and sprinkle it upon the tent, and upon all the vessels, and upon the persons that were there, and upon him that touched a bone, or one slain, or one dead, or a grave:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And the clean person shall sprinkle upon the unclean on the third day, and on the seventh day: and on the seventh day he shall purify himself, and wash his clothes, and bathe himself in water, and shall be clean at even.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| But the man that shall be unclean, and shall not purify himself, that soul shall be cut off from among the congregation, because he hath defiled the sanctuary of the LORD: the water of separation hath not been sprinkled upon him; he is unclean.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And it shall be a perpetual statute unto them, that he that sprinkleth the water of separation shall wash his clothes; and he that toucheth the water of separation shall be unclean until even.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And whatsoever the unclean person toucheth shall be unclean; and the soul that toucheth it shall be unclean until even.}} ==Chapter 20== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| Then came the children of Israel, even the whole congregation, into the desert of Zin in the first month:}} {{font|color=#008888| and the people abode in Kadesh; and Miriam died there, and was buried there.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And there was no water for the congregation: and they gathered themselves together against Moses and against Aaron.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#008888| And the people chode with Moses, and spake, }} {{font|color=#888800|saying, Would God that we had died when our brethren died before the LORD!}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And why have ye brought up the congregation of the LORD into this wilderness, that we and our cattle should die there?}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#008888| And wherefore have ye made us to come up out of Egypt, to bring us in unto this evil place? it is no place of seed, or of figs, or of vines, or of pomegranates; neither is there any water to drink.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And Moses and Aaron went from the presence of the assembly unto the door of the tabernacle of the congregation, and they fell upon their faces: and the glory of the LORD appeared unto them.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| Take the rod, and gather thou the assembly together, thou, and Aaron thy brother, and speak ye unto the rock before their eyes; and it shall give forth his water, and thou shalt bring forth to them water out of the rock: so thou shalt give the congregation and their beasts drink.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And Moses took the rod from before the LORD, as he commanded him.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And Moses and Aaron gathered the congregation together before the rock, and he said unto them, Hear now, ye rebels; must we fetch you water out of this rock?}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And Moses lifted up his hand, and with his rod he smote the rock twice: and the water came out abundantly, and the congregation drank, and their beasts also.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And the LORD spake unto Moses and Aaron, Because ye believed me not, to sanctify me in the eyes of the children of Israel, therefore ye shall not bring this congregation into the land which I have given them.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| This is the water of Meribah; because the children of Israel strove with the LORD, and he was sanctified in them.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#008888| And Moses sent messengers from Kadesh unto the king of Edom, Thus saith thy brother Israel, Thou knowest all the travail that hath befallen us:}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#008888| How our fathers went down into Egypt, and we have dwelt in Egypt a long time; and the Egyptians vexed us, and our fathers:}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#008888| And when we cried unto the LORD, he heard our voice, and sent an angel, and hath brought us forth out of Egypt: and, behold, we are in Kadesh, a city in the uttermost of thy border:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#008888| Let us pass, I pray thee, through thy country: we will not pass through the fields, or through the vineyards, neither will we drink of the water of the wells: we will go by the king's high way, we will not turn to the right hand nor to the left, until we have passed thy borders.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#008888| And Edom said unto him, Thou shalt not pass by me, lest I come out against thee with the sword.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#008888| And the children of Israel said unto him, We will go by the high way: and if I and my cattle drink of thy water, then I will pay for it: I will only, without doing anything else, go through on my feet.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#008888| And he said, Thou shalt not go through. And Edom came out against him with much people, and with a strong hand.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#008888| Thus Edom refused to give Israel passage through his border: wherefore Israel turned away from him.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And the children of Israel, even the whole congregation,}} {{font|color=#008888|journeyed from Kadesh, and}} {{font|color=#888800|came unto mount Hor.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And the LORD spake unto Moses and Aaron in mount Hor, by the coast of the land of Edom, saying,}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| Aaron shall be gathered unto his people: for he shall not enter into the land which I have given unto the children of Israel, because ye rebelled against my word at the water of Meribah.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| Take Aaron and Eleazar his son, and bring them up unto mount Hor:}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And strip Aaron of his garments, and put them upon Eleazar his son: and Aaron shall be gathered unto his people, and shall die there.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And Moses did as the LORD commanded: and they went up into mount Hor in the sight of all the congregation.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| And Moses stripped Aaron of his garments, and put them upon Eleazar his son; and Aaron died there in the top of the mount: and Moses and Eleazar came down from the mount.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| And when all the congregation saw that Aaron was dead, they mourned for Aaron thirty days, even all the house of Israel.}} ==Chapter 21== <small>{{font|color=#0000FF|1}}</small>{{font|color=#000088| And when king Arad the Canaanite, which dwelt in the south, heard tell that Israel came by the way of the spies; then he fought against Israel, and took some of them prisoners.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#000088| And Israel vowed a vow unto the LORD, and said, If thou wilt indeed deliver this people into my hand, then I will utterly destroy their cities.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#000088| And the LORD hearkened to the voice of Israel, and delivered up the Canaanites; and they utterly destroyed them and their cities: and he called the name of the place Hormah.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And they journeyed from mount Hor by the way of the Red sea,}} {{font|color=#008888| to compass the land of Edom: and the soul of the people was much discouraged because of the way.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#008888| And the people spake against God, and against Moses, Wherefore have ye brought us up out of Egypt to die in the wilderness? for there is no bread, neither is there any water;}} {{font|color=#880000|and our soul loatheth this light bread.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#008888| And the LORD sent fiery serpents among the people, and they bit the people; and much people of Israel died.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#008888| Therefore the people came to Moses, and said, We have sinned, for we have spoken against the LORD, and against thee; pray unto the LORD, that he take away the serpents from us. And Moses prayed for the people.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#008888| And the LORD said unto Moses, Make thee a fiery serpent, and set it upon a pole: and it shall come to pass, that every one that is bitten, when he looketh upon it, shall live.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#008888| And Moses made a serpent of brass, and put it upon a pole, and it came to pass, that if a serpent had bitten any man, when he beheld the serpent of brass, he lived.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And the children of Israel set forward, and pitched in Oboth. }} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And they journeyed from Oboth, and pitched at Ijeabarim, in the wilderness which is before Moab, toward the sunrising.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#008888| From thence they removed, and pitched in the valley of Zared.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#008888| From thence they removed, and pitched on the other side of Arnon, which is in the wilderness that cometh out of the coasts of the Amorites: for Arnon is the border of Moab, between Moab and the Amorites.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#008888| Wherefore it is said in the book of the wars of the LORD, What he did in the Red sea, and in the brooks of Arnon,}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#008888| And at the stream of the brooks that goeth down to the dwelling of Ar, and lieth upon the border of Moab.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#000088|And from thence they went to Beer: that is the well whereof the LORD spake unto Moses, Gather the people together, and I will give them water.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#000088| Then Israel sang this song, Spring up, O well; sing ye unto it:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#000088|The princes digged the well, the nobles of the people digged it, by the direction of the lawgiver, with their staves. And from the wilderness they went to Mattanah: }} <small>{{font|color=#0000FF|19}}</small>{{font|color=#000088| And from Mattanah to Nahaliel: and from Nahaliel to Bamoth:}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#000088| And from Bamoth in the valley, that is in the country of Moab, to the top of Pisgah, which looketh toward Jeshimon.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#008888| And Israel sent messengers unto Sihon king of the Amorites, saying,}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#008888| Let me pass through thy land: we will not turn into the fields, or into the vineyards; we will not drink of the waters of the well: but we will go along by the king's high way, until we be past thy borders.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#008888| And Sihon would not suffer Israel to pass through his border: but Sihon gathered all his people together, and went out against Israel into the wilderness: and he came to Jahaz, and fought against Israel.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#008888| And Israel smote him with the edge of the sword, and possessed his land from Arnon unto Jabbok, even unto the children of Ammon: for the border of the children of Ammon was strong.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#008888| And Israel took all these cities: and Israel dwelt in all the cities of the Amorites, in Heshbon, and in all the villages thereof.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#008888| For Heshbon was the city of Sihon the king of the Amorites, who had fought against the former king of Moab, and taken all his land out of his hand, even unto Arnon.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#008888| Wherefore they that speak in proverbs say, Come into Heshbon, let the city of Sihon be built and prepared:}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#008888| For there is a fire gone out of Heshbon, a flame from the city of Sihon: it hath consumed Ar of Moab, and the lords of the high places of Arnon.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#008888| Woe to thee, Moab! thou art undone, O people of Chemosh: he hath given his sons that escaped, and his daughters, into captivity unto Sihon king of the Amorites.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#008888| We have shot at them; Heshbon is perished even unto Dibon, and we have laid them waste even unto Nophah, which reacheth unto Medeba.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#008888| Thus Israel dwelt in the land of the Amorites. }} <small>{{font|color=#0000FF|32}}</small>{{font|color=#008888| And Moses sent to spy out Jaazer, and they took the villages thereof, and drove out the Amorites that were there.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#880000| And they turned and went up by the way of Bashan: and Og the king of Bashan went out against them, he, and all his people, to the battle at Edrei.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#880000| And the LORD said unto Moses, Fear him not: for I have delivered him into thy hand, and all his people, and his land; and thou shalt do to him as thou didst unto Sihon king of the Amorites, which dwelt at Heshbon.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#880000| So they smote him, and his sons, and all his people, until there was none left him alive: and they possessed his land.}} ==Chapter 22== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the children of Israel set forward, and pitched in the plains of Moab on this side Jordan by Jericho.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#880000| And Balak the son of Zippor saw all that Israel had done to the Amorites.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#000088|And Moab was sore afraid of the people, because they were many: and Moab was distressed because of the children of Israel.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#000088|And Moab said}} {{font|color=#880000| unto the elders of Midian,}} {{font|color=#000088|Now shall this company lick up all that are round about us, as the ox licketh up the grass of the field. And Balak the son of Zippor was king of the Moabites at that time.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#000088| He sent messengers therefore unto Balaam the son of Beor to Pethor, which is by the river of the land of the children of his people, to call him, saying, Behold, there is a people come out from Egypt: behold, they cover the face of the earth, and they abide over against me:}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#000088| Come now therefore, I pray thee, curse me this people; for they are too mighty for me: peradventure I shall prevail, that we may smite them, and that I may drive them out of the land: for I wot that he whom thou blessest is blessed, and he whom thou cursest is cursed.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#000088| And the elders of Moab}} {{font|color=#880000|and the elders of Midian}} {{font|color=#000088|departed with the rewards of divination in their hand; and they came unto Balaam, and spake unto him the words of Balak.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#000088| And he said unto them, Lodge here this night, and I will bring you word again, as the LORD shall speak unto me: and the princes of Moab abode with Balaam.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#000088| And God came unto Balaam, and said, What men are these with thee?}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#000088| And Balaam said unto God, Balak the son of Zippor, king of Moab, hath sent unto me saying,}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#000088| Behold, there is a people come out of Egypt, which covereth the face of the earth: come now, curse me them; peradventure I shall be able to overcome them, and drive them out.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#000088| And God said unto Balaam, Thou shalt not go with them; thou shalt not curse the people: for they are blessed.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#000088| And Balaam rose up in the morning, and said unto the princes of Balak, Get you into your land: for the LORD refuseth to give me leave to go with you.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#000088| And the princes of Moab rose up, and they went unto Balak, and said, Balaam refuseth to come with us.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#000088| And Balak sent yet again princes, more, and more honourable than they.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#000088| And they came to Balaam, and said to him, Thus saith Balak the son of Zippor, Let nothing, I pray thee, hinder thee from coming unto me:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#000088| For I will promote thee unto very great honour, and I will do whatsoever thou sayest unto me: come therefore, I pray thee, curse me this people.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#000088| And Balaam answered and said unto the servants of Balak, If Balak would give me his house full of silver and gold, I cannot go beyond the word of the LORD my God, to do less or more.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#000088| Now therefore, I pray you, tarry ye also here this night, that I may know what the LORD will say unto me more.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#000088| And God came unto Balaam at night, and said unto him, If the men come to call thee, rise up, and go with them; but yet the word which I shall say unto thee, that shalt thou do.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#000088| And Balaam rose up in the morning, and saddled his ass, and went with the princes of Moab.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#880000| And God's anger was kindled because he went:}} {{font|color=#000088|and the angel of the LORD stood in the way for an adversary against him. Now he was riding upon his ass, and his two servants were with him.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#000088| And the ass saw the angel of the LORD standing in the way, and his sword drawn in his hand: and the ass turned aside out of the way, and went into the field: and Balaam smote the ass, to turn her into the way.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#000088| But the angel of the LORD stood in a path of the vineyards, a wall being on this side, and a wall on that side.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#000088| And when the ass saw the angel of the LORD, she thrust herself unto the wall, and crushed Balaam's foot against the wall: and he smote her again.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#000088| And the angel of the LORD went further, and stood in a narrow place, where was no way to turn either to the right hand or to the left.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#000088| And when the ass saw the angel of the LORD, she fell down under Balaam: and Balaam's anger was kindled, and he smote the ass with a staff.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#000088| And the LORD opened the mouth of the ass, and she said unto Balaam, What have I done unto thee, that thou hast smitten me these three times?}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#000088| And Balaam said unto the ass, Because thou hast mocked me: I would there were a sword in mine hand, for now would I kill thee.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#000088| And the ass said unto Balaam, Am not I thine ass, upon which thou hast ridden ever since I was thine unto this day? was I ever wont to do so unto thee? And he said, Nay.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#000088| Then the LORD opened the eyes of Balaam, and he saw the angel of the LORD standing in the way, and his sword drawn in his hand: and he bowed down his head, and fell flat on his face.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#000088| And the angel of the LORD said unto him, Wherefore hast thou smitten thine ass these three times? behold, I went out to withstand thee, because thy way is perverse before me:}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#000088| And the ass saw me, and turned from me these three times: unless she had turned from me, surely now also I had slain thee, and saved her alive.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#000088| And Balaam said unto the angel of the LORD, I have sinned; for I knew not that thou stoodest in the way against me: now therefore, if it displease thee, I will get me back again.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#000088| And the angel of the LORD said unto Balaam, Go with the men: but only the word that I shall speak unto thee, that thou shalt speak. So Balaam went with the princes of Balak.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#000088| And when Balak heard that Balaam was come, he went out to meet him unto a city of Moab, which is in the border of Arnon, which is in the utmost coast.}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#000088| And Balak said unto Balaam, Did I not earnestly send unto thee to call thee? wherefore camest thou not unto me? am I not able indeed to promote thee to honour?}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#000088| And Balaam said unto Balak, Lo, I am come unto thee: have I now any power at all to say any thing? the word that God putteth in my mouth, that shall I speak.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#000088| And Balaam went with Balak, and they came unto Kirjathhuzoth.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#000088| And Balak offered oxen and sheep, and sent to Balaam, and to the princes that were with him.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#000088| And it came to pass on the morrow, that Balak took Balaam, and brought him up into the high places of Baal, that thence he might see the utmost part of the people.}} ==Chapter 23== <small>{{font|color=#0000FF|1}}</small>{{font|color=#000088| And Balaam said unto Balak, Build me here seven altars, and prepare me here seven oxen and seven rams.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#000088| And Balak did as Balaam had spoken; and Balak and Balaam offered on every altar a bullock and a ram.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#000088| And Balaam said unto Balak, Stand by thy burnt offering, and I will go: peradventure the LORD will come to meet me: and whatsoever he sheweth me I will tell thee. And he went to an high place.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#000088| And God met Balaam: and he said unto him, I have prepared seven altars, and I have offered upon every altar a bullock and a ram.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#000088| And the LORD put a word in Balaam's mouth, and said, Return unto Balak, and thus thou shalt speak.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#000088| And he returned unto him, and, lo, he stood by his burnt sacrifice, he, and all the princes of Moab.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#000088| And he took up his parable, and said, Balak the king of Moab hath brought me from Aram, out of the mountains of the east, saying, Come, curse me Jacob, and come, defy Israel.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#000088| How shall I curse, whom God hath not cursed? or how shall I defy, whom the LORD hath not defied?}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#000088| For from the top of the rocks I see him, and from the hills I behold him: lo, the people shall dwell alone, and shall not be reckoned among the nations.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#000088| Who can count the dust of Jacob, and the number of the fourth part of Israel? Let me die the death of the righteous, and let my last end be like his!}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#000088| And Balak said unto Balaam, What hast thou done unto me? I took thee to curse mine enemies, and, behold, thou hast blessed them altogether.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#000088| And he answered and said, Must I not take heed to speak that which the LORD hath put in my mouth?}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#000088| And Balak said unto him, Come, I pray thee, with me unto another place, from whence thou mayest see them: thou shalt see but the utmost part of them, and shalt not see them all: and curse me them from thence.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#000088| And he brought him into the field of Zophim, to the top of Pisgah, and built seven altars, and offered a bullock and a ram on every altar.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#000088| And he said unto Balak, Stand here by thy burnt offering, while I meet the LORD yonder.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#000088| And the LORD met Balaam, and put a word in his mouth, and said, Go again unto Balak, and say thus.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#000088| And when he came to him, behold, he stood by his burnt offering, and the princes of Moab with him. And Balak said unto him, What hath the LORD spoken?}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#000088| And he took up his parable, and said, Rise up, Balak, and hear; hearken unto me, thou son of Zippor:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#000088| God is not a man, that he should lie; neither the son of man, that he should repent: hath he said, and shall he not do it? or hath he spoken, and shall he not make it good?}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#000088| Behold, I have received commandment to bless: and he hath blessed; and I cannot reverse it.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#000088| He hath not beheld iniquity in Jacob, neither hath he seen perverseness in Israel: the LORD his God is with him, and the shout of a king is among them.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#000088| God brought them out of Egypt; he hath as it were the strength of an unicorn.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#000088| Surely there is no enchantment against Jacob, neither is there any divination against Israel: according to this time it shall be said of Jacob and of Israel, What hath God wrought!}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#000088| Behold, the people shall rise up as a great lion, and lift up himself as a young lion: he shall not lie down until he eat of the prey, and drink the blood of the slain.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#000088| And Balak said unto Balaam, Neither curse them at all, nor bless them at all.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#000088| But Balaam answered and said unto Balak, Told not I thee, saying, All that the LORD speaketh, that I must do?}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#000088| And Balak said unto Balaam, Come, I pray thee, I will bring thee unto another place; peradventure it will please God that thou mayest curse me them from thence.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#000088| And Balak brought Balaam unto the top of Peor, that looketh toward Jeshimon.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#000088| And Balaam said unto Balak, Build me here seven altars, and prepare me here seven bullocks and seven rams.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#000088| And Balak did as Balaam had said, and offered a bullock and a ram on every altar.}} ==Chapter 24== <small>{{font|color=#0000FF|1}}</small>{{font|color=#000088| And when Balaam saw that it pleased the LORD to bless Israel, he went not, as at other times, to seek for enchantments, but he set his face toward the wilderness.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#000088| And Balaam lifted up his eyes, and he saw Israel abiding in his tents according to their tribes; and the spirit of God came upon him.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#000088| And he took up his parable, and said, Balaam the son of Beor hath said, and the man whose eyes are open hath said:}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#000088| He hath said, which heard the words of God, which saw the vision of the Almighty, falling into a trance, but having his eyes open:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#000088| How goodly are thy tents, O Jacob, and thy tabernacles, O Israel!}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#000088| As the valleys are they spread forth, as gardens by the river's side, as the trees of lign aloes which the LORD hath planted, and as cedar trees beside the waters.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#000088| He shall pour the water out of his buckets, and his seed shall be in many waters, and his king shall be higher than Agag, and his kingdom shall be exalted.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#000088| God brought him forth out of Egypt; he hath as it were the strength of an unicorn: he shall eat up the nations his enemies, and shall break their bones, and pierce them through with his arrows.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#000088| He couched, he lay down as a lion, and as a great lion: who shall stir him up? Blessed is he that blesseth thee, and cursed is he that curseth thee.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#000088| And Balak's anger was kindled against Balaam, and he smote his hands together: and Balak said unto Balaam, I called thee to curse mine enemies, and, behold, thou hast altogether blessed them these three times.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#000088| Therefore now flee thou to thy place: I thought to promote thee unto great honour; but, lo, the LORD hath kept thee back from honour.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#000088| And Balaam said unto Balak, Spake I not also to thy messengers which thou sentest unto me, saying,}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#000088| If Balak would give me his house full of silver and gold, I cannot go beyond the commandment of the LORD, to do either good or bad of mine own mind; but what the LORD saith, that will I speak?}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#880000| And now, behold, I go unto my people: come therefore, and I will advertise thee what this people shall do to thy people in the latter days.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#880000| And he took up his parable, and said, Balaam the son of Beor hath said, and the man whose eyes are open hath said:}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#880000| He hath said, which heard the words of God, and knew the knowledge of the most High, which saw the vision of the Almighty, falling into a trance, but having his eyes open:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#880000| I shall see him, but not now: I shall behold him, but not nigh: there shall come a Star out of Jacob, and a Sceptre shall rise out of Israel, and shall smite the corners of Moab, and destroy all the children of Sheth.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#880000| And Edom shall be a possession, Seir also shall be a possession for his enemies; and Israel shall do valiantly.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#880000| Out of Jacob shall come he that shall have dominion, and shall destroy him that remaineth of the city.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#880000| And when he looked on Amalek, he took up his parable, and said, Amalek was the first of the nations; but his latter end shall be that he perish for ever.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#880000| And he looked on the Kenites, and took up his parable, and said, Strong is thy dwellingplace, and thou puttest thy nest in a rock.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#880000| Nevertheless the Kenite shall be wasted, until Asshur shall carry thee away captive.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#880000| And he took up his parable, and said, Alas, who shall live when God doeth this!}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#880000| And ships shall come from the coast of Chittim, and shall afflict Asshur, and shall afflict Eber, and he also shall perish for ever.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#000088| And Balaam rose up, and went and returned to his place: and Balak also went his way.}} ==Chapter 25== <small>{{font|color=#0000FF|1}}</small>{{font|color=#008888| And Israel abode in Shittim, }} {{font|color=#000088|and the people began to commit whoredom with the daughters of Moab.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#000088| And they called the people unto the sacrifices of their gods: and the people did eat, and bowed down to their gods.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#008888| And Israel joined himself unto Baalpeor: }} {{font|color=#000088|and the anger of the LORD was kindled against Israel.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#000088| And the LORD said unto Moses, Take all the heads of the people, and hang them up before the LORD against the sun, that the fierce anger of the LORD may be turned away from Israel.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#008888| And Moses said unto the judges of Israel, Slay ye every one his men that were joined unto Baalpeor.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And, behold, one of the children of Israel came and brought unto his brethren a Midianitish woman in the sight of Moses, and in the sight of all the congregation of the children of Israel, who were weeping before the door of the tabernacle of the congregation.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And when Phinehas, the son of Eleazar, the son of Aaron the priest, saw it, he rose up from among the congregation, and took a javelin in his hand;}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And he went after the man of Israel into the tent, and thrust both of them through, the man of Israel, and the woman through her belly.}} {{font|color=#880000| So the plague was stayed from the children of Israel.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#880000| And those that died in the plague were twenty and four thousand.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| Phinehas, the son of Eleazar, the son of Aaron the priest, hath turned my wrath away from the children of Israel, while he was zealous for my sake among them, that I consumed not the children of Israel in my jealousy.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| Wherefore say, Behold, I give unto him my covenant of peace:}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And he shall have it, and his seed after him, even the covenant of an everlasting priesthood; because he was zealous for his God, and made an atonement for the children of Israel.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Now the name of the Israelite that was slain, even that was slain with the Midianitish woman, was Zimri, the son of Salu, a prince of a chief house among the Simeonites.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And the name of the Midianitish woman that was slain was Cozbi, the daughter of Zur; he was head over a people, and of a chief house in Midian.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| Vex the Midianites, and smite them:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800|For they vex you with their wiles,}} {{font|color=#880000| wherewith they have beguiled you in the matter of Peor,}} {{font|color=#888800| and in the matter of Cozbi, the daughter of a prince of Midian, their sister, which was slain}} {{font|color=#880000| in the day of the plague for Peor's sake.}} ==Chapter 26== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And it came to pass after the plague, that the LORD spake unto Moses and unto Eleazar the son of Aaron the priest, saying, }} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Take the sum of all the congregation of the children of Israel, from twenty years old and upward, throughout their fathers' house, all that are able to go to war in Israel.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And Moses and Eleazar the priest spake with them in the plains of Moab by Jordan near Jericho, saying,}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| Take the sum of the people, from twenty years old and upward; as the LORD commanded Moses and the children of Israel, which went forth out of the land of Egypt.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| Reuben, the eldest son of Israel: the children of Reuben; Hanoch, of whom cometh the family of the Hanochites: of Pallu, the family of the Palluites:}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Of Hezron, the family of the Hezronites: of Carmi, the family of the Carmites.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| These are the families of the Reubenites: and they that were numbered of them were forty and three thousand and seven hundred and thirty.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And the sons of Pallu; Eliab.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#880000| And the sons of Eliab; Nemuel, and Dathan, and Abiram. This is that Dathan and Abiram, which were famous in the congregation, who strove against Moses and against Aaron in the company of Korah, when they strove against the LORD:}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#880000| And the earth opened her mouth, and swallowed them up together with Korah, when that company died, what time the fire devoured two hundred and fifty men: and they became a sign.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#880000| Notwithstanding the children of Korah died not.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| The sons of Simeon after their families: of Nemuel, the family of the Nemuelites: of Jamin, the family of the Jaminites: of Jachin, the family of the Jachinites:}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Of Zerah, the family of the Zarhites: of Shaul, the family of the Shaulites.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| These are the families of the Simeonites, twenty and two thousand and two hundred.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| The children of Gad after their families: of Zephon, the family of the Zephonites: of Haggi, the family of the Haggites: of Shuni, the family of the Shunites:}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| Of Ozni, the family of the Oznites: of Eri, the family of the Erites:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| Of Arod, the family of the Arodites: of Areli, the family of the Arelites.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| These are the families of the children of Gad according to those that were numbered of them, forty thousand and five hundred.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| The sons of Judah were Er and Onan: and Er and Onan died in the land of Canaan.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And the sons of Judah after their families were; of Shelah, the family of the Shelanites: of Pharez, the family of the Pharzites: of Zerah, the family of the Zarhites.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And the sons of Pharez were; of Hezron, the family of the Hezronites: of Hamul, the family of the Hamulites.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| These are the families of Judah according to those that were numbered of them, threescore and sixteen thousand and five hundred.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| Of the sons of Issachar after their families: of Tola, the family of the Tolaites: of Pua, the family of the Punites:}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| Of Jashub, the family of the Jashubites: of Shimron, the family of the Shimronites.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| These are the families of Issachar according to those that were numbered of them, threescore and four thousand and three hundred.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| Of the sons of Zebulun after their families: of Sered, the family of the Sardites: of Elon, the family of the Elonites: of Jahleel, the family of the Jahleelites.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| These are the families of the Zebulunites according to those that were numbered of them, threescore thousand and five hundred.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| The sons of Joseph after their families were Manasseh and Ephraim.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Of the sons of Manasseh: of Machir, the family of the Machirites: and Machir begat Gilead: of Gilead come the family of the Gileadites.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| These are the sons of Gilead: of Jeezer, the family of the Jeezerites: of Helek, the family of the Helekites:}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| And of Asriel, the family of the Asrielites: and of Shechem, the family of the Shechemites:}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And of Shemida, the family of the Shemidaites: and of Hepher, the family of the Hepherites.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| And Zelophehad the son of Hepher had no sons, but daughters: and the names of the daughters of Zelophehad were Mahlah, and Noah, Hoglah, Milcah, and Tirzah.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| These are the families of Manasseh, and those that were numbered of them, fifty and two thousand and seven hundred.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| These are the sons of Ephraim after their families: of Shuthelah, the family of the Shuthalhites: of Becher, the family of the Bachrites: of Tahan, the family of the Tahanites.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And these are the sons of Shuthelah: of Eran, the family of the Eranites.}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| These are the families of the sons of Ephraim according to those that were numbered of them, thirty and two thousand and five hundred. These are the sons of Joseph after their families.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| The sons of Benjamin after their families: of Bela, the family of the Belaites: of Ashbel, the family of the Ashbelites: of Ahiram, the family of the Ahiramites:}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| Of Shupham, the family of the Shuphamites: of Hupham, the family of the Huphamites.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| And the sons of Bela were Ard and Naaman: of Ard, the family of the Ardites: and of Naaman, the family of the Naamites.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| These are the sons of Benjamin after their families: and they that were numbered of them were forty and five thousand and six hundred.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| These are the sons of Dan after their families: of Shuham, the family of the Shuhamites. These are the families of Dan after their families.}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| All the families of the Shuhamites, according to those that were numbered of them, were threescore and four thousand and four hundred.}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| Of the children of Asher after their families: of Jimna, the family of the Jimnites: of Jesui, the family of the Jesuites: of Beriah, the family of the Beriites.}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| Of the sons of Beriah: of Heber, the family of the Heberites: of Malchiel, the family of the Malchielites.}} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| And the name of the daughter of Asher was Sarah.}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| These are the families of the sons of Asher according to those that were numbered of them; who were fifty and three thousand and four hundred.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| Of the sons of Naphtali after their families: of Jahzeel, the family of the Jahzeelites: of Guni, the family of the Gunites:}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| Of Jezer, the family of the Jezerites: of Shillem, the family of the Shillemites.}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| These are the families of Naphtali according to their families: and they that were numbered of them were forty and five thousand and four hundred.}} <small>{{font|color=#0000FF|51}}</small>{{font|color=#888800| These were the numbered of the children of Israel, six hundred thousand and a thousand seven hundred and thirty.}} <small>{{font|color=#0000FF|52}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|53}}</small>{{font|color=#888800| Unto these the land shall be divided for an inheritance according to the number of names.}} <small>{{font|color=#0000FF|54}}</small>{{font|color=#888800| To many thou shalt give the more inheritance, and to few thou shalt give the less inheritance: to every one shall his inheritance be given according to those that were numbered of him.}} <small>{{font|color=#0000FF|55}}</small>{{font|color=#888800| Notwithstanding the land shall be divided by lot: according to the names of the tribes of their fathers they shall inherit.}} <small>{{font|color=#0000FF|56}}</small>{{font|color=#888800| According to the lot shall the possession thereof be divided between many and few.}} <small>{{font|color=#0000FF|57}}</small>{{font|color=#888800| And these are they that were numbered of the Levites after their families: of Gershon, the family of the Gershonites: of Kohath, the family of the Kohathites: of Merari, the family of the Merarites.}} <small>{{font|color=#0000FF|58}}</small>{{font|color=#888800| These are the families of the Levites: the family of the Libnites, the family of the Hebronites, the family of the Mahlites, the family of the Mushites, the family of the Korathites. And Kohath begat Amram.}} <small>{{font|color=#0000FF|59}}</small>{{font|color=#888800| And the name of Amram's wife was Jochebed, the daughter of Levi, whom her mother bare to Levi in Egypt: and she bare unto Amram Aaron and Moses, and Miriam their sister.}} <small>{{font|color=#0000FF|60}}</small>{{font|color=#888800| And unto Aaron was born Nadab, and Abihu, Eleazar, and Ithamar.}} <small>{{font|color=#0000FF|61}}</small>{{font|color=#888800| And Nadab and Abihu died, when they offered strange fire before the LORD.}} <small>{{font|color=#0000FF|62}}</small>{{font|color=#888800| And those that were numbered of them were twenty and three thousand, all males from a month old and upward: for they were not numbered among the children of Israel, because there was no inheritance given them among the children of Israel.}} <small>{{font|color=#0000FF|63}}</small>{{font|color=#888800| These are they that were numbered by Moses and Eleazar the priest, who numbered the children of Israel in the plains of Moab by Jordan near Jericho.}} <small>{{font|color=#0000FF|64}}</small>{{font|color=#888800| But among these there was not a man of them whom Moses and Aaron the priest numbered, when they numbered the children of Israel in the wilderness of Sinai.}} <small>{{font|color=#0000FF|65}}</small>{{font|color=#888800| For the LORD had said of them, They shall surely die in the wilderness. And there was not left a man of them, save Caleb the son of Jephunneh, and Joshua the son of Nun.}} ==Chapter 27== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| Then came the daughters of Zelophehad, the son of Hepher, the son of Gilead, the son of Machir, the son of Manasseh, of the families of Manasseh the son of Joseph: and these are the names of his daughters; Mahlah, Noah, and Hoglah, and Milcah, and Tirzah.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And they stood before Moses, and before Eleazar the priest, and before the princes and all the congregation, by the door of the tabernacle of the congregation, saying,}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| Our father died in the wilderness, and he was not in the company of them that gathered themselves together against the LORD in the company of Korah; but died in his own sin, and had no sons.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| Why should the name of our father be done away from among his family, because he hath no son? Give unto us therefore a possession among the brethren of our father.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And Moses brought their cause before the LORD.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| The daughters of Zelophehad speak right: thou shalt surely give them a possession of an inheritance among their father's brethren; and thou shalt cause the inheritance of their father to pass unto them.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And thou shalt speak unto the children of Israel, saying, If a man die, and have no son, then ye shall cause his inheritance to pass unto his daughter.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And if he have no daughter, then ye shall give his inheritance unto his brethren.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And if he have no brethren, then ye shall give his inheritance unto his father's brethren.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And if his father have no brethren, then ye shall give his inheritance unto his kinsman that is next to him of his family, and he shall possess it: and it shall be unto the children of Israel a statute of judgment, as the LORD commanded Moses.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And the LORD said unto Moses, Get thee up into this mount Abarim, and see the land which I have given unto the children of Israel.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And when thou hast seen it, thou also shalt be gathered unto thy people, as Aaron thy brother was gathered.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| For ye rebelled against my commandment in the desert of Zin, in the strife of the congregation, to sanctify me at the water before their eyes: that is the water of Meribah in Kadesh in the wilderness of Zin.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And Moses spake unto the LORD, saying,}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| Let the LORD, the God of the spirits of all flesh, set a man over the congregation,}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| Which may go out before them, and which may go in before them, and which may lead them out, and which may bring them in; that the congregation of the LORD be not as sheep which have no shepherd.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And the LORD said unto Moses, Take thee Joshua the son of Nun, a man in whom is the spirit, and lay thine hand upon him;}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And set him before Eleazar the priest, and before all the congregation; and give him a charge in their sight.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And thou shalt put some of thine honour upon him, that all the congregation of the children of Israel may be obedient.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And he shall stand before Eleazar the priest, who shall ask counsel for him after the judgment of Urim before the LORD: at his word shall they go out, and at his word they shall come in, both he, and all the children of Israel with him, even all the congregation.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And Moses did as the LORD commanded him: and he took Joshua, and set him before Eleazar the priest, and before all the congregation:}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And he laid his hands upon him, and gave him a charge, as the LORD commanded by the hand of Moses.}} ==Chapter 28== <small>{{font|color=#0000FF|1}}</small>{{font|color=#880088| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#880088| Command the children of Israel, and say unto them, My offering, and my bread for my sacrifices made by fire, for a sweet savour unto me, shall ye observe to offer unto me in their due season.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#880088| And thou shalt say unto them, This is the offering made by fire which ye shall offer unto the LORD; two lambs of the first year without spot day by day, for a continual burnt offering.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#880088| The one lamb shalt thou offer in the morning, and the other lamb shalt thou offer at even;}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#880088| And a tenth part of an ephah of flour for a meat offering, mingled with the fourth part of an hin of beaten oil.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#880088| It is a continual burnt offering, which was ordained in mount Sinai for a sweet savour, a sacrifice made by fire unto the LORD.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#880088| And the drink offering thereof shall be the fourth part of an hin for the one lamb: in the holy place shalt thou cause the strong wine to be poured unto the LORD for a drink offering.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#880088| And the other lamb shalt thou offer at even: as the meat offering of the morning, and as the drink offering thereof, thou shalt offer it, a sacrifice made by fire, of a sweet savour unto the LORD.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#880088| And on the sabbath day two lambs of the first year without spot, and two tenth deals of flour for a meat offering, mingled with oil, and the drink offering thereof:}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#880088| This is the burnt offering of every sabbath, beside the continual burnt offering, and his drink offering.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#880088| And in the beginnings of your months ye shall offer a burnt offering unto the LORD; two young bullocks, and one ram, seven lambs of the first year without spot;}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#880088| And three tenth deals of flour for a meat offering, mingled with oil, for one bullock; and two tenth deals of flour for a meat offering, mingled with oil, for one ram;}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#880088| And a several tenth deal of flour mingled with oil for a meat offering unto one lamb; for a burnt offering of a sweet savour, a sacrifice made by fire unto the LORD.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#880088| And their drink offerings shall be half an hin of wine unto a bullock, and the third part of an hin unto a ram, and a fourth part of an hin unto a lamb: this is the burnt offering of every month throughout the months of the year.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#880088| And one kid of the goats for a sin offering unto the LORD shall be offered, beside the continual burnt offering, and his drink offering.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#880088| And in the fourteenth day of the first month is the passover of the LORD.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#880088| And in the fifteenth day of this month is the feast: seven days shall unleavened bread be eaten.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#880088| In the first day shall be an holy convocation; ye shall do no manner of servile work therein:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#880088| But ye shall offer a sacrifice made by fire for a burnt offering unto the LORD; two young bullocks, and one ram, and seven lambs of the first year: they shall be unto you without blemish:}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#880088| And their meat offering shall be of flour mingled with oil: three tenth deals shall ye offer for a bullock, and two tenth deals for a ram;}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#880088| A several tenth deal shalt thou offer for every lamb, throughout the seven lambs:}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#880088| And one goat for a sin offering, to make an atonement for you. }} <small>{{font|color=#0000FF|23}}</small>{{font|color=#880088| Ye shall offer these beside the burnt offering in the morning, which is for a continual burnt offering.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#880088| After this manner ye shall offer daily, throughout the seven days, the meat of the sacrifice made by fire, of a sweet savour unto the LORD: it shall be offered beside the continual burnt offering, and his drink offering.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#880088| And on the seventh day ye shall have an holy convocation; ye shall do no servile work.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#880088| Also in the day of the firstfruits, when ye bring a new meat offering unto the LORD, after your weeks be out, ye shall have an holy convocation; ye shall do no servile work:}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#880088| But ye shall offer the burnt offering for a sweet savour unto the LORD; two young bullocks, one ram, seven lambs of the first year;}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#880088| And their meat offering of flour mingled with oil, three tenth deals unto one bullock, two tenth deals unto one ram,}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#880088| A several tenth deal unto one lamb, throughout the seven lambs;}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#880088| And one kid of the goats, to make an atonement for you. }} <small>{{font|color=#0000FF|31}}</small>{{font|color=#880088| Ye shall offer them beside the continual burnt offering, and his meat offering, (they shall be unto you without blemish) and their drink offerings.}} ==Chapter 29== <small>{{font|color=#0000FF|1}}</small>{{font|color=#880088| And in the seventh month, on the first day of the month, ye shall have an holy convocation; ye shall do no servile work it is a day of blowing the trumpets unto you.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#880088| And ye shall offer a burnt offering for a sweet savour unto the LORD; one young bullock, one ram, and seven lambs of the first year without blemish:}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#880088| And their meat offering shall be of flour mingled with oil, three tenth deals for a bullock, and two tenth deals for a ram,}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#880088| And one tenth deal for one lamb, throughout the seven lambs:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#880088| And one kid of the goats for a sin offering, to make an atonement for you:}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#880088| Beside the burnt offering of the month, and his meat offering, and the daily burnt offering, and his meat offering, and their drink offerings, according unto their manner, for a sweet savour, a sacrifice made by fire unto the LORD.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#880088| And ye shall have on the tenth day of this seventh month an holy convocation; and ye shall afflict your souls: ye shall not do any work therein:}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#880088| But ye shall offer a burnt offering unto the LORD for a sweet savour; one young bullock, one ram, and seven lambs of the first year; they shall be unto you without blemish:}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#880088| And their meat offering shall be of flour mingled with oil, three tenth deals to a bullock, and two tenth deals to one ram,}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#880088| A several tenth deal for one lamb, throughout the seven lambs:}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#880088| One kid of the goats for a sin offering; beside the sin offering of atonement, and the continual burnt offering, and the meat offering of it, and their drink offerings.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#880088| And on the fifteenth day of the seventh month ye shall have an holy convocation; ye shall do no servile work, and ye shall keep a feast unto the LORD seven days:}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#880088| And ye shall offer a burnt offering, a sacrifice made by fire, of a sweet savour unto the LORD; thirteen young bullocks, two rams, and fourteen lambs of the first year; they shall be without blemish:}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#880088| And their meat offering shall be of flour mingled with oil, three tenth deals unto every bullock of the thirteen bullocks, two tenth deals to each ram of the two rams,}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#880088| And a several tenth deal to each lamb of the fourteen lambs:}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#880088| And one kid of the goats for a sin offering; beside the continual burnt offering, his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#880088| And on the second day ye shall offer twelve young bullocks, two rams, fourteen lambs of the first year without spot:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#880088| And one kid of the goats for a sin offering; beside the continual burnt offering, and the meat offering thereof, and their drink offerings.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#880088| And on the third day eleven bullocks, two rams, fourteen lambs of the first year without blemish;}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, and his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#880088| And on the fourth day ten bullocks, two rams, and fourteen lambs of the first year without blemish:}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#880088| Their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#880088| And one kid of the goats for a sin offering; beside the continual burnt offering, his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#880088| And on the fifth day nine bullocks, two rams, and fourteen lambs of the first year without spot:}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, and his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#880088| And on the sixth day eight bullocks, two rams, and fourteen lambs of the first year without blemish:}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#880088| And on the seventh day seven bullocks, two rams, and fourteen lambs of the first year without blemish:}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#880088| On the eighth day ye shall have a solemn assembly: ye shall do no servile work therein:}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#880088| But ye shall offer a burnt offering, a sacrifice made by fire, of a sweet savour unto the LORD: one bullock, one ram, seven lambs of the first year without blemish:}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#880088| Their meat offering and their drink offerings for the bullock, for the ram, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, and his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#880088| These things ye shall do unto the LORD in your set feasts, beside your vows, and your freewill offerings, for your burnt offerings, and for your meat offerings, and for your drink offerings, and for your peace offerings.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#880088| And Moses told the children of Israel according to all that the LORD commanded Moses.}} ==Chapter 30== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And Moses spake unto the heads of the tribes concerning the children of Israel, saying, This is the thing which the LORD hath commanded.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| If a man vow a vow unto the LORD, or swear an oath to bind his soul with a bond; he shall not break his word, he shall do according to all that proceedeth out of his mouth.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| If a woman also vow a vow unto the LORD, and bind herself by a bond, being in her father's house in her youth;}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And her father hear her vow, and her bond wherewith she hath bound her soul, and her father shall hold his peace at her; then all her vows shall stand, and every bond wherewith she hath bound her soul shall stand.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| But if her father disallow her in the day that he heareth; not any of her vows, or of her bonds wherewith she hath bound her soul, shall stand: and the LORD shall forgive her, because her father disallowed her.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And if she had at all an husband, when she vowed, or uttered ought out of her lips, wherewith she bound her soul;}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And her husband heard it, and held his peace at her in the day that he heard it: then her vows shall stand, and her bonds wherewith she bound her soul shall stand.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| But if her husband disallowed her on the day that he heard it; then he shall make her vow which she vowed, and that which she uttered with her lips, wherewith she bound her soul, of none effect: and the LORD shall forgive her.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| But every vow of a widow, and of her that is divorced, wherewith they have bound their souls, shall stand against her.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And if she vowed in her husband's house, or bound her soul by a bond with an oath;}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And her husband heard it, and held his peace at her, and disallowed her not: then all her vows shall stand, and every bond wherewith she bound her soul shall stand.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| But if her husband hath utterly made them void on the day he heard them; then whatsoever proceeded out of her lips concerning her vows, or concerning the bond of her soul, shall not stand: her husband hath made them void; and the LORD shall forgive her.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Every vow, and every binding oath to afflict the soul, her husband may establish it, or her husband may make it void.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| But if her husband altogether hold his peace at her from day to day; then he establisheth all her vows, or all her bonds, which are upon her: he confirmeth them, because he held his peace at her in the day that he heard them.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| But if he shall any ways make them void after that he hath heard them; then he shall bear her iniquity.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| These are the statutes, which the LORD commanded Moses, between a man and his wife, between the father and his daughter, being yet in her youth in her father's house.}} ==Chapter 31== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Avenge the children of Israel of the Midianites: afterward shalt thou be gathered unto thy people.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And Moses spake unto the people, saying, Arm some of yourselves unto the war, and let them go against the Midianites, and avenge the LORD of Midian.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| Of every tribe a thousand, throughout all the tribes of Israel, shall ye send to the war.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| So there were delivered out of the thousands of Israel, a thousand of every tribe, twelve thousand armed for war.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And Moses sent them to the war, a thousand of every tribe, them and Phinehas the son of Eleazar the priest, to the war, with the holy instruments, and the trumpets to blow in his hand.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And they warred against the Midianites, as the LORD commanded Moses; and they slew all the males.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And they slew the kings of Midian, beside the rest of them that were slain; namely, Evi, and Rekem, and Zur, and Hur, and Reba, five kings of Midian: Balaam also the son of Beor they slew with the sword.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And the children of Israel took all the women of Midian captives, and their little ones, and took the spoil of all their cattle, and all their flocks, and all their goods.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And they burnt all their cities wherein they dwelt, and all their goodly castles, with fire.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And they took all the spoil, and all the prey, both of men and of beasts.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And they brought the captives, and the prey, and the spoil, unto Moses, and Eleazar the priest, and unto the congregation of the children of Israel, unto the camp at the plains of Moab, which are by Jordan near Jericho.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And Moses, and Eleazar the priest, and all the princes of the congregation, went forth to meet them without the camp.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And Moses was wroth with the officers of the host, with the captains over thousands, and captains over hundreds, which came from the battle.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And Moses said unto them, Have ye saved all the women alive?}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| Behold, these caused the children of Israel, through the counsel of Balaam, to commit trespass against the LORD in the matter of Peor, and there was a plague among the congregation of the LORD.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| Now therefore kill every male among the little ones, and kill every woman that hath known man by lying with him.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| But all the women children, that have not known a man by lying with him, keep alive for yourselves.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And do ye abide without the camp seven days: whosoever hath killed any person, and whosoever hath touched any slain, purify both yourselves and your captives on the third day, and on the seventh day.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And purify all your raiment, and all that is made of skins, and all work of goats' hair, and all things made of wood.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And Eleazar the priest said unto the men of war which went to the battle, This is the ordinance of the law which the LORD commanded Moses;}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Only the gold, and the silver, the brass, the iron, the tin, and the lead,}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| Every thing that may abide the fire, ye shall make it go through the fire, and it shall be clean: nevertheless it shall be purified with the water of separation: and all that abideth not the fire ye shall make go through the water.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| And ye shall wash your clothes on the seventh day, and ye shall be clean, and afterward ye shall come into the camp.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| Take the sum of the prey that was taken, both of man and of beast, thou, and Eleazar the priest, and the chief fathers of the congregation:}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And divide the prey into two parts; between them that took the war upon them, who went out to battle, and between all the congregation:}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| And levy a tribute unto the LORD of the men of war which went out to battle: one soul of five hundred, both of the persons, and of the beeves, and of the asses, and of the sheep:}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Take it of their half, and give it unto Eleazar the priest, for an heave offering of the LORD.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| And of the children of Israel's half, thou shalt take one portion of fifty, of the persons, of the beeves, of the asses, and of the flocks, of all manner of beasts, and give them unto the Levites, which keep the charge of the tabernacle of the LORD.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| And Moses and Eleazar the priest did as the LORD commanded Moses.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And the booty, being the rest of the prey which the men of war had caught, was six hundred thousand and seventy thousand and five thousand sheep,}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| And threescore and twelve thousand beeves,}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| And threescore and one thousand asses,}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| And thirty and two thousand persons in all, of women that had not known man by lying with him.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And the half, which was the portion of them that went out to war, was in number three hundred thousand and seven and thirty thousand and five hundred sheep:}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| And the LORD's tribute of the sheep was six hundred and threescore and fifteen.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| And the beeves were thirty and six thousand; of which the LORD's tribute was threescore and twelve.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| And the asses were thirty thousand and five hundred; of which the LORD's tribute was threescore and one.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| And the persons were sixteen thousand; of which the LORD's tribute was thirty and two persons.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| And Moses gave the tribute, which was the LORD's heave offering, unto Eleazar the priest, as the LORD commanded Moses.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| And of the children of Israel's half, which Moses divided from the men that warred,}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| (Now the half that pertained unto the congregation was three hundred thousand and thirty thousand and seven thousand and five hundred sheep,}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| And thirty and six thousand beeves,}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| And thirty thousand asses and five hundred,}} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| And sixteen thousand persons;)}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| Even of the children of Israel's half, Moses took one portion of fifty, both of man and of beast, and gave them unto the Levites, which kept the charge of the tabernacle of the LORD; as the LORD commanded Moses.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| And the officers which were over thousands of the host, the captains of thousands, and captains of hundreds, came near unto Moses:}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| And they said unto Moses, Thy servants have taken the sum of the men of war which are under our charge, and there lacketh not one man of us.}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| We have therefore brought an oblation for the LORD, what every man hath gotten, of jewels of gold, chains, and bracelets, rings, earrings, and tablets, to make an atonement for our souls before the LORD.}} <small>{{font|color=#0000FF|51}}</small>{{font|color=#888800| And Moses and Eleazar the priest took the gold of them, even all wrought jewels.}} <small>{{font|color=#0000FF|52}}</small>{{font|color=#888800| And all the gold of the offering that they offered up to the LORD, of the captains of thousands, and of the captains of hundreds, was sixteen thousand seven hundred and fifty shekels.}} <small>{{font|color=#0000FF|53}}</small>{{font|color=#888800| (For the men of war had taken spoil, every man for himself.)}} <small>{{font|color=#0000FF|54}}</small>{{font|color=#888800| And Moses and Eleazar the priest took the gold of the captains of thousands and of hundreds, and brought it into the tabernacle of the congregation, for a memorial for the children of Israel before the LORD.}} ==Chapter 32== <small>{{font|color=#0000FF|1}}</small>{{font|color=#008888| Now the children of Reuben and the children of Gad had a very great multitude of cattle: }}{{font|color=#000088|and when they saw the land of Jazer, and the land of Gilead, that, behold, the place was a place for cattle;}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| The children of Gad and the children of Reuben came and spake unto Moses, and to Eleazar the priest, and unto the princes of the congregation, saying,}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| Ataroth, and Dibon, and Jazer, and Nimrah, and Heshbon, and Elealeh, and Shebam, and Nebo, and Beon,}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| Even the country which the LORD smote before the congregation of Israel, is a land for cattle, and thy servants have cattle:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#000088| Wherefore, said they, if we have found grace in thy sight, let this land be given unto thy servants for a possession, and bring us not over Jordan.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#000088| And Moses said unto the children of Gad and to the children of Reuben, Shall your brethren go to war, and shall ye sit here?}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#000088| And wherefore discourage ye the heart of the children of Israel from going over into the land which the LORD hath given them?}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#000088| Thus did your fathers, when I sent them from Kadeshbarnea to see the land.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#000088| For when they went up unto the valley of Eshcol, and saw the land, they discouraged the heart of the children of Israel, that they should not go into the land which the LORD had given them.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#000088| And the LORD's anger was kindled the same time, and he sware, saying,}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#000088| Surely none of the men that came up out of Egypt, from twenty years old and upward, shall see the land which I sware unto Abraham, unto Isaac, and unto Jacob; because they have not wholly followed me:}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#000088| Save Caleb the son of Jephunneh the Kenezite, and Joshua the son of Nun: for they have wholly followed the LORD.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#000088| And the LORD's anger was kindled against Israel, and he made them wander in the wilderness forty years, until all the generation, that had done evil in the sight of the LORD, was consumed.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#000088| And, behold, ye are risen up in your fathers' stead, an increase of sinful men, to augment yet the fierce anger of the LORD toward Israel.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#000088| For if ye turn away from after him, he will yet again leave them in the wilderness; and ye shall destroy all this people.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#008888| And they came near unto him, and said, We will build sheepfolds here for our cattle, and cities for our little ones:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#008888| But we ourselves will go ready armed before the children of Israel, until we have brought them unto their place: and our little ones shall dwell in the fenced cities because of the inhabitants of the land.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#008888| We will not return unto our houses, until the children of Israel have inherited every man his inheritance.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#008888| For we will not inherit with them on yonder side Jordan, or forward; because our inheritance is fallen to us on this side Jordan eastward.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#008888| And Moses said unto them,}}{{font|color=#000088| If ye will do this thing, if ye will go armed before the LORD to war,}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#000088| And will go all of you armed over Jordan before the LORD, until he hath driven out his enemies from before him,}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#000088| And the land be subdued before the LORD: then afterward ye shall return, and be guiltless before the LORD, and before Israel; and this land shall be your possession before the LORD.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#000088| But if ye will not do so, behold, ye have sinned against the LORD: and be sure your sin will find you out.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#008888| Build you cities for your little ones, and folds for your sheep; and do that which hath proceeded out of your mouth.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#000088| And the children of Gad and the children of Reuben spake unto Moses, saying, Thy servants will do as my lord commandeth.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#000088| Our little ones, our wives, our flocks, and all our cattle, shall be there in the cities of Gilead:}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#000088| But thy servants will pass over, every man armed for war, before the Lord to battle, as my lord saith.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| So concerning them Moses commanded Eleazar the priest, and Joshua the son of Nun, and the chief fathers of the tribes of the children of Israel:}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| And Moses said unto them, If the children of Gad and the children of Reuben will pass with you over Jordan, every man armed to battle, before the LORD, and the land shall be subdued before you; then ye shall give them the land of Gilead for a possession:}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| But if they will not pass over with you armed, they shall have possessions among you in the land of Canaan.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| And the children of Gad and the children of Reuben answered, saying, As the LORD hath said unto thy servants, so will we do.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| We will pass over armed before the LORD into the land of Canaan, that the possession of our inheritance on this side Jordan may be our's.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| And Moses gave unto them, even to the children of Gad, and to the children of Reuben, and unto half the tribe of Manasseh the son of Joseph, the kingdom of Sihon king of the Amorites, and the kingdom of Og king of Bashan, the land, with the cities thereof in the coasts, even the cities of the country round about.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#008888| And the children of Gad built Dibon, and Ataroth, and Aroer,}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#008888| And Atroth, Shophan, and Jaazer, and Jogbehah,}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#008888| And Bethnimrah, and Bethharan, fenced cities: and folds for sheep.}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#008888| And the children of Reuben built Heshbon, and Elealeh, and Kirjathaim,}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#008888| And Nebo, and Baalmeon, (their names being changed,) and Shibmah: and gave other names unto the cities which they builded.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#000088| And the children of Machir the son of Manasseh went to Gilead, and took it, and dispossessed the Amorite which was in it.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#008888| And Moses gave Gilead unto Machir the son of Manasseh; and he dwelt therein.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#000088| And Jair the son of Manasseh went and took the small towns thereof, and called them Havothjair.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#000088| And Nobah went and took Kenath, and the villages thereof, and called it Nobah, after his own name.}} ==Chapter 33== <small>{{font|color=#0000FF|1}}</small>{{font|color=#FF8800| These are the journeys of the children of Israel, which went forth out of the land of Egypt with their armies under the hand of Moses and Aaron.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#FF8800| And Moses wrote their goings out according to their journeys by the commandment of the LORD: and these are their journeys according to their goings out.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#FF8800| And they departed from Rameses in the first month, on the fifteenth day of the first month; on the morrow after the passover the children of Israel went out with an high hand in the sight of all the Egyptians.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#FF8800| For the Egyptians buried all their firstborn, which the LORD had smitten among them: upon their gods also the LORD executed judgments.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#FF8800| And the children of Israel removed from Rameses, and pitched in Succoth.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#FF8800| And they departed from Succoth, and pitched in Etham, which is in the edge of the wilderness.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#FF8800| And they removed from Etham, and turned again unto Pihahiroth, which is before Baalzephon: and they pitched before Migdol.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#FF8800| And they departed from before Pihahiroth, and passed through the midst of the sea into the wilderness, and went three days' journey in the wilderness of Etham, and pitched in Marah.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#FF8800| And they removed from Marah, and came unto Elim: and in Elim were twelve fountains of water, and threescore and ten palm trees; and they pitched there.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#FF8800| And they removed from Elim, and encamped by the Red sea.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#FF8800| And they removed from the Red sea, and encamped in the wilderness of Sin.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#FF8800| And they took their journey out of the wilderness of Sin, and encamped in Dophkah.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#FF8800| And they departed from Dophkah, and encamped in Alush. }} <small>{{font|color=#0000FF|14}}</small>{{font|color=#FF8800| And they removed from Alush, and encamped at Rephidim, where was no water for the people to drink.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#FF8800| And they departed from Rephidim, and pitched in the wilderness of Sinai.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#FF8800| And they removed from the desert of Sinai, and pitched at Kibrothhattaavah.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#FF8800| And they departed from Kibrothhattaavah, and encamped at Hazeroth.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#FF8800| And they departed from Hazeroth, and pitched in Rithmah.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#FF8800| And they departed from Rithmah, and pitched at Rimmonparez.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#FF8800| And they departed from Rimmonparez, and pitched in Libnah.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#FF8800| And they removed from Libnah, and pitched at Rissah.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#FF8800| And they journeyed from Rissah, and pitched in Kehelathah.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#FF8800| And they went from Kehelathah, and pitched in mount Shapher.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#FF8800| And they removed from mount Shapher, and encamped in Haradah.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#FF8800| And they removed from Haradah, and pitched in Makheloth.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#FF8800| And they removed from Makheloth, and encamped at Tahath.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#FF8800| And they departed from Tahath, and pitched at Tarah.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#FF8800| And they removed from Tarah, and pitched in Mithcah.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#FF8800| And they went from Mithcah, and pitched in Hashmonah.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#FF8800| And they departed from Hashmonah, and encamped at Moseroth.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#FF8800| And they departed from Moseroth, and pitched in Benejaakan.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#FF8800| And they removed from Benejaakan, and encamped at Horhagidgad.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#FF8800| And they went from Horhagidgad, and pitched in Jotbathah.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#FF8800| And they removed from Jotbathah, and encamped at Ebronah.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#FF8800| And they departed from Ebronah, and encamped at Eziongaber.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#FF8800| And they removed from Eziongaber, and pitched in the wilderness of Zin, which is Kadesh.}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#FF8800| And they removed from Kadesh, and pitched in mount Hor, in the edge of the land of Edom.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#FF0000| And Aaron the priest went up into mount Hor at the commandment of the LORD, and died there, in the fortieth year after the children of Israel were come out of the land of Egypt, in the first day of the fifth month.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#FF0000| And Aaron was an hundred and twenty and three years old when he died in mount Hor.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#FF8800| And king Arad the Canaanite, which dwelt in the south in the land of Canaan, heard of the coming of the children of Israel.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#FF8800| And they departed from mount Hor, and pitched in Zalmonah.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#FF8800| And they departed from Zalmonah, and pitched in Punon.}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#FF8800| And they departed from Punon, and pitched in Oboth.}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#FF8800| And they departed from Oboth, and pitched in Ijeabarim, in the border of Moab.}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#FF8800| And they departed from Iim, and pitched in Dibongad. }} <small>{{font|color=#0000FF|46}}</small>{{font|color=#FF8800| And they removed from Dibongad, and encamped in Almondiblathaim.}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#FF8800| And they removed from Almondiblathaim, and pitched in the mountains of Abarim, before Nebo.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#FF8800| And they departed from the mountains of Abarim, and pitched in the plains of Moab by Jordan near Jericho.}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#FF8800| And they pitched by Jordan, from Bethjesimoth even unto Abelshittim in the plains of Moab.}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| And the LORD spake unto Moses in the plains of Moab by Jordan near Jericho, saying,}} <small>{{font|color=#0000FF|51}}</small>{{font|color=#888800| Speak unto the children of Israel, and say unto them, When ye are passed over Jordan into the land of Canaan;}} <small>{{font|color=#0000FF|52}}</small>{{font|color=#888800| Then ye shall drive out all the inhabitants of the land from before you, and destroy all their pictures, and destroy all their molten images, and quite pluck down all their high places:}} <small>{{font|color=#0000FF|53}}</small>{{font|color=#888800| And ye shall dispossess the inhabitants of the land, and dwell therein: for I have given you the land to possess it.}} <small>{{font|color=#0000FF|54}}</small>{{font|color=#888800| And ye shall divide the land by lot for an inheritance among your families: and to the more ye shall give the more inheritance, and to the fewer ye shall give the less inheritance: every man's inheritance shall be in the place where his lot falleth; according to the tribes of your fathers ye shall inherit.}} <small>{{font|color=#0000FF|55}}</small>{{font|color=#888800| But if ye will not drive out the inhabitants of the land from before you; then it shall come to pass, that those which ye let remain of them shall be pricks in your eyes, and thorns in your sides, and shall vex you in the land wherein ye dwell.}} <small>{{font|color=#0000FF|56}}</small>{{font|color=#888800| Moreover it shall come to pass, that I shall do unto you, as I thought to do unto them.}} ==Chapter 34== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Command the children of Israel, and say unto them, When ye come into the land of Canaan; (this is the land that shall fall unto you for an inheritance, even the land of Canaan with the coasts thereof:)}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| Then your south quarter shall be from the wilderness of Zin along by the coast of Edom, and your south border shall be the outmost coast of the salt sea eastward:}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And your border shall turn from the south to the ascent of Akrabbim, and pass on to Zin: and the going forth thereof shall be from the south to Kadeshbarnea, and shall go on to Hazaraddar, and pass on to Azmon:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And the border shall fetch a compass from Azmon unto the river of Egypt, and the goings out of it shall be at the sea.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And as for the western border, ye shall even have the great sea for a border: this shall be your west border. }} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And this shall be your north border: from the great sea ye shall point out for you mount Hor:}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| From mount Hor ye shall point out your border unto the entrance of Hamath; and the goings forth of the border shall be to Zedad:}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And the border shall go on to Ziphron, and the goings out of it shall be at Hazarenan: this shall be your north border.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And ye shall point out your east border from Hazarenan to Shepham:}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And the coast shall go down from Shepham to Riblah, on the east side of Ain; and the border shall descend, and shall reach unto the side of the sea of Chinnereth eastward:}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And the border shall go down to Jordan, and the goings out of it shall be at the salt sea: this shall be your land with the coasts thereof round about.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And Moses commanded the children of Israel, saying, This is the land which ye shall inherit by lot, which the LORD commanded to give unto the nine tribes, and to the half tribe:}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| For the tribe of the children of Reuben according to the house of their fathers, and the tribe of the children of Gad according to the house of their fathers, have received their inheritance; and half the tribe of Manasseh have received their inheritance:}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| The two tribes and the half tribe have received their inheritance on this side Jordan near Jericho eastward, toward the sunrising.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| These are the names of the men which shall divide the land unto you: Eleazar the priest, and Joshua the son of Nun.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And ye shall take one prince of every tribe, to divide the land by inheritance.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And the names of the men are these: Of the tribe of Judah, Caleb the son of Jephunneh.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And of the tribe of the children of Simeon, Shemuel the son of Ammihud.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Of the tribe of Benjamin, Elidad the son of Chislon.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And the prince of the tribe of the children of Dan, Bukki the son of Jogli.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| The prince of the children of Joseph, for the tribe of the children of Manasseh, Hanniel the son of Ephod.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| And the prince of the tribe of the children of Ephraim, Kemuel the son of Shiphtan.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the prince of the tribe of the children of Zebulun, Elizaphan the son of Parnach.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And the prince of the tribe of the children of Issachar, Paltiel the son of Azzan.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And the prince of the tribe of the children of Asher, Ahihud the son of Shelomi.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| And the prince of the tribe of the children of Naphtali, Pedahel the son of Ammihud.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| These are they whom the LORD commanded to divide the inheritance unto the children of Israel in the land of Canaan.}} ==Chapter 35== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses in the plains of Moab by Jordan near Jericho, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Command the children of Israel, that they give unto the Levites of the inheritance of their possession cities to dwell in; and ye shall give also unto the Levites suburbs for the cities round about them.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And the cities shall they have to dwell in; and the suburbs of them shall be for their cattle, and for their goods, and for all their beasts.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And the suburbs of the cities, which ye shall give unto the Levites, shall reach from the wall of the city and outward a thousand cubits round about.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And ye shall measure from without the city on the east side two thousand cubits, and on the south side two thousand cubits, and on the west side two thousand cubits, and on the north side two thousand cubits; and the city shall be in the midst: this shall be to them the suburbs of the cities.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And among the cities which ye shall give unto the Levites there shall be six cities for refuge, which ye shall appoint for the manslayer, that he may flee thither: and to them ye shall add forty and two cities.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| So all the cities which ye shall give to the Levites shall be forty and eight cities: them shall ye give with their suburbs.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And the cities which ye shall give shall be of the possession of the children of Israel: from them that have many ye shall give many; but from them that have few ye shall give few: every one shall give of his cities unto the Levites according to his inheritance which he inheriteth.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| Speak unto the children of Israel, and say unto them, When ye be come over Jordan into the land of Canaan;}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| Then ye shall appoint you cities to be cities of refuge for you; that the slayer may flee thither, which killeth any person at unawares.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And they shall be unto you cities for refuge from the avenger; that the manslayer die not, until he stand before the congregation in judgment.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And of these cities which ye shall give six cities shall ye have for refuge.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Ye shall give three cities on this side Jordan, and three cities shall ye give in the land of Canaan, which shall be cities of refuge.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| These six cities shall be a refuge, both for the children of Israel, and for the stranger, and for the sojourner among them: that every one that killeth any person unawares may flee thither.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And if he smite him with an instrument of iron, so that he die, he is a murderer: the murderer shall surely be put to death.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And if he smite him with throwing a stone, wherewith he may die, and he die, he is a murderer: the murderer shall surely be put to death.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| Or if he smite him with an hand weapon of wood, wherewith he may die, and he die, he is a murderer: the murderer shall surely be put to death.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| The revenger of blood himself shall slay the murderer: when he meeteth him, he shall slay him.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| But if he thrust him of hatred, or hurl at him by laying of wait, that he die;}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Or in enmity smite him with his hand, that he die: he that smote him shall surely be put to death; for he is a murderer: the revenger of blood shall slay the murderer, when he meeteth him.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| But if he thrust him suddenly without enmity, or have cast upon him any thing without laying of wait,}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| Or with any stone, wherewith a man may die, seeing him not, and cast it upon him, that he die, and was not his enemy, neither sought his harm:}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| Then the congregation shall judge between the slayer and the revenger of blood according to these judgments:}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the congregation shall deliver the slayer out of the hand of the revenger of blood, and the congregation shall restore him to the city of his refuge, whither he was fled: and he shall abide in it unto the death of the high priest, which was anointed with the holy oil.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| But if the slayer shall at any time come without the border of the city of his refuge, whither he was fled;}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And the revenger of blood find him without the borders of the city of his refuge, and the revenger of blood kill the slayer; he shall not be guilty of blood:}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| Because he should have remained in the city of his refuge until the death of the high priest: but after the death of the high priest the slayer shall return into the land of his possession.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| So these things shall be for a statute of judgment unto you throughout your generations in all your dwellings.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| Whoso killeth any person, the murderer shall be put to death by the mouth of witnesses: but one witness shall not testify against any person to cause him to die.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| Moreover ye shall take no satisfaction for the life of a murderer, which is guilty of death: but he shall be surely put to death.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And ye shall take no satisfaction for him that is fled to the city of his refuge, that he should come again to dwell in the land, until the death of the priest.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| So ye shall not pollute the land wherein ye are: for blood it defileth the land: and the land cannot be cleansed of the blood that is shed therein, but by the blood of him that shed it.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| Defile not therefore the land which ye shall inhabit, wherein I dwell: for I the LORD dwell among the children of Israel.}} ==Chapter 36== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the chief fathers of the families of the children of Gilead, the son of Machir, the son of Manasseh, of the families of the sons of Joseph, came near, and spake before Moses, and before the princes, the chief fathers of the children of Israel:}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And they said, The LORD commanded my lord to give the land for an inheritance by lot to the children of Israel: and my lord was commanded by the LORD to give the inheritance of Zelophehad our brother unto his daughters.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And if they be married to any of the sons of the other tribes of the children of Israel, then shall their inheritance be taken from the inheritance of our fathers, and shall be put to the inheritance of the tribe whereunto they are received: so shall it be taken from the lot of our inheritance.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And when the jubile of the children of Israel shall be, then shall their inheritance be put unto the inheritance of the tribe whereunto they are received: so shall their inheritance be taken away from the inheritance of the tribe of our fathers.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And Moses commanded the children of Israel according to the word of the LORD, saying, The tribe of the sons of Joseph hath said well.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| This is the thing which the LORD doth command concerning the daughters of Zelophehad, saying, Let them marry to whom they think best; only to the family of the tribe of their father shall they marry.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| So shall not the inheritance of the children of Israel remove from tribe to tribe: for every one of the children of Israel shall keep himself to the inheritance of the tribe of his fathers.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And every daughter, that possesseth an inheritance in any tribe of the children of Israel, shall be wife unto one of the family of the tribe of her father, that the children of Israel may enjoy every man the inheritance of his fathers.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| Neither shall the inheritance remove from one tribe to another tribe; but every one of the tribes of the children of Israel shall keep himself to his own inheritance.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| Even as the LORD commanded Moses, so did the daughters of Zelophehad:}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| For Mahlah, Tirzah, and Hoglah, and Milcah, and Noah, the daughters of Zelophehad, were married unto their father's brothers' sons:}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And they were married into the families of the sons of Manasseh the son of Joseph, and their inheritance remained in the tribe of the family of their father.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| These are the commandments and the judgments, which the LORD commanded by the hand of Moses unto the children of Israel in the plains of Moab by Jordan near Jericho.}} [[Category:Documentary hypothesis]] jqabm1zi9sndabtvbmmipiywln56iun 2416167 2416166 2022-08-18T14:40:34Z Huz and Buz 2928717 /* Chapter 25 */ wikitext text/x-wiki {{RightTOC}} <[[Bible, English, King James, According to the documentary hypothesis]] According to the [[w:documentary hypothesis|documentary hypothesis]], [[w:Numbers|Numbers]] is composed from a number of originally independant sources joined by a [[w:redaction|redactor]]. There follows the text of Numbers in the [[w:King James Version|King James Version]], with sources highlighted according to the documentary hypothesis *The [[w:Priestly source|"Priestly source"]] is highlighted in {{font|color=#888800|olive yellow}} <small>{{font|color=#0000FF|([[Bible, English, King James, Documentary Hypothesis, Priestly source|view in isolation]])}}</small> *The [[w:Jahwist|"Jahwist source"]] is highlighted in {{font|color=#000088|navy blue}} <small>{{font|color=#0000FF|([[Bible, English, King James, Documentary Hypothesis, JE source, Jahwist source|view in isolation]])}}</small> *The [[w:Elohist|"Elohist source"]] is highlighted in {{font|color=#008888|teal blueish grey}} <small>{{font|color=#0000FF|([[Bible, English, King James, Documentary Hypothesis, JE source, Elohist source|view in isolation]])}}</small> *The [[w:Torah redactor|"Additions by the redactor"]] are highlighted in {{font|color=#880000|maroon red}} ==Chapter 1== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses in the wilderness of Sinai, in the tabernacle of the congregation, on the first day of the second month, in the second year after they were come out of the land of Egypt, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Take ye the sum of all the congregation of the children of Israel, after their families, by the house of their fathers, with the number of their names, every male by their polls;}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| From twenty years old and upward, all that are able to go forth to war in Israel: thou and Aaron shall number them by their armies.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And with you there shall be a man of every tribe; every one head of the house of his fathers.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And these are the names of the men that shall stand with you: of the tribe of Reuben; Elizur the son of Shedeur.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Of Simeon; Shelumiel the son of Zurishaddai.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Of Judah; Nahshon the son of Amminadab.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| Of Issachar; Nethaneel the son of Zuar.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| Of Zebulun; Eliab the son of Helon.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| Of the children of Joseph: of Ephraim; Elishama the son of Ammihud: of Manasseh; Gamaliel the son of Pedahzur.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| Of Benjamin; Abidan the son of Gideoni.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| Of Dan; Ahiezer the son of Ammishaddai.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Of Asher; Pagiel the son of Ocran.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Of Gad; Eliasaph the son of Deuel.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| Of Naphtali; Ahira the son of Enan.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| These were the renowned of the congregation, princes of the tribes of their fathers, heads of thousands in Israel.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And Moses and Aaron took these men which are expressed by their names:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And they assembled all the congregation together on the first day of the second month, and they declared their pedigrees after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, by their polls.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| As the LORD commanded Moses, so he numbered them in the wilderness of Sinai.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And the children of Reuben, Israel's eldest son, by their generations, after their families, by the house of their fathers, according to the number of the names, by their polls, every male from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Reuben, were forty and six thousand and five hundred.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Of the children of Simeon, by their generations, after their families, by the house of their fathers, those that were numbered of them, according to the number of the names, by their polls, every male from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Simeon, were fifty and nine thousand and three hundred.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| Of the children of Gad, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Gad, were forty and five thousand six hundred and fifty.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| Of the children of Judah, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Judah, were threescore and fourteen thousand and six hundred. }} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| Of the children of Issachar, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Issachar, were fifty and four thousand and four hundred.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| Of the children of Zebulun, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Zebulun, were fifty and seven thousand and four hundred.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| Of the children of Joseph, namely, of the children of Ephraim, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Ephraim, were forty thousand and five hundred.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| Of the children of Manasseh, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Manasseh, were thirty and two thousand and two hundred.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| Of the children of Benjamin, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Benjamin, were thirty and five thousand and four hundred.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| Of the children of Dan, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Dan, were threescore and two thousand and seven hundred.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| Of the children of Asher, by their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Asher, were forty and one thousand and five hundred.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| Of the children of Naphtali, throughout their generations, after their families, by the house of their fathers, according to the number of the names, from twenty years old and upward, all that were able to go forth to war;}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| Those that were numbered of them, even of the tribe of Naphtali, were fifty and three thousand and four hundred.}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| These are those that were numbered, which Moses and Aaron numbered, and the princes of Israel, being twelve men: each one was for the house of his fathers.}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| So were all those that were numbered of the children of Israel, by the house of their fathers, from twenty years old and upward, all that were able to go forth to war in Israel; }} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| Even all they that were numbered were six hundred thousand and three thousand and five hundred and fifty.}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| But the Levites after the tribe of their fathers were not numbered among them.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| For the LORD had spoken unto Moses, saying,}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| Only thou shalt not number the tribe of Levi, neither take the sum of them among the children of Israel:}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| But thou shalt appoint the Levites over the tabernacle of testimony, and over all the vessels thereof, and over all things that belong to it: they shall bear the tabernacle, and all the vessels thereof; and they shall minister unto it, and shall encamp round about the tabernacle.}} <small>{{font|color=#0000FF|51}}</small>{{font|color=#888800| And when the tabernacle setteth forward, the Levites shall take it down: and when the tabernacle is to be pitched, the Levites shall set it up: and the stranger that cometh nigh shall be put to death.}} <small>{{font|color=#0000FF|52}}</small>{{font|color=#888800| And the children of Israel shall pitch their tents, every man by his own camp, and every man by his own standard, throughout their hosts.}} <small>{{font|color=#0000FF|53}}</small>{{font|color=#888800| But the Levites shall pitch round about the tabernacle of testimony, that there be no wrath upon the congregation of the children of Israel: and the Levites shall keep the charge of the tabernacle of testimony.}} <small>{{font|color=#0000FF|54}}</small>{{font|color=#888800| And the children of Israel did according to all that the LORD commanded Moses, so did they.}} ==Chapter 2== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Every man of the children of Israel shall pitch by his own standard, with the ensign of their father's house: far off about the tabernacle of the congregation shall they pitch.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And on the east side toward the rising of the sun shall they of the standard of the camp of Judah pitch throughout their armies: and Nahshon the son of Amminadab shall be captain of the children of Judah.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were threescore and fourteen thousand and six hundred.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And those that do pitch next unto him shall be the tribe of Issachar: and Nethaneel the son of Zuar shall be captain of the children of Issachar.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And his host, and those that were numbered thereof, were fifty and four thousand and four hundred.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Then the tribe of Zebulun: and Eliab the son of Helon shall be captain of the children of Zebulun.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And his host, and those that were numbered thereof, were fifty and seven thousand and four hundred.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| All that were numbered in the camp of Judah were an hundred thousand and fourscore thousand and six thousand and four hundred, throughout their armies. These shall first set forth.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| On the south side shall be the standard of the camp of Reuben according to their armies: and the captain of the children of Reuben shall be Elizur the son of Shedeur.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And his host, and those that were numbered thereof, were forty and six thousand and five hundred.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And those which pitch by him shall be the tribe of Simeon: and the captain of the children of Simeon shall be Shelumiel the son of Zurishaddai.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were fifty and nine thousand and three hundred.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Then the tribe of Gad: and the captain of the sons of Gad shall be Eliasaph the son of Reuel.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were forty and five thousand and six hundred and fifty.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| All that were numbered in the camp of Reuben were an hundred thousand and fifty and one thousand and four hundred and fifty, throughout their armies. And they shall set forth in the second rank.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| Then the tabernacle of the congregation shall set forward with the camp of the Levites in the midst of the camp: as they encamp, so shall they set forward, every man in his place by their standards.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| On the west side shall be the standard of the camp of Ephraim according to their armies: and the captain of the sons of Ephraim shall be Elishama the son of Ammihud.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were forty thousand and five hundred.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And by him shall be the tribe of Manasseh: and the captain of the children of Manasseh shall be Gamaliel the son of Pedahzur.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were thirty and two thousand and two hundred.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Then the tribe of Benjamin: and the captain of the sons of Benjamin shall be Abidan the son of Gideoni.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were thirty and five thousand and four hundred.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| All that were numbered of the camp of Ephraim were an hundred thousand and eight thousand and an hundred, throughout their armies. And they shall go forward in the third rank.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| The standard of the camp of Dan shall be on the north side by their armies: and the captain of the children of Dan shall be Ahiezer the son of Ammishaddai.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were threescore and two thousand and seven hundred.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And those that encamp by him shall be the tribe of Asher: and the captain of the children of Asher shall be Pagiel the son of Ocran.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were forty and one thousand and five hundred.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Then the tribe of Naphtali: and the captain of the children of Naphtali shall be Ahira the son of Enan.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| And his host, and those that were numbered of them, were fifty and three thousand and four hundred.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| All they that were numbered in the camp of Dan were an hundred thousand and fifty and seven thousand and six hundred. They shall go hindmost with their standards.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| These are those which were numbered of the children of Israel by the house of their fathers: all those that were numbered of the camps throughout their hosts were six hundred thousand and three thousand and five hundred and fifty.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| But the Levites were not numbered among the children of Israel; as the LORD commanded Moses.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| And the children of Israel did according to all that the LORD commanded Moses: so they pitched by their standards, and so they set forward, every one after their families, according to the house of their fathers.}} ==Chapter 3== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| These also are the generations of Aaron and Moses in the day that the LORD spake with Moses in mount Sinai.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And these are the names of the sons of Aaron; Nadab the firstborn, and Abihu, Eleazar, and Ithamar.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| These are the names of the sons of Aaron, the priests which were anointed, whom he consecrated to minister in the priest's office.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And Nadab and Abihu died before the LORD, when they offered strange fire before the LORD, in the wilderness of Sinai, and they had no children: and Eleazar and Ithamar ministered in the priest's office in the sight of Aaron their father.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Bring the tribe of Levi near, and present them before Aaron the priest, that they may minister unto him.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And they shall keep his charge, and the charge of the whole congregation before the tabernacle of the congregation, to do the service of the tabernacle.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And they shall keep all the instruments of the tabernacle of the congregation, and the charge of the children of Israel, to do the service of the tabernacle.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And thou shalt give the Levites unto Aaron and to his sons: they are wholly given unto him out of the children of Israel.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And thou shalt appoint Aaron and his sons, and they shall wait on their priest's office: and the stranger that cometh nigh shall be put to death.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying, }} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And I, behold, I have taken the Levites from among the children of Israel instead of all the firstborn that openeth the matrix among the children of Israel: therefore the Levites shall be mine;}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Because all the firstborn are mine; for on the day that I smote all the firstborn in the land of Egypt I hallowed unto me all the firstborn in Israel, both man and beast: mine shall they be: I am the LORD.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And the LORD spake unto Moses in the wilderness of Sinai, saying,}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| Number the children of Levi after the house of their fathers, by their families: every male from a month old and upward shalt thou number them.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And Moses numbered them according to the word of the LORD, as he was commanded.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And these were the sons of Levi by their names; Gershon, and Kohath, and Merari.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And these are the names of the sons of Gershon by their families; Libni, and Shimei.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And the sons of Kohath by their families; Amram, and Izehar, Hebron, and Uzziel.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And the sons of Merari by their families; Mahli, and Mushi. These are the families of the Levites according to the house of their fathers.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Of Gershon was the family of the Libnites, and the family of the Shimites: these are the families of the Gershonites.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Those that were numbered of them, according to the number of all the males, from a month old and upward, even those that were numbered of them were seven thousand and five hundred.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| The families of the Gershonites shall pitch behind the tabernacle westward.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| And the chief of the house of the father of the Gershonites shall be Eliasaph the son of Lael.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the charge of the sons of Gershon in the tabernacle of the congregation shall be the tabernacle, and the tent, the covering thereof, and the hanging for the door of the tabernacle of the congregation,}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And the hangings of the court, and the curtain for the door of the court, which is by the tabernacle, and by the altar round about, and the cords of it for all the service thereof.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And of Kohath was the family of the Amramites, and the family of the Izeharites, and the family of the Hebronites, and the family of the Uzzielites: these are the families of the Kohathites.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| In the number of all the males, from a month old and upward, were eight thousand and six hundred, keeping the charge of the sanctuary.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| The families of the sons of Kohath shall pitch on the side of the tabernacle southward.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| And the chief of the house of the father of the families of the Kohathites shall be Elizaphan the son of Uzziel.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| And their charge shall be the ark, and the table, and the candlestick, and the altars, and the vessels of the sanctuary wherewith they minister, and the hanging, and all the service thereof.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And Eleazar the son of Aaron the priest shall be chief over the chief of the Levites, and have the oversight of them that keep the charge of the sanctuary.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| Of Merari was the family of the Mahlites, and the family of the Mushites: these are the families of Merari.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| And those that were numbered of them, according to the number of all the males, from a month old and upward, were six thousand and two hundred.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| And the chief of the house of the father of the families of Merari was Zuriel the son of Abihail: these shall pitch on the side of the tabernacle northward.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And under the custody and charge of the sons of Merari shall be the boards of the tabernacle, and the bars thereof, and the pillars thereof, and the sockets thereof, and all the vessels thereof, and all that serveth thereto,}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| And the pillars of the court round about, and their sockets, and their pins, and their cords.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| But those that encamp before the tabernacle toward the east, even before the tabernacle of the congregation eastward, shall be Moses, and Aaron and his sons, keeping the charge of the sanctuary for the charge of the children of Israel; and the stranger that cometh nigh shall be put to death.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| All that were numbered of the Levites, which Moses and Aaron numbered at the commandment of the LORD, throughout their families, all the males from a month old and upward, were twenty and two thousand.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| And the LORD said unto Moses, Number all the firstborn of the males of the children of Israel from a month old and upward, and take the number of their names.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| And thou shalt take the Levites for me (I am the LORD) instead of all the firstborn among the children of Israel; and the cattle of the Levites instead of all the firstlings among the cattle of the children of Israel.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| And Moses numbered, as the LORD commanded him, all the firstborn among the children of Israel.}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| And all the firstborn males by the number of names, from a month old and upward, of those that were numbered of them, were twenty and two thousand two hundred and threescore and thirteen.}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying, }} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| Take the Levites instead of all the firstborn among the children of Israel, and the cattle of the Levites instead of their cattle; and the Levites shall be mine: I am the LORD.}} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| And for those that are to be redeemed of the two hundred and threescore and thirteen of the firstborn of the children of Israel, which are more than the Levites;}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| Thou shalt even take five shekels apiece by the poll, after the shekel of the sanctuary shalt thou take them: (the shekel is twenty gerahs:)}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| And thou shalt give the money, wherewith the odd number of them is to be redeemed, unto Aaron and to his sons.}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| And Moses took the redemption money of them that were over and above them that were redeemed by the Levites:}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| Of the firstborn of the children of Israel took he the money; a thousand three hundred and threescore and five shekels, after the shekel of the sanctuary:}} <small>{{font|color=#0000FF|51}}</small>{{font|color=#888800| And Moses gave the money of them that were redeemed unto Aaron and to his sons, according to the word of the LORD, as the LORD commanded Moses.}} ==Chapter 4== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying, }} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Take the sum of the sons of Kohath from among the sons of Levi, after their families, by the house of their fathers,}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| From thirty years old and upward even until fifty years old, all that enter into the host, to do the work in the tabernacle of the congregation.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| This shall be the service of the sons of Kohath in the tabernacle of the congregation, about the most holy things:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And when the camp setteth forward, Aaron shall come, and his sons, and they shall take down the covering vail, and cover the ark of testimony with it:}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And shall put thereon the covering of badgers' skins, and shall spread over it a cloth wholly of blue, and shall put in the staves thereof.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And upon the table of shewbread they shall spread a cloth of blue, and put thereon the dishes, and the spoons, and the bowls, and covers to cover withal: and the continual bread shall be thereon:}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And they shall spread upon them a cloth of scarlet, and cover the same with a covering of badgers' skins, and shall put in the staves thereof.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And they shall take a cloth of blue, and cover the candlestick of the light, and his lamps, and his tongs, and his snuffdishes, and all the oil vessels thereof, wherewith they minister unto it:}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And they shall put it and all the vessels thereof within a covering of badgers' skins, and shall put it upon a bar.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And upon the golden altar they shall spread a cloth of blue, and cover it with a covering of badgers' skins, and shall put to the staves thereof:}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And they shall take all the instruments of ministry, wherewith they minister in the sanctuary, and put them in a cloth of blue, and cover them with a covering of badgers' skins, and shall put them on a bar:}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And they shall take away the ashes from the altar, and spread a purple cloth thereon:}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And they shall put upon it all the vessels thereof, wherewith they minister about it, even the censers, the fleshhooks, and the shovels, and the basons, all the vessels of the altar; and they shall spread upon it a covering of badgers' skins, and put to the staves of it.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And when Aaron and his sons have made an end of covering the sanctuary, and all the vessels of the sanctuary, as the camp is to set forward; after that, the sons of Kohath shall come to bear it: but they shall not touch any holy thing, lest they die. These things are the burden of the sons of Kohath in the tabernacle of the congregation.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And to the office of Eleazar the son of Aaron the priest pertaineth the oil for the light, and the sweet incense, and the daily meat offering, and the anointing oil, and the oversight of all the tabernacle, and of all that therein is, in the sanctuary, and in the vessels thereof.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And the LORD spake unto Moses and unto Aaron saying,}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| Cut ye not off the tribe of the families of the Kohathites from among the Levites:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| But thus do unto them, that they may live, and not die, when they approach unto the most holy things: Aaron and his sons shall go in, and appoint them every one to his service and to his burden:}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| But they shall not go in to see when the holy things are covered, lest they die.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Take also the sum of the sons of Gershon, throughout the houses of their fathers, by their families;}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| From thirty years old and upward until fifty years old shalt thou number them; all that enter in to perform the service, to do the work in the tabernacle of the congregation.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| This is the service of the families of the Gershonites, to serve, and for burdens:}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And they shall bear the curtains of the tabernacle, and the tabernacle of the congregation, his covering, and the covering of the badgers' skins that is above upon it, and the hanging for the door of the tabernacle of the congregation,}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And the hangings of the court, and the hanging for the door of the gate of the court, which is by the tabernacle and by the altar round about, and their cords, and all the instruments of their service, and all that is made for them: so shall they serve.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| At the appointment of Aaron and his sons shall be all the service of the sons of the Gershonites, in all their burdens, and in all their service: and ye shall appoint unto them in charge all their burdens.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| This is the service of the families of the sons of Gershon in the tabernacle of the congregation: and their charge shall be under the hand of Ithamar the son of Aaron the priest.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| As for the sons of Merari, thou shalt number them after their families, by the house of their fathers;}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| From thirty years old and upward even unto fifty years old shalt thou number them, every one that entereth into the service, to do the work of the tabernacle of the congregation.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| And this is the charge of their burden, according to all their service in the tabernacle of the congregation; the boards of the tabernacle, and the bars thereof, and the pillars thereof, and sockets thereof,}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And the pillars of the court round about, and their sockets, and their pins, and their cords, with all their instruments, and with all their service: and by name ye shall reckon the instruments of the charge of their burden.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| This is the service of the families of the sons of Merari, according to all their service, in the tabernacle of the congregation, under the hand of Ithamar the son of Aaron the priest.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| And Moses and Aaron and the chief of the congregation numbered the sons of the Kohathites after their families, and after the house of their fathers,}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| From thirty years old and upward even unto fifty years old, every one that entereth into the service, for the work in the tabernacle of the congregation:}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And those that were numbered of them by their families were two thousand seven hundred and fifty.}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| These were they that were numbered of the families of the Kohathites, all that might do service in the tabernacle of the congregation, which Moses and Aaron did number according to the commandment of the LORD by the hand of Moses.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| And those that were numbered of the sons of Gershon, throughout their families, and by the house of their fathers,}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| From thirty years old and upward even unto fifty years old, every one that entereth into the service, for the work in the tabernacle of the congregation,}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| Even those that were numbered of them, throughout their families, by the house of their fathers, were two thousand and six hundred and thirty.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| These are they that were numbered of the families of the sons of Gershon, of all that might do service in the tabernacle of the congregation, whom Moses and Aaron did number according to the commandment of the LORD.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| And those that were numbered of the families of the sons of Merari, throughout their families, by the house of their fathers,}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| From thirty years old and upward even unto fifty years old, every one that entereth into the service, for the work in the tabernacle of the congregation,}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| Even those that were numbered of them after their families, were three thousand and two hundred.}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| These be those that were numbered of the families of the sons of Merari, whom Moses and Aaron numbered according to the word of the LORD by the hand of Moses.}} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| All those that were numbered of the Levites, whom Moses and Aaron and the chief of Israel numbered, after their families, and after the house of their fathers,}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| From thirty years old and upward even unto fifty years old, every one that came to do the service of the ministry, and the service of the burden in the tabernacle of the congregation.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| Even those that were numbered of them, were eight thousand and five hundred and fourscore,}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| According to the commandment of the LORD they were numbered by the hand of Moses, every one according to his service, and according to his burden: thus were they numbered of him, as the LORD commanded Moses.}} ==Chapter 5== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Command the children of Israel, that they put out of the camp every leper, and every one that hath an issue, and whosoever is defiled by the dead:}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| Both male and female shall ye put out, without the camp shall ye put them; that they defile not their camps, in the midst whereof I dwell.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And the children of Israel did so, and put them out without the camp: as the LORD spake unto Moses, so did the children of Israel.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying, }} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Speak unto the children of Israel, When a man or woman shall commit any sin that men commit, to do a trespass against the LORD, and that person be guilty;}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Then they shall confess their sin which they have done: and he shall recompense his trespass with the principal thereof, and add unto it the fifth part thereof, and give it unto him against whom he hath trespassed.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| But if the man have no kinsman to recompense the trespass unto, let the trespass be recompensed unto the LORD, even to the priest; beside the ram of the atonement, whereby an atonement shall be made for him.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And every offering of all the holy things of the children of Israel, which they bring unto the priest, shall be his.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And every man's hallowed things shall be his: whatsoever any man giveth the priest, it shall be his.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| Speak unto the children of Israel, and say unto them, If any man's wife go aside, and commit a trespass against him,}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And a man lie with her carnally, and it be hid from the eyes of her husband, and be kept close, and she be defiled, and there be no witness against her, neither she be taken with the manner;}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And the spirit of jealousy come upon him, and he be jealous of his wife, and she be defiled: or if the spirit of jealousy come upon him, and he be jealous of his wife, and she be not defiled:}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| Then shall the man bring his wife unto the priest, and he shall bring her offering for her, the tenth part of an ephah of barley meal; he shall pour no oil upon it, nor put frankincense thereon; for it is an offering of jealousy, an offering of memorial, bringing iniquity to remembrance.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And the priest shall bring her near, and set her before the LORD:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And the priest shall take holy water in an earthen vessel; and of the dust that is in the floor of the tabernacle the priest shall take, and put it into the water:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And the priest shall set the woman before the LORD, and uncover the woman's head, and put the offering of memorial in her hands, which is the jealousy offering: and the priest shall have in his hand the bitter water that causeth the curse:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And the priest shall charge her by an oath, and say unto the woman, If no man have lain with thee, and if thou hast not gone aside to uncleanness with another instead of thy husband, be thou free from this bitter water that causeth the curse:}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| But if thou hast gone aside to another instead of thy husband, and if thou be defiled, and some man have lain with thee beside thine husband:}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Then the priest shall charge the woman with an oath of cursing, and the priest shall say unto the woman, The LORD make thee a curse and an oath among thy people, when the LORD doth make thy thigh to rot, and thy belly to swell;}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And this water that causeth the curse shall go into thy bowels, to make thy belly to swell, and thy thigh to rot: And the woman shall say, Amen, amen.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And the priest shall write these curses in a book, and he shall blot them out with the bitter water:}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| And he shall cause the woman to drink the bitter water that causeth the curse: and the water that causeth the curse shall enter into her, and become bitter.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| Then the priest shall take the jealousy offering out of the woman's hand, and shall wave the offering before the LORD, and offer it upon the altar:}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And the priest shall take an handful of the offering, even the memorial thereof, and burn it upon the altar, and afterward shall cause the woman to drink the water.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And when he hath made her to drink the water, then it shall come to pass, that, if she be defiled, and have done trespass against her husband, that the water that causeth the curse shall enter into her, and become bitter, and her belly shall swell, and her thigh shall rot: and the woman shall be a curse among her people.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| And if the woman be not defiled, but be clean; then she shall be free, and shall conceive seed.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| This is the law of jealousies, when a wife goeth aside to another instead of her husband, and is defiled;}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| Or when the spirit of jealousy cometh upon him, and he be jealous over his wife, and shall set the woman before the LORD, and the priest shall execute upon her all this law.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| Then shall the man be guiltless from iniquity, and this woman shall bear her iniquity.}} ==Chapter 6== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Speak unto the children of Israel, and say unto them, When either man or woman shall separate themselves to vow a vow of a Nazarite, to separate themselves unto the LORD:}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| He shall separate himself from wine and strong drink, and shall drink no vinegar of wine, or vinegar of strong drink, neither shall he drink any liquor of grapes, nor eat moist grapes, or dried.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| All the days of his separation shall he eat nothing that is made of the vine tree, from the kernels even to the husk.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| All the days of the vow of his separation there shall no razor come upon his head: until the days be fulfilled, in the which he separateth himself unto the LORD, he shall be holy, and shall let the locks of the hair of his head grow.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| All the days that he separateth himself unto the LORD he shall come at no dead body.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| He shall not make himself unclean for his father, or for his mother, for his brother, or for his sister, when they die: because the consecration of his God is upon his head.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| All the days of his separation he is holy unto the LORD.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And if any man die very suddenly by him, and he hath defiled the head of his consecration; then he shall shave his head in the day of his cleansing, on the seventh day shall he shave it.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And on the eighth day he shall bring two turtles, or two young pigeons, to the priest, to the door of the tabernacle of the congregation:}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And the priest shall offer the one for a sin offering, and the other for a burnt offering, and make an atonement for him, for that he sinned by the dead, and shall hallow his head that same day.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And he shall consecrate unto the LORD the days of his separation, and shall bring a lamb of the first year for a trespass offering: but the days that were before shall be lost, because his separation was defiled.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And this is the law of the Nazarite, when the days of his separation are fulfilled: he shall be brought unto the door of the tabernacle of the congregation:}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And he shall offer his offering unto the LORD, one he lamb of the first year without blemish for a burnt offering, and one ewe lamb of the first year without blemish for a sin offering, and one ram without blemish for peace offerings,}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And a basket of unleavened bread, cakes of fine flour mingled with oil, and wafers of unleavened bread anointed with oil, and their meat offering, and their drink offerings.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And the priest shall bring them before the LORD, and shall offer his sin offering, and his burnt offering:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And he shall offer the ram for a sacrifice of peace offerings unto the LORD, with the basket of unleavened bread: the priest shall offer also his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And the Nazarite shall shave the head of his separation at the door of the tabernacle of the congregation, and shall take the hair of the head of his separation, and put it in the fire which is under the sacrifice of the peace offerings.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And the priest shall take the sodden shoulder of the ram, and one unleavened cake out of the basket, and one unleavened wafer, and shall put them upon the hands of the Nazarite, after the hair of his separation is shaven:}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And the priest shall wave them for a wave offering before the LORD: this is holy for the priest, with the wave breast and heave shoulder: and after that the Nazarite may drink wine.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| This is the law of the Nazarite who hath vowed, and of his offering unto the LORD for his separation, beside that that his hand shall get: according to the vow which he vowed, so he must do after the law of his separation.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| Speak unto Aaron and unto his sons, saying, On this wise ye shall bless the children of Israel, saying unto them,}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| The LORD bless thee, and keep thee:}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| The LORD make his face shine upon thee, and be gracious unto thee:}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| The LORD lift up his countenance upon thee, and give thee peace.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And they shall put my name upon the children of Israel, and I will bless them.}} ==Chapter 7== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And it came to pass on the day that Moses had fully set up the tabernacle, and had anointed it, and sanctified it, and all the instruments thereof, both the altar and all the vessels thereof, and had anointed them, and sanctified them;}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| That the princes of Israel, heads of the house of their fathers, who were the princes of the tribes, and were over them that were numbered, offered:}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And they brought their offering before the LORD, six covered wagons, and twelve oxen; a wagon for two of the princes, and for each one an ox: and they brought them before the tabernacle.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| Take it of them, that they may be to do the service of the tabernacle of the congregation; and thou shalt give them unto the Levites, to every man according to his service.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And Moses took the wagons and the oxen, and gave them unto the Levites.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Two wagons and four oxen he gave unto the sons of Gershon, according to their service:}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And four wagons and eight oxen he gave unto the sons of Merari, according unto their service, under the hand of Ithamar the son of Aaron the priest.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| But unto the sons of Kohath he gave none: because the service of the sanctuary belonging unto them was that they should bear upon their shoulders.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And the princes offered for dedicating of the altar in the day that it was anointed, even the princes offered their offering before the altar.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And the LORD said unto Moses, They shall offer their offering, each prince on his day, for the dedicating of the altar.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And he that offered his offering the first day was Nahshon the son of Amminadab, of the tribe of Judah:}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And his offering was one silver charger, the weight thereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them were full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| One spoon of ten shekels of gold, full of incense:}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Nahshon the son of Amminadab.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| On the second day Nethaneel the son of Zuar, prince of Issachar, did offer:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| He offered for his offering one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| One spoon of gold of ten shekels, full of incense:}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Nethaneel the son of Zuar.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| On the third day Eliab the son of Helon, prince of the children of Zebulun, did offer:}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Eliab the son of Helon.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| On the fourth day Elizur the son of Shedeur, prince of the children of Reuben, did offer:}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| His offering was one silver charger of the weight of an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Elizur the son of Shedeur.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| On the fifth day Shelumiel the son of Zurishaddai, prince of the children of Simeon, did offer:}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Shelumiel the son of Zurishaddai.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| On the sixth day Eliasaph the son of Deuel, prince of the children of Gad, offered:}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| His offering was one silver charger of the weight of an hundred and thirty shekels, a silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Eliasaph the son of Deuel.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| On the seventh day Elishama the son of Ammihud, prince of the children of Ephraim, offered:}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|51}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|52}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|53}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Elishama the son of Ammihud.}} <small>{{font|color=#0000FF|54}}</small>{{font|color=#888800| On the eighth day offered Gamaliel the son of Pedahzur, prince of the children of Manasseh:}} <small>{{font|color=#0000FF|55}}</small>{{font|color=#888800| His offering was one silver charger of the weight of an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|56}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|57}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|58}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|59}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Gamaliel the son of Pedahzur.}} <small>{{font|color=#0000FF|60}}</small>{{font|color=#888800| On the ninth day Abidan the son of Gideoni, prince of the children of Benjamin, offered:}} <small>{{font|color=#0000FF|61}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|62}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|63}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|64}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|65}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Abidan the son of Gideoni.}} <small>{{font|color=#0000FF|66}}</small>{{font|color=#888800| On the tenth day Ahiezer the son of Ammishaddai, prince of the children of Dan, offered:}} <small>{{font|color=#0000FF|67}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|68}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|69}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|70}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|71}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Ahiezer the son of Ammishaddai.}} <small>{{font|color=#0000FF|72}}</small>{{font|color=#888800| On the eleventh day Pagiel the son of Ocran, prince of the children of Asher, offered:}} <small>{{font|color=#0000FF|73}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|74}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|75}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|76}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|77}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Pagiel the son of Ocran.}} <small>{{font|color=#0000FF|78}}</small>{{font|color=#888800| On the twelfth day Ahira the son of Enan, prince of the children of Naphtali, offered:}} <small>{{font|color=#0000FF|79}}</small>{{font|color=#888800| His offering was one silver charger, the weight whereof was an hundred and thirty shekels, one silver bowl of seventy shekels, after the shekel of the sanctuary; both of them full of fine flour mingled with oil for a meat offering:}} <small>{{font|color=#0000FF|80}}</small>{{font|color=#888800| One golden spoon of ten shekels, full of incense:}} <small>{{font|color=#0000FF|81}}</small>{{font|color=#888800| One young bullock, one ram, one lamb of the first year, for a burnt offering:}} <small>{{font|color=#0000FF|82}}</small>{{font|color=#888800| One kid of the goats for a sin offering:}} <small>{{font|color=#0000FF|83}}</small>{{font|color=#888800| And for a sacrifice of peace offerings, two oxen, five rams, five he goats, five lambs of the first year: this was the offering of Ahira the son of Enan.}} <small>{{font|color=#0000FF|84}}</small>{{font|color=#888800| This was the dedication of the altar, in the day when it was anointed, by the princes of Israel: twelve chargers of silver, twelve silver bowls, twelve spoons of gold:}} <small>{{font|color=#0000FF|85}}</small>{{font|color=#888800| Each charger of silver weighing an hundred and thirty shekels, each bowl seventy: all the silver vessels weighed two thousand and four hundred shekels, after the shekel of the sanctuary:}} <small>{{font|color=#0000FF|86}}</small>{{font|color=#888800| The golden spoons were twelve, full of incense, weighing ten shekels apiece, after the shekel of the sanctuary: all the gold of the spoons was an hundred and twenty shekels.}} <small>{{font|color=#0000FF|87}}</small>{{font|color=#888800| All the oxen for the burnt offering were twelve bullocks, the rams twelve, the lambs of the first year twelve, with their meat offering: and the kids of the goats for sin offering twelve.}} <small>{{font|color=#0000FF|88}}</small>{{font|color=#888800| And all the oxen for the sacrifice of the peace offerings were twenty and four bullocks, the rams sixty, the he goats sixty, the lambs of the first year sixty. This was the dedication of the altar, after that it was anointed.}} <small>{{font|color=#0000FF|89}}</small>{{font|color=#888800| And when Moses was gone into the tabernacle of the congregation to speak with him, then he heard the voice of one speaking unto him from off the mercy seat that was upon the ark of testimony, from between the two cherubims: and he spake unto him.}} ==Chapter 8== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Speak unto Aaron and say unto him, When thou lightest the lamps, the seven lamps shall give light over against the candlestick.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And Aaron did so; he lighted the lamps thereof over against the candlestick, as the LORD commanded Moses.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And this work of the candlestick was of beaten gold, unto the shaft thereof, unto the flowers thereof, was beaten work: according unto the pattern which the LORD had shewed Moses, so he made the candlestick.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Take the Levites from among the children of Israel, and cleanse them.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And thus shalt thou do unto them, to cleanse them: Sprinkle water of purifying upon them, and let them shave all their flesh, and let them wash their clothes, and so make themselves clean.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| Then let them take a young bullock with his meat offering, even fine flour mingled with oil, and another young bullock shalt thou take for a sin offering.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And thou shalt bring the Levites before the tabernacle of the congregation: and thou shalt gather the whole assembly of the children of Israel together:}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And thou shalt bring the Levites before the LORD: and the children of Israel shall put their hands upon the Levites:}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And Aaron shall offer the Levites before the LORD for an offering of the children of Israel, that they may execute the service of the LORD.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And the Levites shall lay their hands upon the heads of the bullocks: and thou shalt offer the one for a sin offering, and the other for a burnt offering, unto the LORD, to make an atonement for the Levites.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And thou shalt set the Levites before Aaron, and before his sons, and offer them for an offering unto the LORD.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Thus shalt thou separate the Levites from among the children of Israel: and the Levites shall be mine.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And after that shall the Levites go in to do the service of the tabernacle of the congregation: and thou shalt cleanse them, and offer them for an offering.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| For they are wholly given unto me from among the children of Israel; instead of such as open every womb, even instead of the firstborn of all the children of Israel, have I taken them unto me.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| For all the firstborn of the children of Israel are mine, both man and beast: on the day that I smote every firstborn in the land of Egypt I sanctified them for myself.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And I have taken the Levites for all the firstborn of the children of Israel.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And I have given the Levites as a gift to Aaron and to his sons from among the children of Israel, to do the service of the children of Israel in the tabernacle of the congregation, and to make an atonement for the children of Israel: that there be no plague among the children of Israel, when the children of Israel come nigh unto the sanctuary.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And Moses, and Aaron, and all the congregation of the children of Israel, did to the Levites according unto all that the LORD commanded Moses concerning the Levites, so did the children of Israel unto them.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And the Levites were purified, and they washed their clothes; and Aaron offered them as an offering before the LORD; and Aaron made an atonement for them to cleanse them.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And after that went the Levites in to do their service in the tabernacle of the congregation before Aaron, and before his sons: as the LORD had commanded Moses concerning the Levites, so did they unto them.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| This is it that belongeth unto the Levites: from twenty and five years old and upward they shall go in to wait upon the service of the tabernacle of the congregation:}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And from the age of fifty years they shall cease waiting upon the service thereof, and shall serve no more:}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| But shall minister with their brethren in the tabernacle of the congregation, to keep the charge, and shall do no service. Thus shalt thou do unto the Levites touching their charge.}} ==Chapter 9== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses in the wilderness of Sinai, in the first month of the second year after they were come out of the land of Egypt, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Let the children of Israel also keep the passover at his appointed season.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| In the fourteenth day of this month, at even, ye shall keep it in his appointed season: according to all the rites of it, and according to all the ceremonies thereof, shall ye keep it.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And Moses spake unto the children of Israel, that they should keep the passover.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And they kept the passover on the fourteenth day of the first month at even in the wilderness of Sinai: according to all that the LORD commanded Moses, so did the children of Israel.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And there were certain men, who were defiled by the dead body of a man, that they could not keep the passover on that day: and they came before Moses and before Aaron on that day:}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And those men said unto him, We are defiled by the dead body of a man: wherefore are we kept back, that we may not offer an offering of the LORD in his appointed season among the children of Israel?}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And Moses said unto them, Stand still, and I will hear what the LORD will command concerning you.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| Speak unto the children of Israel, saying, If any man of you or of your posterity shall be unclean by reason of a dead body, or be in a journey afar off, yet he shall keep the passover unto the LORD.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| The fourteenth day of the second month at even they shall keep it, and eat it with unleavened bread and bitter herbs.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| They shall leave none of it unto the morning, nor break any bone of it: according to all the ordinances of the passover they shall keep it.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| But the man that is clean, and is not in a journey, and forbeareth to keep the passover, even the same soul shall be cut off from among his people: because he brought not the offering of the LORD in his appointed season, that man shall bear his sin.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And if a stranger shall sojourn among you, and will keep the passover unto the LORD; according to the ordinance of the passover, and according to the manner thereof, so shall he do: ye shall have one ordinance, both for the stranger, and for him that was born in the land.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And on the day that the tabernacle was reared up the cloud covered the tabernacle, namely, the tent of the testimony: and at even there was upon the tabernacle as it were the appearance of fire, until the morning.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| So it was alway: the cloud covered it by day, and the appearance of fire by night.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And when the cloud was taken up from the tabernacle, then after that the children of Israel journeyed: and in the place where the cloud abode, there the children of Israel pitched their tents.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| At the commandment of the LORD the children of Israel journeyed, and at the commandment of the LORD they pitched: as long as the cloud abode upon the tabernacle they rested in their tents.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And when the cloud tarried long upon the tabernacle many days, then the children of Israel kept the charge of the LORD, and journeyed not.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And so it was, when the cloud was a few days upon the tabernacle; according to the commandment of the LORD they abode in their tents, and according to the commandment of the LORD they journeyed.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And so it was, when the cloud abode from even unto the morning, and that the cloud was taken up in the morning, then they journeyed: whether it was by day or by night that the cloud was taken up, they journeyed.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Or whether it were two days, or a month, or a year, that the cloud tarried upon the tabernacle, remaining thereon, the children of Israel abode in their tents, and journeyed not: but when it was taken up, they journeyed.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| At the commandment of the LORD they rested in the tents, and at the commandment of the LORD they journeyed: they kept the charge of the LORD, at the commandment of the LORD by the hand of Moses.}} ==Chapter 10== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Make thee two trumpets of silver; of a whole piece shalt thou make them: that thou mayest use them for the calling of the assembly, and for the journeying of the camps.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And when they shall blow with them, all the assembly shall assemble themselves to thee at the door of the tabernacle of the congregation.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And if they blow but with one trumpet, then the princes, which are heads of the thousands of Israel, shall gather themselves unto thee.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| When ye blow an alarm, then the camps that lie on the east parts shall go forward.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| When ye blow an alarm the second time, then the camps that lie on the south side shall take their journey: they shall blow an alarm for their journeys.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| But when the congregation is to be gathered together, ye shall blow, but ye shall not sound an alarm.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And the sons of Aaron, the priests, shall blow with the trumpets; and they shall be to you for an ordinance for ever throughout your generations.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And if ye go to war in your land against the enemy that oppresseth you, then ye shall blow an alarm with the trumpets; and ye shall be remembered before the LORD your God, and ye shall be saved from your enemies.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| Also in the day of your gladness, and in your solemn days, and in the beginnings of your months, ye shall blow with the trumpets over your burnt offerings, and over the sacrifices of your peace offerings; that they may be to you for a memorial before your God: I am the LORD your God.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And it came to pass on the twentieth day of the second month, in the second year, that the cloud was taken up from off the tabernacle of the testimony.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And the children of Israel took their journeys out of the wilderness of Sinai; and the cloud rested in the wilderness of Paran.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And they first took their journey according to the commandment of the LORD by the hand of Moses.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| In the first place went the standard of the camp of the children of Judah according to their armies: and over his host was Nahshon the son of Amminadab.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And over the host of the tribe of the children of Issachar was Nethaneel the son of Zuar.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And over the host of the tribe of the children of Zebulun was Eliab the son of Helon.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And the tabernacle was taken down; and the sons of Gershon and the sons of Merari set forward, bearing the tabernacle.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And the standard of the camp of Reuben set forward according to their armies: and over his host was Elizur the son of Shedeur.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And over the host of the tribe of the children of Simeon was Shelumiel the son of Zurishaddai.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And over the host of the tribe of the children of Gad was Eliasaph the son of Deuel.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And the Kohathites set forward, bearing the sanctuary: and the other did set up the tabernacle against they came.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And the standard of the camp of the children of Ephraim set forward according to their armies: and over his host was Elishama the son of Ammihud.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And over the host of the tribe of the children of Manasseh was Gamaliel the son of Pedahzur.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| And over the host of the tribe of the children of Benjamin was Abidan the son of Gideoni.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the standard of the camp of the children of Dan set forward, which was the rereward of all the camps throughout their hosts: and over his host was Ahiezer the son of Ammishaddai.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And over the host of the tribe of the children of Asher was Pagiel the son of Ocran.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And over the host of the tribe of the children of Naphtali was Ahira the son of Enan.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| Thus were the journeyings of the children of Israel according to their armies, when they set forward.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#000088| And Moses said unto Hobab, the son of Raguel the Midianite, Moses' father in law, We are journeying unto the place of which the LORD said, I will give it you: come thou with us, and we will do thee good: for the LORD hath spoken good concerning Israel.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#000088| And he said unto him, I will not go; but I will depart to mine own land, and to my kindred.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#000088| And he said, Leave us not, I pray thee; forasmuch as thou knowest how we are to encamp in the wilderness, and thou mayest be to us instead of eyes.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#000088| And it shall be, if thou go with us, yea, it shall be, that what goodness the LORD shall do unto us, the same will we do unto thee.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#000088| And they departed from the mount of the LORD three days' journey: and the ark of the covenant of the LORD went before them in the three days' journey, to search out a resting place for them.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#000088| And the cloud of the LORD was upon them by day, when they went out of the camp.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#000088| And it came to pass, when the ark set forward, that Moses said, Rise up, LORD, and let thine enemies be scattered; and let them that hate thee flee before thee.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#000088| And when it rested, he said, Return, O LORD, unto the many thousands of Israel.}} ==Chapter 11== <small>{{font|color=#0000FF|1}}</small>{{font|color=#000088| And when the people complained, it displeased the LORD: and the LORD heard it; and his anger was kindled; and the fire of the LORD burnt among them, and consumed them that were in the uttermost parts of the camp.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#000088| And the people cried unto Moses; and when Moses prayed unto the LORD, the fire was quenched.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#000088| And he called the name of the place Taberah: because the fire of the LORD burnt among them.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#000088| And the mixt multitude that was among them fell a lusting: and the children of Israel also wept again, and said, Who shall give us flesh to eat?}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#000088| We remember the fish, which we did eat in Egypt freely; the cucumbers, and the melons, and the leeks, and the onions, and the garlick:}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#000088| But now our soul is dried away: there is nothing at all, beside this manna, before our eyes.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#880000| And the manna was as coriander seed, and the colour thereof as the colour of bdellium.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#880000| And the people went about, and gathered it, and ground it in mills, or beat it in a mortar, and baked it in pans, and made cakes of it: and the taste of it was as the taste of fresh oil.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#880000| And when the dew fell upon the camp in the night, the manna fell upon it.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#000088| Then Moses heard the people weep throughout their families, every man in the door of his tent: and the anger of the LORD was kindled greatly; Moses also was displeased.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#008888| And Moses said unto the LORD, Wherefore hast thou afflicted thy servant? and wherefore have I not found favour in thy sight, that thou layest the burden of all this people upon me?}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#008888| Have I conceived all this people? have I begotten them, that thou shouldest say unto me, Carry them in thy bosom, as a nursing father beareth the sucking child, unto the land which thou swarest unto their fathers?}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#000088| Whence should I have flesh to give unto all this people? for they weep unto me, saying, Give us flesh, that we may eat.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#008888| I am not able to bear all this people alone, because it is too heavy for me.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#008888| And if thou deal thus with me, kill me, I pray thee, out of hand, if I have found favour in thy sight; and let me not see my wretchedness.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#008888| And the LORD said unto Moses, Gather unto me seventy men of the elders of Israel, whom thou knowest to be the elders of the people, and officers over them; and bring them unto the tabernacle of the congregation, that they may stand there with thee.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#008888| And I will come down and talk with thee there: and I will take of the spirit which is upon thee, and will put it upon them; and they shall bear the burden of the people with thee, that thou bear it not thyself alone.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#000088| And say thou unto the people, Sanctify yourselves against to morrow, and ye shall eat flesh: for ye have wept in the ears of the LORD, saying, Who shall give us flesh to eat? for it was well with us in Egypt: therefore the LORD will give you flesh, and ye shall eat.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#000088| Ye shall not eat one day, nor two days, nor five days, neither ten days, nor twenty days;}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#000088| But even a whole month, until it come out at your nostrils, and it be loathsome unto you: because that ye have despised the LORD which is among you, and have wept before him, saying, Why came we forth out of Egypt?}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#000088| And Moses said, The people, among whom I am, are six hundred thousand footmen; and thou hast said, I will give them flesh, that they may eat a whole month.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#000088| Shall the flocks and the herds be slain for them, to suffice them? or shall all the fish of the sea be gathered together for them, to suffice them?}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#000088| And the LORD said unto Moses, Is the LORD's hand waxed short? thou shalt see now whether my word shall come to pass unto thee or not.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#000088| And Moses went out, and told the people the words of the LORD, }} {{font|color=#880000|and}} {{font|color=#008888|gathered the seventy men of the elders of the people, and set them round about the tabernacle.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#008888| And the LORD came down in a cloud, and spake unto him, and took of the spirit that was upon him, and gave it unto the seventy elders: and it came to pass, that, when the spirit rested upon them, they prophesied, and did not cease.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#008888| But there remained two of the men in the camp, the name of the one was Eldad, and the name of the other Medad: and the spirit rested upon them; and they were of them that were written, but went not out unto the tabernacle: and they prophesied in the camp.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#008888| And there ran a young man, and told Moses, and said, Eldad and Medad do prophesy in the camp.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#008888| And Joshua the son of Nun, the servant of Moses, one of his young men, answered and said, My lord Moses, forbid them.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#008888| And Moses said unto him, Enviest thou for my sake? would God that all the LORD's people were prophets, and that the LORD would put his spirit upon them!}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#008888| And Moses gat him into the camp, he and the elders of Israel.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#000088| And there went forth a wind from the LORD, and brought quails from the sea, and let them fall by the camp, as it were a day's journey on this side, and as it were a day's journey on the other side, round about the camp, and as it were two cubits high upon the face of the earth.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#000088| And the people stood up all that day, and all that night, and all the next day, and they gathered the quails: he that gathered least gathered ten homers: and they spread them all abroad for themselves round about the camp.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#000088| And while the flesh was yet between their teeth, ere it was chewed, the wrath of the LORD was kindled against the people, and the LORD smote the people with a very great plague.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#000088| And he called the name of that place Kibrothhattaavah: because there they buried the people that lusted.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#880000| And the people journeyed from Kibrothhattaavah unto Hazeroth; and abode at Hazeroth.}} ==Chapter 12== <small>{{font|color=#0000FF|1}}</small>{{font|color=#008888| And Miriam and Aaron spake against Moses}} {{font|color=#880000| because of the Ethiopian woman whom he had married: for he had married an Ethiopian woman.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#008888| And they said, Hath the LORD indeed spoken only by Moses? hath he not spoken also by us? And the LORD heard it.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#008888| (Now the man Moses was very meek, above all the men which were upon the face of the earth.)}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#008888| And the LORD spake suddenly unto Moses, and unto Aaron, and unto Miriam, Come out ye three unto the tabernacle of the congregation. And they three came out.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#008888| And the LORD came down in the pillar of the cloud, and stood in the door of the tabernacle, and called Aaron and Miriam: and they both came forth.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#008888| And he said, Hear now my words: If there be a prophet among you, I the LORD will make myself known unto him in a vision, and will speak unto him in a dream.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#008888| My servant Moses is not so, who is faithful in all mine house. }} <small>{{font|color=#0000FF|8}}</small>{{font|color=#008888| With him will I speak mouth to mouth, even apparently, and not in dark speeches; and the similitude of the LORD shall he behold: wherefore then were ye not afraid to speak against my servant Moses?}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#008888| And the anger of the LORD was kindled against them; and he departed.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#008888| And the cloud departed from off the tabernacle; and, behold, Miriam became leprous, white as snow: and Aaron looked upon Miriam, and, behold, she was leprous.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#008888| And Aaron said unto Moses, Alas, my lord, I beseech thee, lay not the sin upon us, wherein we have done foolishly, and wherein we have sinned.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#008888| Let her not be as one dead, of whom the flesh is half consumed when he cometh out of his mother's womb.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#008888| And Moses cried unto the LORD, saying, Heal her now, O God, I beseech thee.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#008888| And the LORD said unto Moses, If her father had but spit in her face, should she not be ashamed seven days? let her be shut out from the camp seven days, and after that let her be received in again.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#008888| And Miriam was shut out from the camp seven days: and the people journeyed not till Miriam was brought in again.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#008888| And afterward the people removed from Hazeroth, and pitched in the wilderness of Paran.}} ==Chapter 13== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Send thou men, that they may search the land of Canaan, which I give unto the children of Israel: of every tribe of their fathers shall ye send a man, every one a ruler among them.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And Moses by the commandment of the LORD sent them from the wilderness of Paran: all those men were heads of the children of Israel.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And these were their names: of the tribe of Reuben, Shammua the son of Zaccur.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| Of the tribe of Simeon, Shaphat the son of Hori.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Of the tribe of Judah, Caleb the son of Jephunneh.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Of the tribe of Issachar, Igal the son of Joseph.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| Of the tribe of Ephraim, Oshea the son of Nun.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| Of the tribe of Benjamin, Palti the son of Raphu.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| Of the tribe of Zebulun, Gaddiel the son of Sodi.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| Of the tribe of Joseph, namely, of the tribe of Manasseh, Gaddi the son of Susi.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| Of the tribe of Dan, Ammiel the son of Gemalli.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Of the tribe of Asher, Sethur the son of Michael.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Of the tribe of Naphtali, Nahbi the son of Vophsi.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| Of the tribe of Gad, Geuel the son of Machi.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| These are the names of the men which Moses sent to spy out the land. And Moses called Oshea the son of Nun Jehoshua.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And Moses sent them to spy out the land of Canaan, }} {{font|color=#000088|and said unto them, Get you up this way southward, and go up into the mountain:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#000088|And see the land, what it is, and the people that dwelleth therein, whether they be strong or weak, few or many;}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#000088| And what the land is that they dwell in, whether it be good or bad; and what cities they be that they dwell in, whether in tents, or in strong holds;}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#000088| And what the land is, whether it be fat or lean, whether there be wood therein, or not. And be ye of good courage, and bring of the fruit of the land. Now the time was the time of the firstripe grapes.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800|So they went up, and searched the land from the wilderness of Zin unto Rehob, as men come to Hamath.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#000088| And they ascended by the south, and came unto Hebron; where Ahiman, Sheshai, and Talmai, the children of Anak, were. (Now Hebron was built seven years before Zoan in Egypt.)}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#000088| And they came unto the brook of Eshcol, and cut down from thence a branch with one cluster of grapes, and they bare it between two upon a staff; and they brought of the pomegranates, and of the figs.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#000088| The place was called the brook Eshcol, because of the cluster of grapes which the children of Israel cut down from thence.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And they returned from searching of the land after forty days.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And they went and came to Moses, and to Aaron, and to all the congregation of the children of Israel, unto the wilderness of Paran,}} {{font|color=#000088|to Kadesh; and brought back word unto them, and unto all the congregation, and shewed them the fruit of the land.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#000088| And they told him, and said, We came unto the land whither thou sentest us, and surely it floweth with milk and honey; and this is the fruit of it.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#000088| Nevertheless the people be strong that dwell in the land, and the cities are walled, and very great: and moreover we saw the children of Anak there.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#000088| The Amalekites dwell in the land of the south: and the Hittites, and the Jebusites, and the Amorites, dwell in the mountains: and the Canaanites dwell by the sea, and by the coast of Jordan.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#000088| And Caleb stilled the people before Moses, and said, Let us go up at once, and possess it; for we are well able to overcome it.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#000088| But the men that went up with him said, We be not able to go up against the people; for they are stronger than we.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And they brought up an evil report of the land which they had searched unto the children of Israel, saying, The land, through which we have gone to search it, is a land that eateth up the inhabitants thereof; and all the people that we saw in it are men of a great stature.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#000088| And there we saw the giants, the sons of Anak, which come of the giants: and we were in our own sight as grasshoppers, and so we were in their sight.}} ==Chapter 14== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And all the congregation lifted up their voice, and cried;}} {{font|color=#000088| and the people wept that night.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And all the children of Israel murmured against Moses and against Aaron: and the whole congregation said unto them, Would God that we had died in the land of Egypt! or would God we had died in this wilderness!}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And wherefore hath the LORD brought us unto this land, to fall by the sword, that our wives and our children should be a prey? were it not better for us to return into Egypt?}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And they said one to another, Let us make a captain, and let us return into Egypt.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| Then Moses and Aaron fell on their faces before all the assembly of the congregation of the children of Israel.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And Joshua the son of Nun, and Caleb the son of Jephunneh, which were of them that searched the land, rent their clothes:}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800|And they spake unto all the company of the children of Israel, saying, The land, which we passed through to search it, is an exceeding good land.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| If the LORD delight in us, then he will bring us into this land, and give it us; a land which floweth with milk and honey.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| Only rebel not ye against the LORD, neither fear ye the people of the land; for they are bread for us: their defence is departed from them, and the LORD is with us: fear them not.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| But all the congregation bade stone them with stones. And the glory of the LORD appeared in the tabernacle of the congregation before all the children of Israel.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#000088| And the LORD said unto Moses, How long will this people provoke me? and how long will it be ere they believe me, for all the signs which I have shewed among them?}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#000088| I will smite them with the pestilence, and disinherit them, and will make of thee a greater nation and mightier than they.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#000088| And Moses said unto the LORD, Then the Egyptians shall hear it, (for thou broughtest up this people in thy might from among them;)}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#000088| And they will tell it to the inhabitants of this land: for they have heard that thou LORD art among this people, that thou LORD art seen face to face, and that thy cloud standeth over them, and that thou goest before them, by day time in a pillar of a cloud, and in a pillar of fire by night.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#000088| Now if thou shalt kill all this people as one man, then the nations which have heard the fame of thee will speak, saying,}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#000088| Because the LORD was not able to bring this people into the land which he sware unto them, therefore he hath slain them in the wilderness.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#000088| And now, I beseech thee, let the power of my Lord be great, according as thou hast spoken, saying,}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#000088| The LORD is longsuffering, and of great mercy, forgiving iniquity and transgression, and by no means clearing the guilty, visiting the iniquity of the fathers upon the children unto the third and fourth generation.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#000088| Pardon, I beseech thee, the iniquity of this people according unto the greatness of thy mercy, and as thou hast forgiven this people, from Egypt even until now.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#000088| And the LORD said, I have pardoned according to thy word: }} <small>{{font|color=#0000FF|21}}</small>{{font|color=#000088| But as truly as I live, all the earth shall be filled with the glory of the LORD.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#000088| Because all those men which have seen my glory, and my miracles, which I did in Egypt and in the wilderness, and have tempted me now these ten times, and have not hearkened to my voice;}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#000088| Surely they shall not see the land which I sware unto their fathers, neither shall any of them that provoked me see it:}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#000088| But my servant Caleb, because he had another spirit with him, and hath followed me fully, him will I bring into the land whereinto he went; and his seed shall possess it.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#000088| (Now the Amalekites and the Canaanites dwelt in the valley.) Tomorrow turn you, and get you into the wilderness by the way of the Red sea.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying, }} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| How long shall I bear with this evil congregation, which murmur against me? I have heard the murmurings of the children of Israel, which they murmur against me.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| Say unto them, As truly as I live, saith the LORD, as ye have spoken in mine ears, so will I do to you:}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Your carcases shall fall in this wilderness; and all that were numbered of you, according to your whole number, from twenty years old and upward which have murmured against me.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| Doubtless ye shall not come into the land, concerning which I sware to make you dwell therein, save Caleb the son of Jephunneh, and Joshua the son of Nun.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| But your little ones, which ye said should be a prey, them will I bring in, and they shall know the land which ye have despised.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| But as for you, your carcases, they shall fall in this wilderness.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| And your children shall wander in the wilderness forty years, and bear your whoredoms, until your carcases be wasted in the wilderness.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| After the number of the days in which ye searched the land, even forty days, each day for a year, shall ye bear your iniquities, even forty years, and ye shall know my breach of promise.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| I the LORD have said, I will surely do it unto all this evil congregation, that are gathered together against me: in this wilderness they shall be consumed, and there they shall die.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And the men, which Moses sent to search the land, who returned, and made all the congregation to murmur against him, by bringing up a slander upon the land,}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| Even those men that did bring up the evil report upon the land, died by the plague before the LORD.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| But Joshua the son of Nun, and Caleb the son of Jephunneh, which were of the men that went to search the land, lived still.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#000088| And Moses told these sayings unto all the children of Israel: and the people mourned greatly.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#000088| And they rose up early in the morning, and gat them up into the top of the mountain, saying, Lo, we be here, and will go up unto the place which the LORD hath promised: for we have sinned.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#000088| And Moses said, Wherefore now do ye transgress the commandment of the LORD? but it shall not prosper.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#000088| Go not up, for the LORD is not among you; that ye be not smitten before your enemies.}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#000088| For the Amalekites and the Canaanites are there before you, and ye shall fall by the sword: because ye are turned away from the LORD, therefore the LORD will not be with you.}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#000088| But they presumed to go up unto the hill top: nevertheless the ark of the covenant of the LORD, and Moses, departed not out of the camp.}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#000088| Then the Amalekites came down, and the Canaanites which dwelt in that hill, and smote them, and discomfited them, even unto Hormah.}} ==Chapter 15== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Speak unto the children of Israel, and say unto them, When ye be come into the land of your habitations, which I give unto you,}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And will make an offering by fire unto the LORD, a burnt offering, or a sacrifice in performing a vow, or in a freewill offering, or in your solemn feasts, to make a sweet savour unto the LORD, of the herd or of the flock:}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| Then shall he that offereth his offering unto the LORD bring a meat offering of a tenth deal of flour mingled with the fourth part of an hin of oil.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And the fourth part of an hin of wine for a drink offering shalt thou prepare with the burnt offering or sacrifice, for one lamb.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Or for a ram, thou shalt prepare for a meat offering two tenth deals of flour mingled with the third part of an hin of oil.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And for a drink offering thou shalt offer the third part of an hin of wine, for a sweet savour unto the LORD.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And when thou preparest a bullock for a burnt offering, or for a sacrifice in performing a vow, or peace offerings unto the LORD:}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| Then shall he bring with a bullock a meat offering of three tenth deals of flour mingled with half an hin of oil.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And thou shalt bring for a drink offering half an hin of wine, for an offering made by fire, of a sweet savour unto the LORD.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| Thus shall it be done for one bullock, or for one ram, or for a lamb, or a kid.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| According to the number that ye shall prepare, so shall ye do to every one according to their number.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| All that are born of the country shall do these things after this manner, in offering an offering made by fire, of a sweet savour unto the LORD.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And if a stranger sojourn with you, or whosoever be among you in your generations, and will offer an offering made by fire, of a sweet savour unto the LORD; as ye do, so he shall do.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| One ordinance shall be both for you of the congregation, and also for the stranger that sojourneth with you, an ordinance for ever in your generations: as ye are, so shall the stranger be before the LORD.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| One law and one manner shall be for you, and for the stranger that sojourneth with you.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| Speak unto the children of Israel, and say unto them, When ye come into the land whither I bring you,}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| Then it shall be, that, when ye eat of the bread of the land, ye shall offer up an heave offering unto the LORD.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| Ye shall offer up a cake of the first of your dough for an heave offering: as ye do the heave offering of the threshingfloor, so shall ye heave it.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Of the first of your dough ye shall give unto the LORD an heave offering in your generations.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And if ye have erred, and not observed all these commandments, which the LORD hath spoken unto Moses,}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| Even all that the LORD hath commanded you by the hand of Moses, from the day that the LORD commanded Moses, and henceforward among your generations;}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| Then it shall be, if ought be committed by ignorance without the knowledge of the congregation, that all the congregation shall offer one young bullock for a burnt offering, for a sweet savour unto the LORD, with his meat offering, and his drink offering, according to the manner, and one kid of the goats for a sin offering.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the priest shall make an atonement for all the congregation of the children of Israel, and it shall be forgiven them; for it is ignorance: and they shall bring their offering, a sacrifice made by fire unto the LORD, and their sin offering before the LORD, for their ignorance:}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And it shall be forgiven all the congregation of the children of Israel, and the stranger that sojourneth among them; seeing all the people were in ignorance.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And if any soul sin through ignorance, then he shall bring a she goat of the first year for a sin offering.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| And the priest shall make an atonement for the soul that sinneth ignorantly, when he sinneth by ignorance before the LORD, to make an atonement for him; and it shall be forgiven him.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Ye shall have one law for him that sinneth through ignorance, both for him that is born among the children of Israel, and for the stranger that sojourneth among them.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| But the soul that doeth ought presumptuously, whether he be born in the land, or a stranger, the same reproacheth the LORD; and that soul shall be cut off from among his people.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| Because he hath despised the word of the LORD, and hath broken his commandment, that soul shall utterly be cut off; his iniquity shall be upon him.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And while the children of Israel were in the wilderness, they found a man that gathered sticks upon the sabbath day.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| And they that found him gathering sticks brought him unto Moses and Aaron, and unto all the congregation.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| And they put him in ward, because it was not declared what should be done to him.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| And the LORD said unto Moses, The man shall be surely put to death: all the congregation shall stone him with stones without the camp.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And all the congregation brought him without the camp, and stoned him with stones, and he died; as the LORD commanded Moses.}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| Speak unto the children of Israel, and bid them that they make them fringes in the borders of their garments throughout their generations, and that they put upon the fringe of the borders a ribband of blue:}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| And it shall be unto you for a fringe, that ye may look upon it, and remember all the commandments of the LORD, and do them; and that ye seek not after your own heart and your own eyes, after which ye use to go a whoring:}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| That ye may remember, and do all my commandments, and be holy unto your God.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| I am the LORD your God, which brought you out of the land of Egypt, to be your God: I am the LORD your God.}} ==Chapter 16== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| Now Korah, the son of Izhar, the son of Kohath, the son of Levi,}} {{font|color=#008888| and Dathan and Abiram, the sons of Eliab,}} {{font|color=#880000|and On, the son of Peleth,}} {{font|color=#008888| son}}{{font|color=#888800|s}} {{font|color=#008888|of Reuben, took men:}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And they rose up before Moses, with certain of the children of Israel, two hundred and fifty princes of the assembly, famous in the congregation, men of renown:}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And they gathered themselves together against Moses and against Aaron, and said unto them, Ye take too much upon you, seeing all the congregation are holy, every one of them, and the LORD is among them: wherefore then lift ye up yourselves above the congregation of the LORD?}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And when Moses heard it, he fell upon his face:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And he spake unto Korah and unto all his company, saying, Even to morrow the LORD will shew who are his, and who is holy; and will cause him to come near unto him: even him whom he hath chosen will he cause to come near unto him.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| This do; Take you censers, Korah, and all his company;}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And put fire therein, and put incense in them before the LORD to morrow: and it shall be that the man whom the LORD doth choose, he shall be holy: ye take too much upon you, ye sons of Levi.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And Moses said unto Korah, Hear, I pray you, ye sons of Levi:}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| Seemeth it but a small thing unto you, that the God of Israel hath separated you from the congregation of Israel, to bring you near to himself to do the service of the tabernacle of the LORD, and to stand before the congregation to minister unto them?}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And he hath brought thee near to him, and all thy brethren the sons of Levi with thee: and seek ye the priesthood also?}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| For which cause both thou and all thy company are gathered together against the LORD: and what is Aaron, that ye murmur against him?}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#008888| And Moses sent to call Dathan and Abiram, the sons of Eliab: which said, We will not come up:}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#008888| Is it a small thing that thou hast brought us up out of a land that floweth with milk and honey, to kill us in the wilderness, except thou make thyself altogether a prince over us?}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#008888| Moreover thou hast not brought us into a land that floweth with milk and honey, or given us inheritance of fields and vineyards: wilt thou put out the eyes of these men? we will not come up.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#008888| And Moses was very wroth, and said unto the LORD, Respect not thou their offering: I have not taken one ass from them, neither have I hurt one of them.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And Moses said unto Korah, Be thou and all thy company before the LORD, thou, and they, and Aaron, to morrow:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And take every man his censer, and put incense in them, and bring ye before the LORD every man his censer, two hundred and fifty censers; thou also, and Aaron, each of you his censer.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And they took every man his censer, and put fire in them, and laid incense thereon, and stood in the door of the tabernacle of the congregation with Moses and Aaron.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And Korah gathered all the congregation against them unto the door of the tabernacle of the congregation: and the glory of the LORD appeared unto all the congregation.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying, }} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Separate yourselves from among this congregation, that I may consume them in a moment.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And they fell upon their faces, and said, O God, the God of the spirits of all flesh, shall one man sin, and wilt thou be wroth with all the congregation?}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| Speak unto the congregation, saying, Get you up from about the tabernacle}} {{font|color=#880000|of Korah, Dathan, and Abiram.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#008888| And Moses rose up and went unto Dathan and Abiram; and the elders of Israel followed him.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And he spake unto the congregation,}} {{font|color=#008888| saying, Depart, I pray you, from the tents of these wicked men, and touch nothing of their's, lest ye be consumed in all their sins.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| So they gat up from the tabernacle }} {{font|color=#880000| of Korah, Dathan, and Abiram,}} {{font|color=#888800| on every side:}} {{font|color=#008888| and Dathan and Abiram came out, and stood in the door of their tents, and their wives, and their sons, and their little children.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#008888| And Moses said, Hereby ye shall know that the LORD hath sent me to do all these works; for I have not done them of mine own mind.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#008888| If these men die the common death of all men, or if they be visited after the visitation of all men; then the LORD hath not sent me.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#008888| But if the LORD make a new thing, and the earth open her mouth, and swallow them up, with all that appertain unto them, and they go down quick into the pit; then ye shall understand that these men have provoked the LORD.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#008888| And it came to pass, as he had made an end of speaking all these words, that the ground clave asunder that was under them:}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#008888| And the earth opened her mouth, and swallowed them up, and their houses,}} {{font|color=#880000| and all the men that appertained unto Korah,}} {{font|color=#008888| and all their goods.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#008888| They, and all that appertained to them, went down alive into the pit, and the earth closed upon them: and they perished from among the congregation.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#008888| And all Israel that were round about them fled at the cry of them: for they said, Lest the earth swallow us up also.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| And there came out a fire from the LORD, and consumed the two hundred and fifty men that offered incense.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying, }} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| Speak unto Eleazar the son of Aaron the priest, that he take up the censers out of the burning, and scatter thou the fire yonder; for they are hallowed.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| The censers of these sinners against their own souls, let them make them broad plates for a covering of the altar: for they offered them before the LORD, therefore they are hallowed: and they shall be a sign unto the children of Israel.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| And Eleazar the priest took the brasen censers, wherewith they that were burnt had offered; and they were made broad plates for a covering of the altar:}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| To be a memorial unto the children of Israel, that no stranger, which is not of the seed of Aaron, come near to offer incense before the LORD; that he be not as Korah, and as his company: as the LORD said to him by the hand of Moses.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| But on the morrow all the congregation of the children of Israel murmured against Moses and against Aaron, saying, Ye have killed the people of the LORD.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| And it came to pass, when the congregation was gathered against Moses and against Aaron, that they looked toward the tabernacle of the congregation: and, behold, the cloud covered it, and the glory of the LORD appeared.}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| And Moses and Aaron came before the tabernacle of the congregation.}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| Get you up from among this congregation, that I may consume them as in a moment. And they fell upon their faces. }} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| And Moses said unto Aaron, Take a censer, and put fire therein from off the altar, and put on incense, and go quickly unto the congregation, and make an atonement for them: for there is wrath gone out from the LORD; the plague is begun.}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| And Aaron took as Moses commanded, and ran into the midst of the congregation; and, behold, the plague was begun among the people: and he put on incense, and made an atonement for the people.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| And he stood between the dead and the living; and the plague was stayed.}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| Now they that died in the plague were fourteen thousand and seven hundred, beside them that died about the matter of Korah.}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| And Aaron returned unto Moses unto the door of the tabernacle of the congregation: and the plague was stayed.}} ==Chapter 17== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Speak unto the children of Israel, and take of every one of them a rod according to the house of their fathers, of all their princes according to the house of their fathers twelve rods: write thou every man's name upon his rod.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And thou shalt write Aaron's name upon the rod of Levi: for one rod shall be for the head of the house of their fathers.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And thou shalt lay them up in the tabernacle of the congregation before the testimony, where I will meet with you.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And it shall come to pass, that the man's rod, whom I shall choose, shall blossom: and I will make to cease from me the murmurings of the children of Israel, whereby they murmur against you.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And Moses spake unto the children of Israel, and every one of their princes gave him a rod apiece, for each prince one, according to their fathers' houses, even twelve rods: and the rod of Aaron was among their rods.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And Moses laid up the rods before the LORD in the tabernacle of witness.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And it came to pass, that on the morrow Moses went into the tabernacle of witness; and, behold, the rod of Aaron for the house of Levi was budded, and brought forth buds, and bloomed blossoms, and yielded almonds.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And Moses brought out all the rods from before the LORD unto all the children of Israel: and they looked, and took every man his rod.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And the LORD said unto Moses, Bring Aaron's rod again before the testimony, to be kept for a token against the rebels; and thou shalt quite take away their murmurings from me, that they die not.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And Moses did so: as the LORD commanded him, so did he. }} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And the children of Israel spake unto Moses, saying, Behold, we die, we perish, we all perish.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Whosoever cometh any thing near unto the tabernacle of the LORD shall die: shall we be consumed with dying?}} ==Chapter 18== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD said unto Aaron, Thou and thy sons and thy father's house with thee shall bear the iniquity of the sanctuary: and thou and thy sons with thee shall bear the iniquity of your priesthood.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And thy brethren also of the tribe of Levi, the tribe of thy father, bring thou with thee, that they may be joined unto thee, and minister unto thee: but thou and thy sons with thee shall minister before the tabernacle of witness.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And they shall keep thy charge, and the charge of all the tabernacle: only they shall not come nigh the vessels of the sanctuary and the altar, that neither they, nor ye also, die.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And they shall be joined unto thee, and keep the charge of the tabernacle of the congregation, for all the service of the tabernacle: and a stranger shall not come nigh unto you.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And ye shall keep the charge of the sanctuary, and the charge of the altar: that there be no wrath any more upon the children of Israel.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And I, behold, I have taken your brethren the Levites from among the children of Israel: to you they are given as a gift for the LORD, to do the service of the tabernacle of the congregation.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Therefore thou and thy sons with thee shall keep your priest's office for everything of the altar, and within the vail; and ye shall serve: I have given your priest's office unto you as a service of gift: and the stranger that cometh nigh shall be put to death.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And the LORD spake unto Aaron, Behold, I also have given thee the charge of mine heave offerings of all the hallowed things of the children of Israel; unto thee have I given them by reason of the anointing, and to thy sons, by an ordinance for ever.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| This shall be thine of the most holy things, reserved from the fire: every oblation of their's, every meat offering of their's, and every sin offering of their's, and every trespass offering of their's which they shall render unto me, shall be most holy for thee and for thy sons.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| In the most holy place shalt thou eat it; every male shall eat it: it shall be holy unto thee.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And this is thine; the heave offering of their gift, with all the wave offerings of the children of Israel: I have given them unto thee, and to thy sons and to thy daughters with thee, by a statute for ever: every one that is clean in thy house shall eat of it.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| All the best of the oil, and all the best of the wine, and of the wheat, the firstfruits of them which they shall offer unto the LORD, them have I given thee.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And whatsoever is first ripe in the land, which they shall bring unto the LORD, shall be thine; every one that is clean in thine house shall eat of it.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Every thing devoted in Israel shall be thine.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| Every thing that openeth the matrix in all flesh, which they bring unto the LORD, whether it be of men or beasts, shall be thine: nevertheless the firstborn of man shalt thou surely redeem, and the firstling of unclean beasts shalt thou redeem.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And those that are to be redeemed from a month old shalt thou redeem, according to thine estimation, for the money of five shekels, after the shekel of the sanctuary, which is twenty gerahs.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| But the firstling of a cow, or the firstling of a sheep, or the firstling of a goat, thou shalt not redeem; they are holy: thou shalt sprinkle their blood upon the altar, and shalt burn their fat for an offering made by fire, for a sweet savour unto the LORD.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And the flesh of them shall be thine, as the wave breast and as the right shoulder are thine.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| All the heave offerings of the holy things, which the children of Israel offer unto the LORD, have I given thee, and thy sons and thy daughters with thee, by a statute for ever: it is a covenant of salt for ever before the LORD unto thee and to thy seed with thee.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And the LORD spake unto Aaron, Thou shalt have no inheritance in their land, neither shalt thou have any part among them: I am thy part and thine inheritance among the children of Israel.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And, behold, I have given the children of Levi all the tenth in Israel for an inheritance, for their service which they serve, even the service of the tabernacle of the congregation.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Neither must the children of Israel henceforth come nigh the tabernacle of the congregation, lest they bear sin, and die.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| But the Levites shall do the service of the tabernacle of the congregation, and they shall bear their iniquity: it shall be a statute for ever throughout your generations, that among the children of Israel they have no inheritance.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| But the tithes of the children of Israel, which they offer as an heave offering unto the LORD, I have given to the Levites to inherit: therefore I have said unto them, Among the children of Israel they shall have no inheritance.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| Thus speak unto the Levites, and say unto them, When ye take of the children of Israel the tithes which I have given you from them for your inheritance, then ye shall offer up an heave offering of it for the LORD, even a tenth part of the tithe.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And this your heave offering shall be reckoned unto you, as though it were the corn of the threshingfloor, and as the fulness of the winepress.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| Thus ye also shall offer an heave offering unto the LORD of all your tithes, which ye receive of the children of Israel; and ye shall give thereof the LORD's heave offering to Aaron the priest.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Out of all your gifts ye shall offer every heave offering of the LORD, of all the best thereof, even the hallowed part thereof out of it.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| Therefore thou shalt say unto them, When ye have heaved the best thereof from it, then it shall be counted unto the Levites as the increase of the threshingfloor, and as the increase of the winepress.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| And ye shall eat it in every place, ye and your households: for it is your reward for your service in the tabernacle of the congregation.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And ye shall bear no sin by reason of it, when ye have heaved from it the best of it: neither shall ye pollute the holy things of the children of Israel, lest ye die.}} ==Chapter 19== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses and unto Aaron, saying, }} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| This is the ordinance of the law which the LORD hath commanded, saying, Speak unto the children of Israel, that they bring thee a red heifer without spot, wherein is no blemish, and upon which never came yoke:}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And ye shall give her unto Eleazar the priest, that he may bring her forth without the camp, and one shall slay her before his face:}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And Eleazar the priest shall take of her blood with his finger, and sprinkle of her blood directly before the tabernacle of the congregation seven times:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And one shall burn the heifer in his sight; her skin, and her flesh, and her blood, with her dung, shall he burn:}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And the priest shall take cedar wood, and hyssop, and scarlet, and cast it into the midst of the burning of the heifer.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| Then the priest shall wash his clothes, and he shall bathe his flesh in water, and afterward he shall come into the camp, and the priest shall be unclean until the even.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And he that burneth her shall wash his clothes in water, and bathe his flesh in water, and shall be unclean until the even.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And a man that is clean shall gather up the ashes of the heifer, and lay them up without the camp in a clean place, and it shall be kept for the congregation of the children of Israel for a water of separation: it is a purification for sin.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And he that gathereth the ashes of the heifer shall wash his clothes, and be unclean until the even: and it shall be unto the children of Israel, and unto the stranger that sojourneth among them, for a statute for ever.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| He that toucheth the dead body of any man shall be unclean seven days.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| He shall purify himself with it on the third day, and on the seventh day he shall be clean: but if he purify not himself the third day, then the seventh day he shall not be clean.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Whosoever toucheth the dead body of any man that is dead, and purifieth not himself, defileth the tabernacle of the LORD; and that soul shall be cut off from Israel: because the water of separation was not sprinkled upon him, he shall be unclean; his uncleanness is yet upon him.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| This is the law, when a man dieth in a tent: all that come into the tent, and all that is in the tent, shall be unclean seven days.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And every open vessel, which hath no covering bound upon it, is unclean.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And whosoever toucheth one that is slain with a sword in the open fields, or a dead body, or a bone of a man, or a grave, shall be unclean seven days.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And for an unclean person they shall take of the ashes of the burnt heifer of purification for sin, and running water shall be put thereto in a vessel:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And a clean person shall take hyssop, and dip it in the water, and sprinkle it upon the tent, and upon all the vessels, and upon the persons that were there, and upon him that touched a bone, or one slain, or one dead, or a grave:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And the clean person shall sprinkle upon the unclean on the third day, and on the seventh day: and on the seventh day he shall purify himself, and wash his clothes, and bathe himself in water, and shall be clean at even.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| But the man that shall be unclean, and shall not purify himself, that soul shall be cut off from among the congregation, because he hath defiled the sanctuary of the LORD: the water of separation hath not been sprinkled upon him; he is unclean.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And it shall be a perpetual statute unto them, that he that sprinkleth the water of separation shall wash his clothes; and he that toucheth the water of separation shall be unclean until even.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And whatsoever the unclean person toucheth shall be unclean; and the soul that toucheth it shall be unclean until even.}} ==Chapter 20== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| Then came the children of Israel, even the whole congregation, into the desert of Zin in the first month:}} {{font|color=#008888| and the people abode in Kadesh; and Miriam died there, and was buried there.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And there was no water for the congregation: and they gathered themselves together against Moses and against Aaron.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#008888| And the people chode with Moses, and spake, }} {{font|color=#888800|saying, Would God that we had died when our brethren died before the LORD!}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And why have ye brought up the congregation of the LORD into this wilderness, that we and our cattle should die there?}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#008888| And wherefore have ye made us to come up out of Egypt, to bring us in unto this evil place? it is no place of seed, or of figs, or of vines, or of pomegranates; neither is there any water to drink.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And Moses and Aaron went from the presence of the assembly unto the door of the tabernacle of the congregation, and they fell upon their faces: and the glory of the LORD appeared unto them.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| Take the rod, and gather thou the assembly together, thou, and Aaron thy brother, and speak ye unto the rock before their eyes; and it shall give forth his water, and thou shalt bring forth to them water out of the rock: so thou shalt give the congregation and their beasts drink.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And Moses took the rod from before the LORD, as he commanded him.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And Moses and Aaron gathered the congregation together before the rock, and he said unto them, Hear now, ye rebels; must we fetch you water out of this rock?}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And Moses lifted up his hand, and with his rod he smote the rock twice: and the water came out abundantly, and the congregation drank, and their beasts also.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And the LORD spake unto Moses and Aaron, Because ye believed me not, to sanctify me in the eyes of the children of Israel, therefore ye shall not bring this congregation into the land which I have given them.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| This is the water of Meribah; because the children of Israel strove with the LORD, and he was sanctified in them.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#008888| And Moses sent messengers from Kadesh unto the king of Edom, Thus saith thy brother Israel, Thou knowest all the travail that hath befallen us:}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#008888| How our fathers went down into Egypt, and we have dwelt in Egypt a long time; and the Egyptians vexed us, and our fathers:}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#008888| And when we cried unto the LORD, he heard our voice, and sent an angel, and hath brought us forth out of Egypt: and, behold, we are in Kadesh, a city in the uttermost of thy border:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#008888| Let us pass, I pray thee, through thy country: we will not pass through the fields, or through the vineyards, neither will we drink of the water of the wells: we will go by the king's high way, we will not turn to the right hand nor to the left, until we have passed thy borders.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#008888| And Edom said unto him, Thou shalt not pass by me, lest I come out against thee with the sword.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#008888| And the children of Israel said unto him, We will go by the high way: and if I and my cattle drink of thy water, then I will pay for it: I will only, without doing anything else, go through on my feet.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#008888| And he said, Thou shalt not go through. And Edom came out against him with much people, and with a strong hand.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#008888| Thus Edom refused to give Israel passage through his border: wherefore Israel turned away from him.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And the children of Israel, even the whole congregation,}} {{font|color=#008888|journeyed from Kadesh, and}} {{font|color=#888800|came unto mount Hor.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And the LORD spake unto Moses and Aaron in mount Hor, by the coast of the land of Edom, saying,}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| Aaron shall be gathered unto his people: for he shall not enter into the land which I have given unto the children of Israel, because ye rebelled against my word at the water of Meribah.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| Take Aaron and Eleazar his son, and bring them up unto mount Hor:}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And strip Aaron of his garments, and put them upon Eleazar his son: and Aaron shall be gathered unto his people, and shall die there.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And Moses did as the LORD commanded: and they went up into mount Hor in the sight of all the congregation.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| And Moses stripped Aaron of his garments, and put them upon Eleazar his son; and Aaron died there in the top of the mount: and Moses and Eleazar came down from the mount.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| And when all the congregation saw that Aaron was dead, they mourned for Aaron thirty days, even all the house of Israel.}} ==Chapter 21== <small>{{font|color=#0000FF|1}}</small>{{font|color=#000088| And when king Arad the Canaanite, which dwelt in the south, heard tell that Israel came by the way of the spies; then he fought against Israel, and took some of them prisoners.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#000088| And Israel vowed a vow unto the LORD, and said, If thou wilt indeed deliver this people into my hand, then I will utterly destroy their cities.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#000088| And the LORD hearkened to the voice of Israel, and delivered up the Canaanites; and they utterly destroyed them and their cities: and he called the name of the place Hormah.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And they journeyed from mount Hor by the way of the Red sea,}} {{font|color=#008888| to compass the land of Edom: and the soul of the people was much discouraged because of the way.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#008888| And the people spake against God, and against Moses, Wherefore have ye brought us up out of Egypt to die in the wilderness? for there is no bread, neither is there any water;}} {{font|color=#880000|and our soul loatheth this light bread.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#008888| And the LORD sent fiery serpents among the people, and they bit the people; and much people of Israel died.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#008888| Therefore the people came to Moses, and said, We have sinned, for we have spoken against the LORD, and against thee; pray unto the LORD, that he take away the serpents from us. And Moses prayed for the people.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#008888| And the LORD said unto Moses, Make thee a fiery serpent, and set it upon a pole: and it shall come to pass, that every one that is bitten, when he looketh upon it, shall live.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#008888| And Moses made a serpent of brass, and put it upon a pole, and it came to pass, that if a serpent had bitten any man, when he beheld the serpent of brass, he lived.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And the children of Israel set forward, and pitched in Oboth. }} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And they journeyed from Oboth, and pitched at Ijeabarim, in the wilderness which is before Moab, toward the sunrising.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#008888| From thence they removed, and pitched in the valley of Zared.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#008888| From thence they removed, and pitched on the other side of Arnon, which is in the wilderness that cometh out of the coasts of the Amorites: for Arnon is the border of Moab, between Moab and the Amorites.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#008888| Wherefore it is said in the book of the wars of the LORD, What he did in the Red sea, and in the brooks of Arnon,}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#008888| And at the stream of the brooks that goeth down to the dwelling of Ar, and lieth upon the border of Moab.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#000088|And from thence they went to Beer: that is the well whereof the LORD spake unto Moses, Gather the people together, and I will give them water.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#000088| Then Israel sang this song, Spring up, O well; sing ye unto it:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#000088|The princes digged the well, the nobles of the people digged it, by the direction of the lawgiver, with their staves. And from the wilderness they went to Mattanah: }} <small>{{font|color=#0000FF|19}}</small>{{font|color=#000088| And from Mattanah to Nahaliel: and from Nahaliel to Bamoth:}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#000088| And from Bamoth in the valley, that is in the country of Moab, to the top of Pisgah, which looketh toward Jeshimon.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#008888| And Israel sent messengers unto Sihon king of the Amorites, saying,}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#008888| Let me pass through thy land: we will not turn into the fields, or into the vineyards; we will not drink of the waters of the well: but we will go along by the king's high way, until we be past thy borders.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#008888| And Sihon would not suffer Israel to pass through his border: but Sihon gathered all his people together, and went out against Israel into the wilderness: and he came to Jahaz, and fought against Israel.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#008888| And Israel smote him with the edge of the sword, and possessed his land from Arnon unto Jabbok, even unto the children of Ammon: for the border of the children of Ammon was strong.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#008888| And Israel took all these cities: and Israel dwelt in all the cities of the Amorites, in Heshbon, and in all the villages thereof.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#008888| For Heshbon was the city of Sihon the king of the Amorites, who had fought against the former king of Moab, and taken all his land out of his hand, even unto Arnon.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#008888| Wherefore they that speak in proverbs say, Come into Heshbon, let the city of Sihon be built and prepared:}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#008888| For there is a fire gone out of Heshbon, a flame from the city of Sihon: it hath consumed Ar of Moab, and the lords of the high places of Arnon.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#008888| Woe to thee, Moab! thou art undone, O people of Chemosh: he hath given his sons that escaped, and his daughters, into captivity unto Sihon king of the Amorites.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#008888| We have shot at them; Heshbon is perished even unto Dibon, and we have laid them waste even unto Nophah, which reacheth unto Medeba.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#008888| Thus Israel dwelt in the land of the Amorites. }} <small>{{font|color=#0000FF|32}}</small>{{font|color=#008888| And Moses sent to spy out Jaazer, and they took the villages thereof, and drove out the Amorites that were there.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#880000| And they turned and went up by the way of Bashan: and Og the king of Bashan went out against them, he, and all his people, to the battle at Edrei.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#880000| And the LORD said unto Moses, Fear him not: for I have delivered him into thy hand, and all his people, and his land; and thou shalt do to him as thou didst unto Sihon king of the Amorites, which dwelt at Heshbon.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#880000| So they smote him, and his sons, and all his people, until there was none left him alive: and they possessed his land.}} ==Chapter 22== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the children of Israel set forward, and pitched in the plains of Moab on this side Jordan by Jericho.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#880000| And Balak the son of Zippor saw all that Israel had done to the Amorites.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#000088|And Moab was sore afraid of the people, because they were many: and Moab was distressed because of the children of Israel.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#000088|And Moab said}} {{font|color=#880000| unto the elders of Midian,}} {{font|color=#000088|Now shall this company lick up all that are round about us, as the ox licketh up the grass of the field. And Balak the son of Zippor was king of the Moabites at that time.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#000088| He sent messengers therefore unto Balaam the son of Beor to Pethor, which is by the river of the land of the children of his people, to call him, saying, Behold, there is a people come out from Egypt: behold, they cover the face of the earth, and they abide over against me:}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#000088| Come now therefore, I pray thee, curse me this people; for they are too mighty for me: peradventure I shall prevail, that we may smite them, and that I may drive them out of the land: for I wot that he whom thou blessest is blessed, and he whom thou cursest is cursed.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#000088| And the elders of Moab}} {{font|color=#880000|and the elders of Midian}} {{font|color=#000088|departed with the rewards of divination in their hand; and they came unto Balaam, and spake unto him the words of Balak.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#000088| And he said unto them, Lodge here this night, and I will bring you word again, as the LORD shall speak unto me: and the princes of Moab abode with Balaam.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#000088| And God came unto Balaam, and said, What men are these with thee?}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#000088| And Balaam said unto God, Balak the son of Zippor, king of Moab, hath sent unto me saying,}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#000088| Behold, there is a people come out of Egypt, which covereth the face of the earth: come now, curse me them; peradventure I shall be able to overcome them, and drive them out.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#000088| And God said unto Balaam, Thou shalt not go with them; thou shalt not curse the people: for they are blessed.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#000088| And Balaam rose up in the morning, and said unto the princes of Balak, Get you into your land: for the LORD refuseth to give me leave to go with you.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#000088| And the princes of Moab rose up, and they went unto Balak, and said, Balaam refuseth to come with us.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#000088| And Balak sent yet again princes, more, and more honourable than they.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#000088| And they came to Balaam, and said to him, Thus saith Balak the son of Zippor, Let nothing, I pray thee, hinder thee from coming unto me:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#000088| For I will promote thee unto very great honour, and I will do whatsoever thou sayest unto me: come therefore, I pray thee, curse me this people.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#000088| And Balaam answered and said unto the servants of Balak, If Balak would give me his house full of silver and gold, I cannot go beyond the word of the LORD my God, to do less or more.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#000088| Now therefore, I pray you, tarry ye also here this night, that I may know what the LORD will say unto me more.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#000088| And God came unto Balaam at night, and said unto him, If the men come to call thee, rise up, and go with them; but yet the word which I shall say unto thee, that shalt thou do.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#000088| And Balaam rose up in the morning, and saddled his ass, and went with the princes of Moab.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#880000| And God's anger was kindled because he went:}} {{font|color=#000088|and the angel of the LORD stood in the way for an adversary against him. Now he was riding upon his ass, and his two servants were with him.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#000088| And the ass saw the angel of the LORD standing in the way, and his sword drawn in his hand: and the ass turned aside out of the way, and went into the field: and Balaam smote the ass, to turn her into the way.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#000088| But the angel of the LORD stood in a path of the vineyards, a wall being on this side, and a wall on that side.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#000088| And when the ass saw the angel of the LORD, she thrust herself unto the wall, and crushed Balaam's foot against the wall: and he smote her again.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#000088| And the angel of the LORD went further, and stood in a narrow place, where was no way to turn either to the right hand or to the left.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#000088| And when the ass saw the angel of the LORD, she fell down under Balaam: and Balaam's anger was kindled, and he smote the ass with a staff.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#000088| And the LORD opened the mouth of the ass, and she said unto Balaam, What have I done unto thee, that thou hast smitten me these three times?}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#000088| And Balaam said unto the ass, Because thou hast mocked me: I would there were a sword in mine hand, for now would I kill thee.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#000088| And the ass said unto Balaam, Am not I thine ass, upon which thou hast ridden ever since I was thine unto this day? was I ever wont to do so unto thee? And he said, Nay.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#000088| Then the LORD opened the eyes of Balaam, and he saw the angel of the LORD standing in the way, and his sword drawn in his hand: and he bowed down his head, and fell flat on his face.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#000088| And the angel of the LORD said unto him, Wherefore hast thou smitten thine ass these three times? behold, I went out to withstand thee, because thy way is perverse before me:}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#000088| And the ass saw me, and turned from me these three times: unless she had turned from me, surely now also I had slain thee, and saved her alive.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#000088| And Balaam said unto the angel of the LORD, I have sinned; for I knew not that thou stoodest in the way against me: now therefore, if it displease thee, I will get me back again.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#000088| And the angel of the LORD said unto Balaam, Go with the men: but only the word that I shall speak unto thee, that thou shalt speak. So Balaam went with the princes of Balak.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#000088| And when Balak heard that Balaam was come, he went out to meet him unto a city of Moab, which is in the border of Arnon, which is in the utmost coast.}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#000088| And Balak said unto Balaam, Did I not earnestly send unto thee to call thee? wherefore camest thou not unto me? am I not able indeed to promote thee to honour?}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#000088| And Balaam said unto Balak, Lo, I am come unto thee: have I now any power at all to say any thing? the word that God putteth in my mouth, that shall I speak.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#000088| And Balaam went with Balak, and they came unto Kirjathhuzoth.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#000088| And Balak offered oxen and sheep, and sent to Balaam, and to the princes that were with him.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#000088| And it came to pass on the morrow, that Balak took Balaam, and brought him up into the high places of Baal, that thence he might see the utmost part of the people.}} ==Chapter 23== <small>{{font|color=#0000FF|1}}</small>{{font|color=#000088| And Balaam said unto Balak, Build me here seven altars, and prepare me here seven oxen and seven rams.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#000088| And Balak did as Balaam had spoken; and Balak and Balaam offered on every altar a bullock and a ram.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#000088| And Balaam said unto Balak, Stand by thy burnt offering, and I will go: peradventure the LORD will come to meet me: and whatsoever he sheweth me I will tell thee. And he went to an high place.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#000088| And God met Balaam: and he said unto him, I have prepared seven altars, and I have offered upon every altar a bullock and a ram.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#000088| And the LORD put a word in Balaam's mouth, and said, Return unto Balak, and thus thou shalt speak.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#000088| And he returned unto him, and, lo, he stood by his burnt sacrifice, he, and all the princes of Moab.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#000088| And he took up his parable, and said, Balak the king of Moab hath brought me from Aram, out of the mountains of the east, saying, Come, curse me Jacob, and come, defy Israel.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#000088| How shall I curse, whom God hath not cursed? or how shall I defy, whom the LORD hath not defied?}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#000088| For from the top of the rocks I see him, and from the hills I behold him: lo, the people shall dwell alone, and shall not be reckoned among the nations.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#000088| Who can count the dust of Jacob, and the number of the fourth part of Israel? Let me die the death of the righteous, and let my last end be like his!}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#000088| And Balak said unto Balaam, What hast thou done unto me? I took thee to curse mine enemies, and, behold, thou hast blessed them altogether.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#000088| And he answered and said, Must I not take heed to speak that which the LORD hath put in my mouth?}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#000088| And Balak said unto him, Come, I pray thee, with me unto another place, from whence thou mayest see them: thou shalt see but the utmost part of them, and shalt not see them all: and curse me them from thence.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#000088| And he brought him into the field of Zophim, to the top of Pisgah, and built seven altars, and offered a bullock and a ram on every altar.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#000088| And he said unto Balak, Stand here by thy burnt offering, while I meet the LORD yonder.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#000088| And the LORD met Balaam, and put a word in his mouth, and said, Go again unto Balak, and say thus.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#000088| And when he came to him, behold, he stood by his burnt offering, and the princes of Moab with him. And Balak said unto him, What hath the LORD spoken?}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#000088| And he took up his parable, and said, Rise up, Balak, and hear; hearken unto me, thou son of Zippor:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#000088| God is not a man, that he should lie; neither the son of man, that he should repent: hath he said, and shall he not do it? or hath he spoken, and shall he not make it good?}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#000088| Behold, I have received commandment to bless: and he hath blessed; and I cannot reverse it.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#000088| He hath not beheld iniquity in Jacob, neither hath he seen perverseness in Israel: the LORD his God is with him, and the shout of a king is among them.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#000088| God brought them out of Egypt; he hath as it were the strength of an unicorn.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#000088| Surely there is no enchantment against Jacob, neither is there any divination against Israel: according to this time it shall be said of Jacob and of Israel, What hath God wrought!}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#000088| Behold, the people shall rise up as a great lion, and lift up himself as a young lion: he shall not lie down until he eat of the prey, and drink the blood of the slain.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#000088| And Balak said unto Balaam, Neither curse them at all, nor bless them at all.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#000088| But Balaam answered and said unto Balak, Told not I thee, saying, All that the LORD speaketh, that I must do?}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#000088| And Balak said unto Balaam, Come, I pray thee, I will bring thee unto another place; peradventure it will please God that thou mayest curse me them from thence.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#000088| And Balak brought Balaam unto the top of Peor, that looketh toward Jeshimon.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#000088| And Balaam said unto Balak, Build me here seven altars, and prepare me here seven bullocks and seven rams.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#000088| And Balak did as Balaam had said, and offered a bullock and a ram on every altar.}} ==Chapter 24== <small>{{font|color=#0000FF|1}}</small>{{font|color=#000088| And when Balaam saw that it pleased the LORD to bless Israel, he went not, as at other times, to seek for enchantments, but he set his face toward the wilderness.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#000088| And Balaam lifted up his eyes, and he saw Israel abiding in his tents according to their tribes; and the spirit of God came upon him.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#000088| And he took up his parable, and said, Balaam the son of Beor hath said, and the man whose eyes are open hath said:}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#000088| He hath said, which heard the words of God, which saw the vision of the Almighty, falling into a trance, but having his eyes open:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#000088| How goodly are thy tents, O Jacob, and thy tabernacles, O Israel!}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#000088| As the valleys are they spread forth, as gardens by the river's side, as the trees of lign aloes which the LORD hath planted, and as cedar trees beside the waters.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#000088| He shall pour the water out of his buckets, and his seed shall be in many waters, and his king shall be higher than Agag, and his kingdom shall be exalted.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#000088| God brought him forth out of Egypt; he hath as it were the strength of an unicorn: he shall eat up the nations his enemies, and shall break their bones, and pierce them through with his arrows.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#000088| He couched, he lay down as a lion, and as a great lion: who shall stir him up? Blessed is he that blesseth thee, and cursed is he that curseth thee.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#000088| And Balak's anger was kindled against Balaam, and he smote his hands together: and Balak said unto Balaam, I called thee to curse mine enemies, and, behold, thou hast altogether blessed them these three times.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#000088| Therefore now flee thou to thy place: I thought to promote thee unto great honour; but, lo, the LORD hath kept thee back from honour.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#000088| And Balaam said unto Balak, Spake I not also to thy messengers which thou sentest unto me, saying,}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#000088| If Balak would give me his house full of silver and gold, I cannot go beyond the commandment of the LORD, to do either good or bad of mine own mind; but what the LORD saith, that will I speak?}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#880000| And now, behold, I go unto my people: come therefore, and I will advertise thee what this people shall do to thy people in the latter days.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#880000| And he took up his parable, and said, Balaam the son of Beor hath said, and the man whose eyes are open hath said:}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#880000| He hath said, which heard the words of God, and knew the knowledge of the most High, which saw the vision of the Almighty, falling into a trance, but having his eyes open:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#880000| I shall see him, but not now: I shall behold him, but not nigh: there shall come a Star out of Jacob, and a Sceptre shall rise out of Israel, and shall smite the corners of Moab, and destroy all the children of Sheth.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#880000| And Edom shall be a possession, Seir also shall be a possession for his enemies; and Israel shall do valiantly.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#880000| Out of Jacob shall come he that shall have dominion, and shall destroy him that remaineth of the city.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#880000| And when he looked on Amalek, he took up his parable, and said, Amalek was the first of the nations; but his latter end shall be that he perish for ever.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#880000| And he looked on the Kenites, and took up his parable, and said, Strong is thy dwellingplace, and thou puttest thy nest in a rock.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#880000| Nevertheless the Kenite shall be wasted, until Asshur shall carry thee away captive.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#880000| And he took up his parable, and said, Alas, who shall live when God doeth this!}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#880000| And ships shall come from the coast of Chittim, and shall afflict Asshur, and shall afflict Eber, and he also shall perish for ever.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#000088| And Balaam rose up, and went and returned to his place: and Balak also went his way.}} ==Chapter 25== <small>{{font|color=#0000FF|1}}</small>{{font|color=#008888| And Israel abode in Shittim, }} {{font|color=#000088|and the people began to commit whoredom with the daughters of Moab.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#000088| And they called the people unto the sacrifices of their gods: and the people did eat, and bowed down to their gods.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#008888| And Israel joined himself unto Baalpeor: }} {{font|color=#000088|and the anger of the LORD was kindled against Israel.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#000088| And the LORD said unto Moses, Take all the heads of the people, and hang them up before the LORD against the sun, that the fierce anger of the LORD may be turned away from Israel.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#008888| And Moses said unto the judges of Israel, Slay ye every one his men that were joined unto Baalpeor.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And, behold, one of the children of Israel came and brought unto his brethren a Midianitish woman in the sight of Moses, and in the sight of all the congregation of the children of Israel, who were weeping before the door of the tabernacle of the congregation.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And when Phinehas, the son of Eleazar, the son of Aaron the priest, saw it, he rose up from among the congregation, and took a javelin in his hand;}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And he went after the man of Israel into the tent, and thrust both of them through, the man of Israel, and the woman through her belly.}} {{font|color=#880000| So the plague was stayed from the children of Israel.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#880000| And those that died in the plague were twenty and four thousand.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| Phinehas, the son of Eleazar, the son of Aaron the priest, hath turned my wrath away from the children of Israel, while he was zealous for my sake among them, that I consumed not the children of Israel in my jealousy.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| Wherefore say, Behold, I give unto him my covenant of peace:}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And he shall have it, and his seed after him, even the covenant of an everlasting priesthood; because he was zealous for his God, and made an atonement for the children of Israel.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Now the name of the Israelite that was slain, even that was slain with the Midianitish woman, was Zimri, the son of Salu, a prince of a chief house among the Simeonites.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And the name of the Midianitish woman that was slain was Cozbi, the daughter of Zur; he was head over a people, and of a chief house in Midian.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| Vex the Midianites, and smite them:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800|For they vex you with their wiles,}} {{font|color=#880000| wherewith they have beguiled you in the matter of Peor, and}} {{font|color=#888800| in the matter of Cozbi, the daughter of a prince of Midian, their sister, which was slain}} {{font|color=#880000| in the day of the plague for Peor's sake.}} ==Chapter 26== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And it came to pass after the plague, that the LORD spake unto Moses and unto Eleazar the son of Aaron the priest, saying, }} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Take the sum of all the congregation of the children of Israel, from twenty years old and upward, throughout their fathers' house, all that are able to go to war in Israel.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And Moses and Eleazar the priest spake with them in the plains of Moab by Jordan near Jericho, saying,}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| Take the sum of the people, from twenty years old and upward; as the LORD commanded Moses and the children of Israel, which went forth out of the land of Egypt.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| Reuben, the eldest son of Israel: the children of Reuben; Hanoch, of whom cometh the family of the Hanochites: of Pallu, the family of the Palluites:}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| Of Hezron, the family of the Hezronites: of Carmi, the family of the Carmites.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| These are the families of the Reubenites: and they that were numbered of them were forty and three thousand and seven hundred and thirty.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And the sons of Pallu; Eliab.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#880000| And the sons of Eliab; Nemuel, and Dathan, and Abiram. This is that Dathan and Abiram, which were famous in the congregation, who strove against Moses and against Aaron in the company of Korah, when they strove against the LORD:}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#880000| And the earth opened her mouth, and swallowed them up together with Korah, when that company died, what time the fire devoured two hundred and fifty men: and they became a sign.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#880000| Notwithstanding the children of Korah died not.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| The sons of Simeon after their families: of Nemuel, the family of the Nemuelites: of Jamin, the family of the Jaminites: of Jachin, the family of the Jachinites:}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Of Zerah, the family of the Zarhites: of Shaul, the family of the Shaulites.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| These are the families of the Simeonites, twenty and two thousand and two hundred.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| The children of Gad after their families: of Zephon, the family of the Zephonites: of Haggi, the family of the Haggites: of Shuni, the family of the Shunites:}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| Of Ozni, the family of the Oznites: of Eri, the family of the Erites:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| Of Arod, the family of the Arodites: of Areli, the family of the Arelites.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| These are the families of the children of Gad according to those that were numbered of them, forty thousand and five hundred.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| The sons of Judah were Er and Onan: and Er and Onan died in the land of Canaan.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And the sons of Judah after their families were; of Shelah, the family of the Shelanites: of Pharez, the family of the Pharzites: of Zerah, the family of the Zarhites.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And the sons of Pharez were; of Hezron, the family of the Hezronites: of Hamul, the family of the Hamulites.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| These are the families of Judah according to those that were numbered of them, threescore and sixteen thousand and five hundred.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| Of the sons of Issachar after their families: of Tola, the family of the Tolaites: of Pua, the family of the Punites:}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| Of Jashub, the family of the Jashubites: of Shimron, the family of the Shimronites.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| These are the families of Issachar according to those that were numbered of them, threescore and four thousand and three hundred.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| Of the sons of Zebulun after their families: of Sered, the family of the Sardites: of Elon, the family of the Elonites: of Jahleel, the family of the Jahleelites.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| These are the families of the Zebulunites according to those that were numbered of them, threescore thousand and five hundred.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| The sons of Joseph after their families were Manasseh and Ephraim.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Of the sons of Manasseh: of Machir, the family of the Machirites: and Machir begat Gilead: of Gilead come the family of the Gileadites.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| These are the sons of Gilead: of Jeezer, the family of the Jeezerites: of Helek, the family of the Helekites:}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| And of Asriel, the family of the Asrielites: and of Shechem, the family of the Shechemites:}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And of Shemida, the family of the Shemidaites: and of Hepher, the family of the Hepherites.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| And Zelophehad the son of Hepher had no sons, but daughters: and the names of the daughters of Zelophehad were Mahlah, and Noah, Hoglah, Milcah, and Tirzah.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| These are the families of Manasseh, and those that were numbered of them, fifty and two thousand and seven hundred.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| These are the sons of Ephraim after their families: of Shuthelah, the family of the Shuthalhites: of Becher, the family of the Bachrites: of Tahan, the family of the Tahanites.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And these are the sons of Shuthelah: of Eran, the family of the Eranites.}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| These are the families of the sons of Ephraim according to those that were numbered of them, thirty and two thousand and five hundred. These are the sons of Joseph after their families.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| The sons of Benjamin after their families: of Bela, the family of the Belaites: of Ashbel, the family of the Ashbelites: of Ahiram, the family of the Ahiramites:}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| Of Shupham, the family of the Shuphamites: of Hupham, the family of the Huphamites.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| And the sons of Bela were Ard and Naaman: of Ard, the family of the Ardites: and of Naaman, the family of the Naamites.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| These are the sons of Benjamin after their families: and they that were numbered of them were forty and five thousand and six hundred.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| These are the sons of Dan after their families: of Shuham, the family of the Shuhamites. These are the families of Dan after their families.}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| All the families of the Shuhamites, according to those that were numbered of them, were threescore and four thousand and four hundred.}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| Of the children of Asher after their families: of Jimna, the family of the Jimnites: of Jesui, the family of the Jesuites: of Beriah, the family of the Beriites.}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| Of the sons of Beriah: of Heber, the family of the Heberites: of Malchiel, the family of the Malchielites.}} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| And the name of the daughter of Asher was Sarah.}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| These are the families of the sons of Asher according to those that were numbered of them; who were fifty and three thousand and four hundred.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| Of the sons of Naphtali after their families: of Jahzeel, the family of the Jahzeelites: of Guni, the family of the Gunites:}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| Of Jezer, the family of the Jezerites: of Shillem, the family of the Shillemites.}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| These are the families of Naphtali according to their families: and they that were numbered of them were forty and five thousand and four hundred.}} <small>{{font|color=#0000FF|51}}</small>{{font|color=#888800| These were the numbered of the children of Israel, six hundred thousand and a thousand seven hundred and thirty.}} <small>{{font|color=#0000FF|52}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|53}}</small>{{font|color=#888800| Unto these the land shall be divided for an inheritance according to the number of names.}} <small>{{font|color=#0000FF|54}}</small>{{font|color=#888800| To many thou shalt give the more inheritance, and to few thou shalt give the less inheritance: to every one shall his inheritance be given according to those that were numbered of him.}} <small>{{font|color=#0000FF|55}}</small>{{font|color=#888800| Notwithstanding the land shall be divided by lot: according to the names of the tribes of their fathers they shall inherit.}} <small>{{font|color=#0000FF|56}}</small>{{font|color=#888800| According to the lot shall the possession thereof be divided between many and few.}} <small>{{font|color=#0000FF|57}}</small>{{font|color=#888800| And these are they that were numbered of the Levites after their families: of Gershon, the family of the Gershonites: of Kohath, the family of the Kohathites: of Merari, the family of the Merarites.}} <small>{{font|color=#0000FF|58}}</small>{{font|color=#888800| These are the families of the Levites: the family of the Libnites, the family of the Hebronites, the family of the Mahlites, the family of the Mushites, the family of the Korathites. And Kohath begat Amram.}} <small>{{font|color=#0000FF|59}}</small>{{font|color=#888800| And the name of Amram's wife was Jochebed, the daughter of Levi, whom her mother bare to Levi in Egypt: and she bare unto Amram Aaron and Moses, and Miriam their sister.}} <small>{{font|color=#0000FF|60}}</small>{{font|color=#888800| And unto Aaron was born Nadab, and Abihu, Eleazar, and Ithamar.}} <small>{{font|color=#0000FF|61}}</small>{{font|color=#888800| And Nadab and Abihu died, when they offered strange fire before the LORD.}} <small>{{font|color=#0000FF|62}}</small>{{font|color=#888800| And those that were numbered of them were twenty and three thousand, all males from a month old and upward: for they were not numbered among the children of Israel, because there was no inheritance given them among the children of Israel.}} <small>{{font|color=#0000FF|63}}</small>{{font|color=#888800| These are they that were numbered by Moses and Eleazar the priest, who numbered the children of Israel in the plains of Moab by Jordan near Jericho.}} <small>{{font|color=#0000FF|64}}</small>{{font|color=#888800| But among these there was not a man of them whom Moses and Aaron the priest numbered, when they numbered the children of Israel in the wilderness of Sinai.}} <small>{{font|color=#0000FF|65}}</small>{{font|color=#888800| For the LORD had said of them, They shall surely die in the wilderness. And there was not left a man of them, save Caleb the son of Jephunneh, and Joshua the son of Nun.}} ==Chapter 27== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| Then came the daughters of Zelophehad, the son of Hepher, the son of Gilead, the son of Machir, the son of Manasseh, of the families of Manasseh the son of Joseph: and these are the names of his daughters; Mahlah, Noah, and Hoglah, and Milcah, and Tirzah.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And they stood before Moses, and before Eleazar the priest, and before the princes and all the congregation, by the door of the tabernacle of the congregation, saying,}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| Our father died in the wilderness, and he was not in the company of them that gathered themselves together against the LORD in the company of Korah; but died in his own sin, and had no sons.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| Why should the name of our father be done away from among his family, because he hath no son? Give unto us therefore a possession among the brethren of our father.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And Moses brought their cause before the LORD.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| The daughters of Zelophehad speak right: thou shalt surely give them a possession of an inheritance among their father's brethren; and thou shalt cause the inheritance of their father to pass unto them.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And thou shalt speak unto the children of Israel, saying, If a man die, and have no son, then ye shall cause his inheritance to pass unto his daughter.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And if he have no daughter, then ye shall give his inheritance unto his brethren.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And if he have no brethren, then ye shall give his inheritance unto his father's brethren.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And if his father have no brethren, then ye shall give his inheritance unto his kinsman that is next to him of his family, and he shall possess it: and it shall be unto the children of Israel a statute of judgment, as the LORD commanded Moses.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And the LORD said unto Moses, Get thee up into this mount Abarim, and see the land which I have given unto the children of Israel.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And when thou hast seen it, thou also shalt be gathered unto thy people, as Aaron thy brother was gathered.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| For ye rebelled against my commandment in the desert of Zin, in the strife of the congregation, to sanctify me at the water before their eyes: that is the water of Meribah in Kadesh in the wilderness of Zin.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And Moses spake unto the LORD, saying,}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| Let the LORD, the God of the spirits of all flesh, set a man over the congregation,}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| Which may go out before them, and which may go in before them, and which may lead them out, and which may bring them in; that the congregation of the LORD be not as sheep which have no shepherd.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And the LORD said unto Moses, Take thee Joshua the son of Nun, a man in whom is the spirit, and lay thine hand upon him;}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And set him before Eleazar the priest, and before all the congregation; and give him a charge in their sight.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And thou shalt put some of thine honour upon him, that all the congregation of the children of Israel may be obedient.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And he shall stand before Eleazar the priest, who shall ask counsel for him after the judgment of Urim before the LORD: at his word shall they go out, and at his word they shall come in, both he, and all the children of Israel with him, even all the congregation.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And Moses did as the LORD commanded him: and he took Joshua, and set him before Eleazar the priest, and before all the congregation:}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| And he laid his hands upon him, and gave him a charge, as the LORD commanded by the hand of Moses.}} ==Chapter 28== <small>{{font|color=#0000FF|1}}</small>{{font|color=#880088| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#880088| Command the children of Israel, and say unto them, My offering, and my bread for my sacrifices made by fire, for a sweet savour unto me, shall ye observe to offer unto me in their due season.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#880088| And thou shalt say unto them, This is the offering made by fire which ye shall offer unto the LORD; two lambs of the first year without spot day by day, for a continual burnt offering.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#880088| The one lamb shalt thou offer in the morning, and the other lamb shalt thou offer at even;}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#880088| And a tenth part of an ephah of flour for a meat offering, mingled with the fourth part of an hin of beaten oil.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#880088| It is a continual burnt offering, which was ordained in mount Sinai for a sweet savour, a sacrifice made by fire unto the LORD.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#880088| And the drink offering thereof shall be the fourth part of an hin for the one lamb: in the holy place shalt thou cause the strong wine to be poured unto the LORD for a drink offering.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#880088| And the other lamb shalt thou offer at even: as the meat offering of the morning, and as the drink offering thereof, thou shalt offer it, a sacrifice made by fire, of a sweet savour unto the LORD.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#880088| And on the sabbath day two lambs of the first year without spot, and two tenth deals of flour for a meat offering, mingled with oil, and the drink offering thereof:}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#880088| This is the burnt offering of every sabbath, beside the continual burnt offering, and his drink offering.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#880088| And in the beginnings of your months ye shall offer a burnt offering unto the LORD; two young bullocks, and one ram, seven lambs of the first year without spot;}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#880088| And three tenth deals of flour for a meat offering, mingled with oil, for one bullock; and two tenth deals of flour for a meat offering, mingled with oil, for one ram;}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#880088| And a several tenth deal of flour mingled with oil for a meat offering unto one lamb; for a burnt offering of a sweet savour, a sacrifice made by fire unto the LORD.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#880088| And their drink offerings shall be half an hin of wine unto a bullock, and the third part of an hin unto a ram, and a fourth part of an hin unto a lamb: this is the burnt offering of every month throughout the months of the year.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#880088| And one kid of the goats for a sin offering unto the LORD shall be offered, beside the continual burnt offering, and his drink offering.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#880088| And in the fourteenth day of the first month is the passover of the LORD.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#880088| And in the fifteenth day of this month is the feast: seven days shall unleavened bread be eaten.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#880088| In the first day shall be an holy convocation; ye shall do no manner of servile work therein:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#880088| But ye shall offer a sacrifice made by fire for a burnt offering unto the LORD; two young bullocks, and one ram, and seven lambs of the first year: they shall be unto you without blemish:}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#880088| And their meat offering shall be of flour mingled with oil: three tenth deals shall ye offer for a bullock, and two tenth deals for a ram;}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#880088| A several tenth deal shalt thou offer for every lamb, throughout the seven lambs:}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#880088| And one goat for a sin offering, to make an atonement for you. }} <small>{{font|color=#0000FF|23}}</small>{{font|color=#880088| Ye shall offer these beside the burnt offering in the morning, which is for a continual burnt offering.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#880088| After this manner ye shall offer daily, throughout the seven days, the meat of the sacrifice made by fire, of a sweet savour unto the LORD: it shall be offered beside the continual burnt offering, and his drink offering.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#880088| And on the seventh day ye shall have an holy convocation; ye shall do no servile work.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#880088| Also in the day of the firstfruits, when ye bring a new meat offering unto the LORD, after your weeks be out, ye shall have an holy convocation; ye shall do no servile work:}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#880088| But ye shall offer the burnt offering for a sweet savour unto the LORD; two young bullocks, one ram, seven lambs of the first year;}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#880088| And their meat offering of flour mingled with oil, three tenth deals unto one bullock, two tenth deals unto one ram,}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#880088| A several tenth deal unto one lamb, throughout the seven lambs;}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#880088| And one kid of the goats, to make an atonement for you. }} <small>{{font|color=#0000FF|31}}</small>{{font|color=#880088| Ye shall offer them beside the continual burnt offering, and his meat offering, (they shall be unto you without blemish) and their drink offerings.}} ==Chapter 29== <small>{{font|color=#0000FF|1}}</small>{{font|color=#880088| And in the seventh month, on the first day of the month, ye shall have an holy convocation; ye shall do no servile work it is a day of blowing the trumpets unto you.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#880088| And ye shall offer a burnt offering for a sweet savour unto the LORD; one young bullock, one ram, and seven lambs of the first year without blemish:}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#880088| And their meat offering shall be of flour mingled with oil, three tenth deals for a bullock, and two tenth deals for a ram,}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#880088| And one tenth deal for one lamb, throughout the seven lambs:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#880088| And one kid of the goats for a sin offering, to make an atonement for you:}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#880088| Beside the burnt offering of the month, and his meat offering, and the daily burnt offering, and his meat offering, and their drink offerings, according unto their manner, for a sweet savour, a sacrifice made by fire unto the LORD.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#880088| And ye shall have on the tenth day of this seventh month an holy convocation; and ye shall afflict your souls: ye shall not do any work therein:}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#880088| But ye shall offer a burnt offering unto the LORD for a sweet savour; one young bullock, one ram, and seven lambs of the first year; they shall be unto you without blemish:}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#880088| And their meat offering shall be of flour mingled with oil, three tenth deals to a bullock, and two tenth deals to one ram,}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#880088| A several tenth deal for one lamb, throughout the seven lambs:}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#880088| One kid of the goats for a sin offering; beside the sin offering of atonement, and the continual burnt offering, and the meat offering of it, and their drink offerings.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#880088| And on the fifteenth day of the seventh month ye shall have an holy convocation; ye shall do no servile work, and ye shall keep a feast unto the LORD seven days:}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#880088| And ye shall offer a burnt offering, a sacrifice made by fire, of a sweet savour unto the LORD; thirteen young bullocks, two rams, and fourteen lambs of the first year; they shall be without blemish:}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#880088| And their meat offering shall be of flour mingled with oil, three tenth deals unto every bullock of the thirteen bullocks, two tenth deals to each ram of the two rams,}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#880088| And a several tenth deal to each lamb of the fourteen lambs:}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#880088| And one kid of the goats for a sin offering; beside the continual burnt offering, his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#880088| And on the second day ye shall offer twelve young bullocks, two rams, fourteen lambs of the first year without spot:}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#880088| And one kid of the goats for a sin offering; beside the continual burnt offering, and the meat offering thereof, and their drink offerings.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#880088| And on the third day eleven bullocks, two rams, fourteen lambs of the first year without blemish;}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, and his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#880088| And on the fourth day ten bullocks, two rams, and fourteen lambs of the first year without blemish:}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#880088| Their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#880088| And one kid of the goats for a sin offering; beside the continual burnt offering, his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#880088| And on the fifth day nine bullocks, two rams, and fourteen lambs of the first year without spot:}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, and his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#880088| And on the sixth day eight bullocks, two rams, and fourteen lambs of the first year without blemish:}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#880088| And on the seventh day seven bullocks, two rams, and fourteen lambs of the first year without blemish:}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#880088| And their meat offering and their drink offerings for the bullocks, for the rams, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#880088| On the eighth day ye shall have a solemn assembly: ye shall do no servile work therein:}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#880088| But ye shall offer a burnt offering, a sacrifice made by fire, of a sweet savour unto the LORD: one bullock, one ram, seven lambs of the first year without blemish:}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#880088| Their meat offering and their drink offerings for the bullock, for the ram, and for the lambs, shall be according to their number, after the manner:}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#880088| And one goat for a sin offering; beside the continual burnt offering, and his meat offering, and his drink offering.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#880088| These things ye shall do unto the LORD in your set feasts, beside your vows, and your freewill offerings, for your burnt offerings, and for your meat offerings, and for your drink offerings, and for your peace offerings.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#880088| And Moses told the children of Israel according to all that the LORD commanded Moses.}} ==Chapter 30== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And Moses spake unto the heads of the tribes concerning the children of Israel, saying, This is the thing which the LORD hath commanded.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| If a man vow a vow unto the LORD, or swear an oath to bind his soul with a bond; he shall not break his word, he shall do according to all that proceedeth out of his mouth.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| If a woman also vow a vow unto the LORD, and bind herself by a bond, being in her father's house in her youth;}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And her father hear her vow, and her bond wherewith she hath bound her soul, and her father shall hold his peace at her; then all her vows shall stand, and every bond wherewith she hath bound her soul shall stand.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| But if her father disallow her in the day that he heareth; not any of her vows, or of her bonds wherewith she hath bound her soul, shall stand: and the LORD shall forgive her, because her father disallowed her.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And if she had at all an husband, when she vowed, or uttered ought out of her lips, wherewith she bound her soul;}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And her husband heard it, and held his peace at her in the day that he heard it: then her vows shall stand, and her bonds wherewith she bound her soul shall stand.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| But if her husband disallowed her on the day that he heard it; then he shall make her vow which she vowed, and that which she uttered with her lips, wherewith she bound her soul, of none effect: and the LORD shall forgive her.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| But every vow of a widow, and of her that is divorced, wherewith they have bound their souls, shall stand against her.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And if she vowed in her husband's house, or bound her soul by a bond with an oath;}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And her husband heard it, and held his peace at her, and disallowed her not: then all her vows shall stand, and every bond wherewith she bound her soul shall stand.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| But if her husband hath utterly made them void on the day he heard them; then whatsoever proceeded out of her lips concerning her vows, or concerning the bond of her soul, shall not stand: her husband hath made them void; and the LORD shall forgive her.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| Every vow, and every binding oath to afflict the soul, her husband may establish it, or her husband may make it void.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| But if her husband altogether hold his peace at her from day to day; then he establisheth all her vows, or all her bonds, which are upon her: he confirmeth them, because he held his peace at her in the day that he heard them.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| But if he shall any ways make them void after that he hath heard them; then he shall bear her iniquity.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| These are the statutes, which the LORD commanded Moses, between a man and his wife, between the father and his daughter, being yet in her youth in her father's house.}} ==Chapter 31== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Avenge the children of Israel of the Midianites: afterward shalt thou be gathered unto thy people.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And Moses spake unto the people, saying, Arm some of yourselves unto the war, and let them go against the Midianites, and avenge the LORD of Midian.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| Of every tribe a thousand, throughout all the tribes of Israel, shall ye send to the war.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| So there were delivered out of the thousands of Israel, a thousand of every tribe, twelve thousand armed for war.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And Moses sent them to the war, a thousand of every tribe, them and Phinehas the son of Eleazar the priest, to the war, with the holy instruments, and the trumpets to blow in his hand.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And they warred against the Midianites, as the LORD commanded Moses; and they slew all the males.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And they slew the kings of Midian, beside the rest of them that were slain; namely, Evi, and Rekem, and Zur, and Hur, and Reba, five kings of Midian: Balaam also the son of Beor they slew with the sword.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And the children of Israel took all the women of Midian captives, and their little ones, and took the spoil of all their cattle, and all their flocks, and all their goods.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And they burnt all their cities wherein they dwelt, and all their goodly castles, with fire.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And they took all the spoil, and all the prey, both of men and of beasts.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And they brought the captives, and the prey, and the spoil, unto Moses, and Eleazar the priest, and unto the congregation of the children of Israel, unto the camp at the plains of Moab, which are by Jordan near Jericho.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And Moses, and Eleazar the priest, and all the princes of the congregation, went forth to meet them without the camp.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| And Moses was wroth with the officers of the host, with the captains over thousands, and captains over hundreds, which came from the battle.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| And Moses said unto them, Have ye saved all the women alive?}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| Behold, these caused the children of Israel, through the counsel of Balaam, to commit trespass against the LORD in the matter of Peor, and there was a plague among the congregation of the LORD.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| Now therefore kill every male among the little ones, and kill every woman that hath known man by lying with him.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| But all the women children, that have not known a man by lying with him, keep alive for yourselves.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And do ye abide without the camp seven days: whosoever hath killed any person, and whosoever hath touched any slain, purify both yourselves and your captives on the third day, and on the seventh day.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And purify all your raiment, and all that is made of skins, and all work of goats' hair, and all things made of wood.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| And Eleazar the priest said unto the men of war which went to the battle, This is the ordinance of the law which the LORD commanded Moses;}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| Only the gold, and the silver, the brass, the iron, the tin, and the lead,}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| Every thing that may abide the fire, ye shall make it go through the fire, and it shall be clean: nevertheless it shall be purified with the water of separation: and all that abideth not the fire ye shall make go through the water.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| And ye shall wash your clothes on the seventh day, and ye shall be clean, and afterward ye shall come into the camp.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| Take the sum of the prey that was taken, both of man and of beast, thou, and Eleazar the priest, and the chief fathers of the congregation:}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And divide the prey into two parts; between them that took the war upon them, who went out to battle, and between all the congregation:}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| And levy a tribute unto the LORD of the men of war which went out to battle: one soul of five hundred, both of the persons, and of the beeves, and of the asses, and of the sheep:}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| Take it of their half, and give it unto Eleazar the priest, for an heave offering of the LORD.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| And of the children of Israel's half, thou shalt take one portion of fifty, of the persons, of the beeves, of the asses, and of the flocks, of all manner of beasts, and give them unto the Levites, which keep the charge of the tabernacle of the LORD.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| And Moses and Eleazar the priest did as the LORD commanded Moses.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And the booty, being the rest of the prey which the men of war had caught, was six hundred thousand and seventy thousand and five thousand sheep,}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| And threescore and twelve thousand beeves,}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| And threescore and one thousand asses,}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#888800| And thirty and two thousand persons in all, of women that had not known man by lying with him.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#888800| And the half, which was the portion of them that went out to war, was in number three hundred thousand and seven and thirty thousand and five hundred sheep:}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#888800| And the LORD's tribute of the sheep was six hundred and threescore and fifteen.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#888800| And the beeves were thirty and six thousand; of which the LORD's tribute was threescore and twelve.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#888800| And the asses were thirty thousand and five hundred; of which the LORD's tribute was threescore and one.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#888800| And the persons were sixteen thousand; of which the LORD's tribute was thirty and two persons.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#888800| And Moses gave the tribute, which was the LORD's heave offering, unto Eleazar the priest, as the LORD commanded Moses.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#888800| And of the children of Israel's half, which Moses divided from the men that warred,}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#888800| (Now the half that pertained unto the congregation was three hundred thousand and thirty thousand and seven thousand and five hundred sheep,}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#888800| And thirty and six thousand beeves,}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#888800| And thirty thousand asses and five hundred,}} <small>{{font|color=#0000FF|46}}</small>{{font|color=#888800| And sixteen thousand persons;)}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#888800| Even of the children of Israel's half, Moses took one portion of fifty, both of man and of beast, and gave them unto the Levites, which kept the charge of the tabernacle of the LORD; as the LORD commanded Moses.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#888800| And the officers which were over thousands of the host, the captains of thousands, and captains of hundreds, came near unto Moses:}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#888800| And they said unto Moses, Thy servants have taken the sum of the men of war which are under our charge, and there lacketh not one man of us.}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| We have therefore brought an oblation for the LORD, what every man hath gotten, of jewels of gold, chains, and bracelets, rings, earrings, and tablets, to make an atonement for our souls before the LORD.}} <small>{{font|color=#0000FF|51}}</small>{{font|color=#888800| And Moses and Eleazar the priest took the gold of them, even all wrought jewels.}} <small>{{font|color=#0000FF|52}}</small>{{font|color=#888800| And all the gold of the offering that they offered up to the LORD, of the captains of thousands, and of the captains of hundreds, was sixteen thousand seven hundred and fifty shekels.}} <small>{{font|color=#0000FF|53}}</small>{{font|color=#888800| (For the men of war had taken spoil, every man for himself.)}} <small>{{font|color=#0000FF|54}}</small>{{font|color=#888800| And Moses and Eleazar the priest took the gold of the captains of thousands and of hundreds, and brought it into the tabernacle of the congregation, for a memorial for the children of Israel before the LORD.}} ==Chapter 32== <small>{{font|color=#0000FF|1}}</small>{{font|color=#008888| Now the children of Reuben and the children of Gad had a very great multitude of cattle: }}{{font|color=#000088|and when they saw the land of Jazer, and the land of Gilead, that, behold, the place was a place for cattle;}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| The children of Gad and the children of Reuben came and spake unto Moses, and to Eleazar the priest, and unto the princes of the congregation, saying,}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| Ataroth, and Dibon, and Jazer, and Nimrah, and Heshbon, and Elealeh, and Shebam, and Nebo, and Beon,}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| Even the country which the LORD smote before the congregation of Israel, is a land for cattle, and thy servants have cattle:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#000088| Wherefore, said they, if we have found grace in thy sight, let this land be given unto thy servants for a possession, and bring us not over Jordan.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#000088| And Moses said unto the children of Gad and to the children of Reuben, Shall your brethren go to war, and shall ye sit here?}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#000088| And wherefore discourage ye the heart of the children of Israel from going over into the land which the LORD hath given them?}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#000088| Thus did your fathers, when I sent them from Kadeshbarnea to see the land.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#000088| For when they went up unto the valley of Eshcol, and saw the land, they discouraged the heart of the children of Israel, that they should not go into the land which the LORD had given them.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#000088| And the LORD's anger was kindled the same time, and he sware, saying,}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#000088| Surely none of the men that came up out of Egypt, from twenty years old and upward, shall see the land which I sware unto Abraham, unto Isaac, and unto Jacob; because they have not wholly followed me:}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#000088| Save Caleb the son of Jephunneh the Kenezite, and Joshua the son of Nun: for they have wholly followed the LORD.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#000088| And the LORD's anger was kindled against Israel, and he made them wander in the wilderness forty years, until all the generation, that had done evil in the sight of the LORD, was consumed.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#000088| And, behold, ye are risen up in your fathers' stead, an increase of sinful men, to augment yet the fierce anger of the LORD toward Israel.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#000088| For if ye turn away from after him, he will yet again leave them in the wilderness; and ye shall destroy all this people.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#008888| And they came near unto him, and said, We will build sheepfolds here for our cattle, and cities for our little ones:}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#008888| But we ourselves will go ready armed before the children of Israel, until we have brought them unto their place: and our little ones shall dwell in the fenced cities because of the inhabitants of the land.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#008888| We will not return unto our houses, until the children of Israel have inherited every man his inheritance.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#008888| For we will not inherit with them on yonder side Jordan, or forward; because our inheritance is fallen to us on this side Jordan eastward.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#008888| And Moses said unto them,}}{{font|color=#000088| If ye will do this thing, if ye will go armed before the LORD to war,}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#000088| And will go all of you armed over Jordan before the LORD, until he hath driven out his enemies from before him,}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#000088| And the land be subdued before the LORD: then afterward ye shall return, and be guiltless before the LORD, and before Israel; and this land shall be your possession before the LORD.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#000088| But if ye will not do so, behold, ye have sinned against the LORD: and be sure your sin will find you out.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#008888| Build you cities for your little ones, and folds for your sheep; and do that which hath proceeded out of your mouth.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#000088| And the children of Gad and the children of Reuben spake unto Moses, saying, Thy servants will do as my lord commandeth.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#000088| Our little ones, our wives, our flocks, and all our cattle, shall be there in the cities of Gilead:}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#000088| But thy servants will pass over, every man armed for war, before the Lord to battle, as my lord saith.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| So concerning them Moses commanded Eleazar the priest, and Joshua the son of Nun, and the chief fathers of the tribes of the children of Israel:}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| And Moses said unto them, If the children of Gad and the children of Reuben will pass with you over Jordan, every man armed to battle, before the LORD, and the land shall be subdued before you; then ye shall give them the land of Gilead for a possession:}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| But if they will not pass over with you armed, they shall have possessions among you in the land of Canaan.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| And the children of Gad and the children of Reuben answered, saying, As the LORD hath said unto thy servants, so will we do.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| We will pass over armed before the LORD into the land of Canaan, that the possession of our inheritance on this side Jordan may be our's.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| And Moses gave unto them, even to the children of Gad, and to the children of Reuben, and unto half the tribe of Manasseh the son of Joseph, the kingdom of Sihon king of the Amorites, and the kingdom of Og king of Bashan, the land, with the cities thereof in the coasts, even the cities of the country round about.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#008888| And the children of Gad built Dibon, and Ataroth, and Aroer,}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#008888| And Atroth, Shophan, and Jaazer, and Jogbehah,}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#008888| And Bethnimrah, and Bethharan, fenced cities: and folds for sheep.}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#008888| And the children of Reuben built Heshbon, and Elealeh, and Kirjathaim,}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#008888| And Nebo, and Baalmeon, (their names being changed,) and Shibmah: and gave other names unto the cities which they builded.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#000088| And the children of Machir the son of Manasseh went to Gilead, and took it, and dispossessed the Amorite which was in it.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#008888| And Moses gave Gilead unto Machir the son of Manasseh; and he dwelt therein.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#000088| And Jair the son of Manasseh went and took the small towns thereof, and called them Havothjair.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#000088| And Nobah went and took Kenath, and the villages thereof, and called it Nobah, after his own name.}} ==Chapter 33== <small>{{font|color=#0000FF|1}}</small>{{font|color=#FF8800| These are the journeys of the children of Israel, which went forth out of the land of Egypt with their armies under the hand of Moses and Aaron.}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#FF8800| And Moses wrote their goings out according to their journeys by the commandment of the LORD: and these are their journeys according to their goings out.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#FF8800| And they departed from Rameses in the first month, on the fifteenth day of the first month; on the morrow after the passover the children of Israel went out with an high hand in the sight of all the Egyptians.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#FF8800| For the Egyptians buried all their firstborn, which the LORD had smitten among them: upon their gods also the LORD executed judgments.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#FF8800| And the children of Israel removed from Rameses, and pitched in Succoth.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#FF8800| And they departed from Succoth, and pitched in Etham, which is in the edge of the wilderness.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#FF8800| And they removed from Etham, and turned again unto Pihahiroth, which is before Baalzephon: and they pitched before Migdol.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#FF8800| And they departed from before Pihahiroth, and passed through the midst of the sea into the wilderness, and went three days' journey in the wilderness of Etham, and pitched in Marah.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#FF8800| And they removed from Marah, and came unto Elim: and in Elim were twelve fountains of water, and threescore and ten palm trees; and they pitched there.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#FF8800| And they removed from Elim, and encamped by the Red sea.}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#FF8800| And they removed from the Red sea, and encamped in the wilderness of Sin.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#FF8800| And they took their journey out of the wilderness of Sin, and encamped in Dophkah.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#FF8800| And they departed from Dophkah, and encamped in Alush. }} <small>{{font|color=#0000FF|14}}</small>{{font|color=#FF8800| And they removed from Alush, and encamped at Rephidim, where was no water for the people to drink.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#FF8800| And they departed from Rephidim, and pitched in the wilderness of Sinai.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#FF8800| And they removed from the desert of Sinai, and pitched at Kibrothhattaavah.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#FF8800| And they departed from Kibrothhattaavah, and encamped at Hazeroth.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#FF8800| And they departed from Hazeroth, and pitched in Rithmah.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#FF8800| And they departed from Rithmah, and pitched at Rimmonparez.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#FF8800| And they departed from Rimmonparez, and pitched in Libnah.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#FF8800| And they removed from Libnah, and pitched at Rissah.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#FF8800| And they journeyed from Rissah, and pitched in Kehelathah.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#FF8800| And they went from Kehelathah, and pitched in mount Shapher.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#FF8800| And they removed from mount Shapher, and encamped in Haradah.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#FF8800| And they removed from Haradah, and pitched in Makheloth.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#FF8800| And they removed from Makheloth, and encamped at Tahath.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#FF8800| And they departed from Tahath, and pitched at Tarah.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#FF8800| And they removed from Tarah, and pitched in Mithcah.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#FF8800| And they went from Mithcah, and pitched in Hashmonah.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#FF8800| And they departed from Hashmonah, and encamped at Moseroth.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#FF8800| And they departed from Moseroth, and pitched in Benejaakan.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#FF8800| And they removed from Benejaakan, and encamped at Horhagidgad.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#FF8800| And they went from Horhagidgad, and pitched in Jotbathah.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#FF8800| And they removed from Jotbathah, and encamped at Ebronah.}} <small>{{font|color=#0000FF|35}}</small>{{font|color=#FF8800| And they departed from Ebronah, and encamped at Eziongaber.}} <small>{{font|color=#0000FF|36}}</small>{{font|color=#FF8800| And they removed from Eziongaber, and pitched in the wilderness of Zin, which is Kadesh.}} <small>{{font|color=#0000FF|37}}</small>{{font|color=#FF8800| And they removed from Kadesh, and pitched in mount Hor, in the edge of the land of Edom.}} <small>{{font|color=#0000FF|38}}</small>{{font|color=#FF0000| And Aaron the priest went up into mount Hor at the commandment of the LORD, and died there, in the fortieth year after the children of Israel were come out of the land of Egypt, in the first day of the fifth month.}} <small>{{font|color=#0000FF|39}}</small>{{font|color=#FF0000| And Aaron was an hundred and twenty and three years old when he died in mount Hor.}} <small>{{font|color=#0000FF|40}}</small>{{font|color=#FF8800| And king Arad the Canaanite, which dwelt in the south in the land of Canaan, heard of the coming of the children of Israel.}} <small>{{font|color=#0000FF|41}}</small>{{font|color=#FF8800| And they departed from mount Hor, and pitched in Zalmonah.}} <small>{{font|color=#0000FF|42}}</small>{{font|color=#FF8800| And they departed from Zalmonah, and pitched in Punon.}} <small>{{font|color=#0000FF|43}}</small>{{font|color=#FF8800| And they departed from Punon, and pitched in Oboth.}} <small>{{font|color=#0000FF|44}}</small>{{font|color=#FF8800| And they departed from Oboth, and pitched in Ijeabarim, in the border of Moab.}} <small>{{font|color=#0000FF|45}}</small>{{font|color=#FF8800| And they departed from Iim, and pitched in Dibongad. }} <small>{{font|color=#0000FF|46}}</small>{{font|color=#FF8800| And they removed from Dibongad, and encamped in Almondiblathaim.}} <small>{{font|color=#0000FF|47}}</small>{{font|color=#FF8800| And they removed from Almondiblathaim, and pitched in the mountains of Abarim, before Nebo.}} <small>{{font|color=#0000FF|48}}</small>{{font|color=#FF8800| And they departed from the mountains of Abarim, and pitched in the plains of Moab by Jordan near Jericho.}} <small>{{font|color=#0000FF|49}}</small>{{font|color=#FF8800| And they pitched by Jordan, from Bethjesimoth even unto Abelshittim in the plains of Moab.}} <small>{{font|color=#0000FF|50}}</small>{{font|color=#888800| And the LORD spake unto Moses in the plains of Moab by Jordan near Jericho, saying,}} <small>{{font|color=#0000FF|51}}</small>{{font|color=#888800| Speak unto the children of Israel, and say unto them, When ye are passed over Jordan into the land of Canaan;}} <small>{{font|color=#0000FF|52}}</small>{{font|color=#888800| Then ye shall drive out all the inhabitants of the land from before you, and destroy all their pictures, and destroy all their molten images, and quite pluck down all their high places:}} <small>{{font|color=#0000FF|53}}</small>{{font|color=#888800| And ye shall dispossess the inhabitants of the land, and dwell therein: for I have given you the land to possess it.}} <small>{{font|color=#0000FF|54}}</small>{{font|color=#888800| And ye shall divide the land by lot for an inheritance among your families: and to the more ye shall give the more inheritance, and to the fewer ye shall give the less inheritance: every man's inheritance shall be in the place where his lot falleth; according to the tribes of your fathers ye shall inherit.}} <small>{{font|color=#0000FF|55}}</small>{{font|color=#888800| But if ye will not drive out the inhabitants of the land from before you; then it shall come to pass, that those which ye let remain of them shall be pricks in your eyes, and thorns in your sides, and shall vex you in the land wherein ye dwell.}} <small>{{font|color=#0000FF|56}}</small>{{font|color=#888800| Moreover it shall come to pass, that I shall do unto you, as I thought to do unto them.}} ==Chapter 34== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Command the children of Israel, and say unto them, When ye come into the land of Canaan; (this is the land that shall fall unto you for an inheritance, even the land of Canaan with the coasts thereof:)}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| Then your south quarter shall be from the wilderness of Zin along by the coast of Edom, and your south border shall be the outmost coast of the salt sea eastward:}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And your border shall turn from the south to the ascent of Akrabbim, and pass on to Zin: and the going forth thereof shall be from the south to Kadeshbarnea, and shall go on to Hazaraddar, and pass on to Azmon:}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And the border shall fetch a compass from Azmon unto the river of Egypt, and the goings out of it shall be at the sea.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And as for the western border, ye shall even have the great sea for a border: this shall be your west border. }} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| And this shall be your north border: from the great sea ye shall point out for you mount Hor:}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| From mount Hor ye shall point out your border unto the entrance of Hamath; and the goings forth of the border shall be to Zedad:}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And the border shall go on to Ziphron, and the goings out of it shall be at Hazarenan: this shall be your north border.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| And ye shall point out your east border from Hazarenan to Shepham:}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| And the coast shall go down from Shepham to Riblah, on the east side of Ain; and the border shall descend, and shall reach unto the side of the sea of Chinnereth eastward:}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And the border shall go down to Jordan, and the goings out of it shall be at the salt sea: this shall be your land with the coasts thereof round about.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And Moses commanded the children of Israel, saying, This is the land which ye shall inherit by lot, which the LORD commanded to give unto the nine tribes, and to the half tribe:}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| For the tribe of the children of Reuben according to the house of their fathers, and the tribe of the children of Gad according to the house of their fathers, have received their inheritance; and half the tribe of Manasseh have received their inheritance:}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| The two tribes and the half tribe have received their inheritance on this side Jordan near Jericho eastward, toward the sunrising.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| These are the names of the men which shall divide the land unto you: Eleazar the priest, and Joshua the son of Nun.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| And ye shall take one prince of every tribe, to divide the land by inheritance.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| And the names of the men are these: Of the tribe of Judah, Caleb the son of Jephunneh.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| And of the tribe of the children of Simeon, Shemuel the son of Ammihud.}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Of the tribe of Benjamin, Elidad the son of Chislon.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| And the prince of the tribe of the children of Dan, Bukki the son of Jogli.}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| The prince of the children of Joseph, for the tribe of the children of Manasseh, Hanniel the son of Ephod.}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| And the prince of the tribe of the children of Ephraim, Kemuel the son of Shiphtan.}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the prince of the tribe of the children of Zebulun, Elizaphan the son of Parnach.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| And the prince of the tribe of the children of Issachar, Paltiel the son of Azzan.}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And the prince of the tribe of the children of Asher, Ahihud the son of Shelomi.}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| And the prince of the tribe of the children of Naphtali, Pedahel the son of Ammihud.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| These are they whom the LORD commanded to divide the inheritance unto the children of Israel in the land of Canaan.}} ==Chapter 35== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the LORD spake unto Moses in the plains of Moab by Jordan near Jericho, saying,}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| Command the children of Israel, that they give unto the Levites of the inheritance of their possession cities to dwell in; and ye shall give also unto the Levites suburbs for the cities round about them.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And the cities shall they have to dwell in; and the suburbs of them shall be for their cattle, and for their goods, and for all their beasts.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And the suburbs of the cities, which ye shall give unto the Levites, shall reach from the wall of the city and outward a thousand cubits round about.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And ye shall measure from without the city on the east side two thousand cubits, and on the south side two thousand cubits, and on the west side two thousand cubits, and on the north side two thousand cubits; and the city shall be in the midst: this shall be to them the suburbs of the cities.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| And among the cities which ye shall give unto the Levites there shall be six cities for refuge, which ye shall appoint for the manslayer, that he may flee thither: and to them ye shall add forty and two cities.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| So all the cities which ye shall give to the Levites shall be forty and eight cities: them shall ye give with their suburbs.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And the cities which ye shall give shall be of the possession of the children of Israel: from them that have many ye shall give many; but from them that have few ye shall give few: every one shall give of his cities unto the Levites according to his inheritance which he inheriteth.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| And the LORD spake unto Moses, saying,}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| Speak unto the children of Israel, and say unto them, When ye be come over Jordan into the land of Canaan;}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| Then ye shall appoint you cities to be cities of refuge for you; that the slayer may flee thither, which killeth any person at unawares.}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And they shall be unto you cities for refuge from the avenger; that the manslayer die not, until he stand before the congregation in judgment.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| And of these cities which ye shall give six cities shall ye have for refuge.}} <small>{{font|color=#0000FF|14}}</small>{{font|color=#888800| Ye shall give three cities on this side Jordan, and three cities shall ye give in the land of Canaan, which shall be cities of refuge.}} <small>{{font|color=#0000FF|15}}</small>{{font|color=#888800| These six cities shall be a refuge, both for the children of Israel, and for the stranger, and for the sojourner among them: that every one that killeth any person unawares may flee thither.}} <small>{{font|color=#0000FF|16}}</small>{{font|color=#888800| And if he smite him with an instrument of iron, so that he die, he is a murderer: the murderer shall surely be put to death.}} <small>{{font|color=#0000FF|17}}</small>{{font|color=#888800| And if he smite him with throwing a stone, wherewith he may die, and he die, he is a murderer: the murderer shall surely be put to death.}} <small>{{font|color=#0000FF|18}}</small>{{font|color=#888800| Or if he smite him with an hand weapon of wood, wherewith he may die, and he die, he is a murderer: the murderer shall surely be put to death.}} <small>{{font|color=#0000FF|19}}</small>{{font|color=#888800| The revenger of blood himself shall slay the murderer: when he meeteth him, he shall slay him.}} <small>{{font|color=#0000FF|20}}</small>{{font|color=#888800| But if he thrust him of hatred, or hurl at him by laying of wait, that he die;}} <small>{{font|color=#0000FF|21}}</small>{{font|color=#888800| Or in enmity smite him with his hand, that he die: he that smote him shall surely be put to death; for he is a murderer: the revenger of blood shall slay the murderer, when he meeteth him.}} <small>{{font|color=#0000FF|22}}</small>{{font|color=#888800| But if he thrust him suddenly without enmity, or have cast upon him any thing without laying of wait,}} <small>{{font|color=#0000FF|23}}</small>{{font|color=#888800| Or with any stone, wherewith a man may die, seeing him not, and cast it upon him, that he die, and was not his enemy, neither sought his harm:}} <small>{{font|color=#0000FF|24}}</small>{{font|color=#888800| Then the congregation shall judge between the slayer and the revenger of blood according to these judgments:}} <small>{{font|color=#0000FF|25}}</small>{{font|color=#888800| And the congregation shall deliver the slayer out of the hand of the revenger of blood, and the congregation shall restore him to the city of his refuge, whither he was fled: and he shall abide in it unto the death of the high priest, which was anointed with the holy oil.}} <small>{{font|color=#0000FF|26}}</small>{{font|color=#888800| But if the slayer shall at any time come without the border of the city of his refuge, whither he was fled;}} <small>{{font|color=#0000FF|27}}</small>{{font|color=#888800| And the revenger of blood find him without the borders of the city of his refuge, and the revenger of blood kill the slayer; he shall not be guilty of blood:}} <small>{{font|color=#0000FF|28}}</small>{{font|color=#888800| Because he should have remained in the city of his refuge until the death of the high priest: but after the death of the high priest the slayer shall return into the land of his possession.}} <small>{{font|color=#0000FF|29}}</small>{{font|color=#888800| So these things shall be for a statute of judgment unto you throughout your generations in all your dwellings.}} <small>{{font|color=#0000FF|30}}</small>{{font|color=#888800| Whoso killeth any person, the murderer shall be put to death by the mouth of witnesses: but one witness shall not testify against any person to cause him to die.}} <small>{{font|color=#0000FF|31}}</small>{{font|color=#888800| Moreover ye shall take no satisfaction for the life of a murderer, which is guilty of death: but he shall be surely put to death.}} <small>{{font|color=#0000FF|32}}</small>{{font|color=#888800| And ye shall take no satisfaction for him that is fled to the city of his refuge, that he should come again to dwell in the land, until the death of the priest.}} <small>{{font|color=#0000FF|33}}</small>{{font|color=#888800| So ye shall not pollute the land wherein ye are: for blood it defileth the land: and the land cannot be cleansed of the blood that is shed therein, but by the blood of him that shed it.}} <small>{{font|color=#0000FF|34}}</small>{{font|color=#888800| Defile not therefore the land which ye shall inhabit, wherein I dwell: for I the LORD dwell among the children of Israel.}} ==Chapter 36== <small>{{font|color=#0000FF|1}}</small>{{font|color=#888800| And the chief fathers of the families of the children of Gilead, the son of Machir, the son of Manasseh, of the families of the sons of Joseph, came near, and spake before Moses, and before the princes, the chief fathers of the children of Israel:}} <small>{{font|color=#0000FF|2}}</small>{{font|color=#888800| And they said, The LORD commanded my lord to give the land for an inheritance by lot to the children of Israel: and my lord was commanded by the LORD to give the inheritance of Zelophehad our brother unto his daughters.}} <small>{{font|color=#0000FF|3}}</small>{{font|color=#888800| And if they be married to any of the sons of the other tribes of the children of Israel, then shall their inheritance be taken from the inheritance of our fathers, and shall be put to the inheritance of the tribe whereunto they are received: so shall it be taken from the lot of our inheritance.}} <small>{{font|color=#0000FF|4}}</small>{{font|color=#888800| And when the jubile of the children of Israel shall be, then shall their inheritance be put unto the inheritance of the tribe whereunto they are received: so shall their inheritance be taken away from the inheritance of the tribe of our fathers.}} <small>{{font|color=#0000FF|5}}</small>{{font|color=#888800| And Moses commanded the children of Israel according to the word of the LORD, saying, The tribe of the sons of Joseph hath said well.}} <small>{{font|color=#0000FF|6}}</small>{{font|color=#888800| This is the thing which the LORD doth command concerning the daughters of Zelophehad, saying, Let them marry to whom they think best; only to the family of the tribe of their father shall they marry.}} <small>{{font|color=#0000FF|7}}</small>{{font|color=#888800| So shall not the inheritance of the children of Israel remove from tribe to tribe: for every one of the children of Israel shall keep himself to the inheritance of the tribe of his fathers.}} <small>{{font|color=#0000FF|8}}</small>{{font|color=#888800| And every daughter, that possesseth an inheritance in any tribe of the children of Israel, shall be wife unto one of the family of the tribe of her father, that the children of Israel may enjoy every man the inheritance of his fathers.}} <small>{{font|color=#0000FF|9}}</small>{{font|color=#888800| Neither shall the inheritance remove from one tribe to another tribe; but every one of the tribes of the children of Israel shall keep himself to his own inheritance.}} <small>{{font|color=#0000FF|10}}</small>{{font|color=#888800| Even as the LORD commanded Moses, so did the daughters of Zelophehad:}} <small>{{font|color=#0000FF|11}}</small>{{font|color=#888800| For Mahlah, Tirzah, and Hoglah, and Milcah, and Noah, the daughters of Zelophehad, were married unto their father's brothers' sons:}} <small>{{font|color=#0000FF|12}}</small>{{font|color=#888800| And they were married into the families of the sons of Manasseh the son of Joseph, and their inheritance remained in the tribe of the family of their father.}} <small>{{font|color=#0000FF|13}}</small>{{font|color=#888800| These are the commandments and the judgments, which the LORD commanded by the hand of Moses unto the children of Israel in the plains of Moab by Jordan near Jericho.}} [[Category:Documentary hypothesis]] 1ipzlm7qo3rc5iqe9mujymrklbdhhi5 0 57577 2416467 2391817 2022-08-19T09:36:01Z 122.173.30.234 /* References */ wikitext text/x-wiki [[Category:Computer science]] [[Category:Mathematics]] [[Category:Science]] [[Category:Statistics]] {{it}} {{secondary}} {{75%done}} {{clear}} Data are values of qualitative or quantitative variables, belonging to a set of items.<ref>[[Wikipedia: Data]]</ref> ==Introduction== One way to think about data is to consider how some of the organs of our bodies detect changes to their environment. These are called "Sensory organs", and through our Sensory organs, we are aware of all aspects of our environment. The senses include sight, hearing, smell, taste and touch. We make decisions based on images, sounds, odors, tastes and touches that are received by our eyes, ears, nose, tongue and skin. All images, sounds, odors, tastes and touches from our environment may be thought of as "Data". ==Data== In science all images, sounds, odors, tastes and touches of our environment convert to "Alphabet", "Numbers "and "Symbols", therefore all facts and events may be represented by (A...Z),(0...9) and (@,[,\,<, >, ?, *, %, ...) or some combination thereof. For example: * John is 12 years old (combination of "Alphabet" and "Numbers) * 2+3=5 (combination of "Numbers "and "Symbols") * My email address is myname@somedomain.com (combination of "Alphabet" and "Symbols") * Etc. In computer science, "Alphabet", "Numbers " and "Symbols" convert to a unique numeric code that is assigned to each letter, number, or symbol. For example: *Alphabet (A,...Z): *unique code for " A " is → 01000001 in binary or 65 in decimal *..... *unique code for " Z " is → 01011010 in binary or 90 in decimal Numbers (0,…9) *unique code for 0 is → 00110000 in binary or 48 in decimal *..... *unique code for 9 is → 00111001 in binary or 57 in decimal Symbols (@,[,\,<,>,?,*,%,…): *unique code for " @ " is → 01000000 in binary or 64 in decimal *unique code for " [ " is → 01011011 in binary or 91 in decimal *..... {{RoundBoxTop|theme=2}} <big> The definition of facts and events by "Alphabet","Numbers "and "Symbols" in computer science are set of unique values known as "Data".</big> {{RoundBoxBottom}} Datum is the singular form of data. Data typically consists of measurements often recorded as a collection of numbers or characters, but may also include images, sounds, odors, tastes and touches In science, data is obtained systematically in order to test theories and hypotheses. ==Information== Data itself is not associated with any context, and therefore it has no inherent meaning. For example, the number 12 is an element of datum, on its own it has no meaning. However when placed into the context of "John is 12 years old" we understand the meaning. 12 → "John is 12 years old" → 12 years old.In the example: * 12 is a datum * "John is 12 years old" is a context * 12 years old is meaning 12 → "there are 12 months in a year" → 12 months * 12 is a datum * "there are 12 months in a year" is a context * 12 months is meaning Therefore: '''datum → context → aware ''' In science there are a lot of datum (data) and with a context then it give meaning to the results. Therefore: '''data → context → result '' In computer science, data converted with a mathematical or logical operator is referred to as a process and generates information. Therefore: ''' data → process → information ''' {{RoundBoxTop|theme=2}} <big> Data converted by a process is called "Information".</big> {{RoundBoxBottom}} For example: 2+3=5 or {2,3} → + → 5 or 00000010,00000011 → + → 00000101 (unique binary values) {{RoundBoxTop|theme=4}} Note: Information from one process may become data for the next process. {{RoundBoxBottom}} Data1 → process1 → information1 ... (information1) data2 → process2 → information2 ... (information2) data3 → process3 → information3 ... [[File:Data and information.png|center|Distinction between "Data" and "Information"]] ==See also== {{wikipedia}} * [[Wikipedia: ASCII]] * [[Data analysis]] * [[Data screening]] * [[Missing data]] == References == {{reflist}}2. [https://caesaranalytics.com/ Caesar Analytics] prruq09dzked3u71zvhjk5ir9egjemq 2416468 2416467 2022-08-19T09:37:40Z 122.173.30.234 /* References */ wikitext text/x-wiki [[Category:Computer science]] [[Category:Mathematics]] [[Category:Science]] [[Category:Statistics]] {{it}} {{secondary}} {{75%done}} {{clear}} Data are values of qualitative or quantitative variables, belonging to a set of items.<ref>[[Wikipedia: Data]]</ref> ==Introduction== One way to think about data is to consider how some of the organs of our bodies detect changes to their environment. These are called "Sensory organs", and through our Sensory organs, we are aware of all aspects of our environment. The senses include sight, hearing, smell, taste and touch. We make decisions based on images, sounds, odors, tastes and touches that are received by our eyes, ears, nose, tongue and skin. All images, sounds, odors, tastes and touches from our environment may be thought of as "Data". ==Data== In science all images, sounds, odors, tastes and touches of our environment convert to "Alphabet", "Numbers "and "Symbols", therefore all facts and events may be represented by (A...Z),(0...9) and (@,[,\,<, >, ?, *, %, ...) or some combination thereof. For example: * John is 12 years old (combination of "Alphabet" and "Numbers) * 2+3=5 (combination of "Numbers "and "Symbols") * My email address is myname@somedomain.com (combination of "Alphabet" and "Symbols") * Etc. In computer science, "Alphabet", "Numbers " and "Symbols" convert to a unique numeric code that is assigned to each letter, number, or symbol. For example: *Alphabet (A,...Z): *unique code for " A " is → 01000001 in binary or 65 in decimal *..... *unique code for " Z " is → 01011010 in binary or 90 in decimal Numbers (0,…9) *unique code for 0 is → 00110000 in binary or 48 in decimal *..... *unique code for 9 is → 00111001 in binary or 57 in decimal Symbols (@,[,\,<,>,?,*,%,…): *unique code for " @ " is → 01000000 in binary or 64 in decimal *unique code for " [ " is → 01011011 in binary or 91 in decimal *..... {{RoundBoxTop|theme=2}} <big> The definition of facts and events by "Alphabet","Numbers "and "Symbols" in computer science are set of unique values known as "Data".</big> {{RoundBoxBottom}} Datum is the singular form of data. Data typically consists of measurements often recorded as a collection of numbers or characters, but may also include images, sounds, odors, tastes and touches In science, data is obtained systematically in order to test theories and hypotheses. ==Information== Data itself is not associated with any context, and therefore it has no inherent meaning. For example, the number 12 is an element of datum, on its own it has no meaning. However when placed into the context of "John is 12 years old" we understand the meaning. 12 → "John is 12 years old" → 12 years old.In the example: * 12 is a datum * "John is 12 years old" is a context * 12 years old is meaning 12 → "there are 12 months in a year" → 12 months * 12 is a datum * "there are 12 months in a year" is a context * 12 months is meaning Therefore: '''datum → context → aware ''' In science there are a lot of datum (data) and with a context then it give meaning to the results. Therefore: '''data → context → result '' In computer science, data converted with a mathematical or logical operator is referred to as a process and generates information. Therefore: ''' data → process → information ''' {{RoundBoxTop|theme=2}} <big> Data converted by a process is called "Information".</big> {{RoundBoxBottom}} For example: 2+3=5 or {2,3} → + → 5 or 00000010,00000011 → + → 00000101 (unique binary values) {{RoundBoxTop|theme=4}} Note: Information from one process may become data for the next process. {{RoundBoxBottom}} Data1 → process1 → information1 ... (information1) data2 → process2 → information2 ... (information2) data3 → process3 → information3 ... [[File:Data and information.png|center|Distinction between "Data" and "Information"]] ==See also== {{wikipedia}} * [[Wikipedia: ASCII]] * [[Data analysis]] * [[Data screening]] * [[Missing data]] == References == {{reflist}}2. [https://caesaranalytics.com/ Caesar Analytics: Data] 5ofmjsudjx1fffrd9xe7ju3co4eg4kz 0 105038 2416208 2135384 2022-08-18T17:09:26Z Margob28 2943256 I have created a content on the topic: Psycholinguistics wikitext text/x-wiki {{linguistics}} {{tertiary}} {{nearlythere}} {{wikipedia}} '''Defining Psycholinguistics''' Psycholinguistics is an approach to language study just like phonology, morphology, syntax, semantics, pragmatics, and sociolinguistics. Considering the morphological components of the word "Psycholinguistics", two terms are traceable-- "psycho" and "linguistics". The former term, which is "psycho" has it's etymology from the Greek word, "'''psȳchḗ"''' which means "mind". The later concept, "linguistics", has language as its root. Hence, the concept of Psycholinguistics simply describes the relationship between '''''language''''' and the '''''human mind'''''. Human is emphasised in the description of the concept of Psycholinguistics because of the peculiarity of language as an attribute of humans, animals use paralinguistic cues. '''Scope of Psycholinguistics''' '''Psycholinguistics''' is the study of language from a [[Psychology|psychological]] perspective. To be more precise, studying the psychology of language in relation to its use is the core of Psycholinguistics. Defining the scope of Psycholinguistics revolves around the following questions: i. how do language start? ii. what are the components of language? iii. what are the processes of communication? ===Participants=== * [[Psycholinguistics/Participants|List of Participants]] ===Foundations=== * [[Psycholinguistics/History and Major Theories|History and Major Theories]] - [[user:Charliekramer | Charlie Kramer]] * [[Psycholinguistics/Language and Thought|Language and Thought]] - [[User:Midas|Midas]] * [[Psycholinguistics/Language and the Brain|Language and the Brain]] - [[User:Jchetwynd|Jennifer Chetwynd]] ** [[Psycholinguistics/Hemispheric Lateralization of Language|Hemispheric Lateralization of Language]] - [[User:Jennica Lounsbury|Jennica Lounsbury]] * [[Psycholinguistics/Connectionist Models|Connectionist Models]] - [[user:bcorrigan|Ben Corrigan]] ===Language Perception=== * Speech Perception ** [[Psycholinguistics/Components of Speech|Components of Speech]] - [[user:Chantel | Chantel Deck]] ** [[Psycholinguistics/Articulatory Phonetics|Articulatory Phonetics]] - [[user:Aericanwizard | Garrett Nicolai]] ** [[Psycholinguistics/Acoustic Phonetics|Acoustic Phonetics]] - [[User:grechukk|Katelyn Grechuk]] ** [[Psycholinguistics/Perception of Continuous Speech|Perception of Continuous Speech]] - [[user:Akvdh | Andrea VanderHeyden]] ** [[Psycholinguistics/Models of Speech Perception|Models of Speech Perception]] - [[user:Megsmith|Meg Smith]] **[[Psycholinguistics/Neural Basis of Speech Perception|Neural Basis of Speech Perception]] - [[User:Suzanne_Walsh11|Suzanne Walsh]] * [[Psycholinguistics/Reading|Reading]]- [[User:Jnvincent|Jacqueline Vincent]] * [[Psycholinguistics/Morphology|Morphology]] - [[user:Starfield|Jillian Baker]] * Words, Meaning, and Word Recognition ** [[Psycholinguistics/What is a Word?|What is a Word?]] - [[user:Egreer|Eleanor Greer]] ** [[Psycholinguistics/The Mental Lexicon|The Mental Lexicon]] - [[user:Nathanael Crawford|Nathanael Crawford]] ** [[Psycholinguistics/Semantics in the Brain|Semantics in the Brain]] - [[user:Hdgallant|Heather Gallant]] ** [[Psycholinguistics/Neural Bases of Lexical Access|Neural Bases of Lexical Access]] - [[user:Ramz Aziz|Ramz Aziz]] * Sentence Processing (Syntax) ** [[Psycholinguistics/Formal Models of Grammar|Formal Models of Grammar]] ** [[Psycholinguistics/Parsing|Parsing]] - [[user:Ectoombs|Elaine Toombs]] ** [[Psycholinguistics/Individual Differences in Syntactic Processing|Individual Differences in Syntactic Processing]] - [[User:Ju wen Ou|Ju wen Ou]] ** [[Psycholinguistics/Syntax in the Brain|Syntax in the Brain]] - [[user:Alicataroo]] * [[Psycholinguistics/Prosody|Prosody]] - [[user:keegancolville|Keegan Colville]] * [[Psycholinguistics/Discourse|Discourse]] - [[user:CatherineMcIntyre| Catherine McIntyre]] ===Language and Other Human Cognitive Artefacts=== * [[Psycholinguistics/Language and Music|Language and Music]] - [[User:cbrousseau|Claire Brousseau]] * [[Psycholinguistics/Language and Arithmetic|Language and Arithmetic]] - [[User:Samanathaarseneau| Samantha Arseneau]] ===Language Output=== *[[Psycholinguistics/Speech Errors|Speech Errors]] - [[user:Johnboudreau|John Boudreau]] * [[Psycholinguistics/Chronometry|Chronometry]] - [[user:B-Parks|Ben Parker]] * [[Psycholinguistics/Models of Speech Production|Models of Speech Production]] - [[user:Moraghjang|Moragh Jang]] * [[Psycholinguistics/Neural Components of Speech Production|Neural Components of Speech Production]] - [[user:em736458|Emma Horner]] * [[Psycholinguistics/Gesture|Gesture]] - [[user:Kfraser|Kathryn Fraser]] * [[Psycholinguistics/Writing|Writing]] - [[user:Hilaryy|Hilary Van Loon]] ===Language Acquisition=== * [[Psycholinguistics/Theories and Models of Language Acquisition|Theories and Models of Language Acquisition]]- [[User:Megan J Mac Donald|Megan J Mac Donald]] * [[Psycholinguistics/Pidgins, Creoles, and Home Sign|Pidgins, Creoles, and Home Sign]] - [[User:Emilypmcguire|Emily McGuire]] * [[Psycholinguistics/Development of Speech Perception|Development of Speech Perception]] - [[User:Bengisug|Bengisu Gonul]] * [[Psycholinguistics/Development of Speech Production|Development of Speech Production]] - [[User:Jennfleur|Jennifer Browning]] * [[Psycholinguistics/Acquisition of More than One Language|Acquisition of More than One Language]] - [[User:Doaney|Laura Doane]] * [[Psycholinguistics/A comparison of language processing in Chinese and English|A comparison of language processing in Chinese and English]] - [[user:minsun|Minzhou Sun]] * [[Psycholinguistics/Neural Basis of Multi-Lingualism|Neural Basis of Multi-Lingualism]] - [[User:Joel Sardinha|Joel Sardinha]] ===Language Disorders=== * [[Psycholinguistics/Aphasia|Aphasia]] - [[User:Aslwin|Amie Windle]] ** [[Psycholinguistics/Aphasia Rehabilitation|Aphasia Rehabilitation]] - [[User:Tayd|Taylor Dyon]] * [[Psycholinguistics/Dyslexia|Dyslexia]] - [[User:TheEleventhDoc|Matt Orr]] * [[Psycholinguistics/DownsSyndromeLanguage|Language in Down's Syndrome]] [[Category:Courses]] [[Category:Psycholinguistics| ]] e1rysalpbpyvt0kozvcmutkshrtxrru 102 115400 2416457 2364341 2022-08-19T08:42:38Z Yeongokssi 2948526 /* Old Course */ wikitext text/x-wiki <big>Welcome to the Department of Latin</big> Please be aware there are currently three courses available here The objective of this department is to teach Latin. ==Laws of the department== *Students please write your name in the student section *Teachers please write your name in the teacher section *DO NOT deface the department in any way *Help the department grow!!! * BE BOLD!! ==Courses== ===Simple Latin course=== This is by far the most complete course available on this site, with over 100 lessons covering most basic grammar topics. It is aimed at a conversational style of language learning so includes many everyday terms and in some cases some neologisms. These are clearly indicated in the notes. It is supported by two Memrise courses for vocabulary and sentence composition practice. * Available at [[Latin]] === Old Course === # Latin introduction (Latin 1) #* [[Latin/Introduction]] #* [[Latin/Chapter 1]] #* [[Latin/Chapter 2]] #* Chapter 3 #* Chapter 4 # Latin as used in English (Latin 2) # Caesar's chronicles (Latin 3) # Latin conversations (Latin 4) === New Course === # Grammar & Getting Started #* [[Latin/Subjects and Objects|Subjects and Objects]] {{75Percent}} #* [[Latin/An Introduction to Verbs|An Introduction to Verbs]] {{0Percent}} ==Plans for the department== Future plans for the department include a brand new Latin course building upon the old one coming in the next few months shown in the links above. In the mean time aspects of the old course will still be available for all to access. ==Students== === Old Course === Katherine Walker, Glenmark Villanueva, Aidan Karl Athendorus Devaney, Geethanjana Wanigasekara, Charles Fisher, Davinder K Joan E, Lisbeth L. Annalisa O.D., George Chemmala Dorothy Grace, H. Kim === New Course === ''There are currently no students taking the New Course'' ==Teachers== ===Old Course=== ''There are currently no teachers for the old course.'' ===New Course=== Zac Copeland-Greene ==See also== * [[Latin stream]] ==External links== * [https://translate.google.com/ Google translate] [[it:Materia:Grammatica latina per le superiori 1]] [[Category:Latin|*]] [[Category:Language learning portals|Latin]] f4uluy3q7uds2q65p2cp1onmby5b04o 3 137364 2416145 2416109 2022-08-18T12:56:34Z Dave Braunschweig 426084 /* New user receiving error message during workshop */ Reply wikitext text/x-wiki {{Archive box|[[/2013/]] · [[/2014/]] · [[/2015/]] · [[/2016/]] · [[/2017/]] · [[/2018/]] · [[/2019/]] · [[/2020/]] · [[/2021/]] · [[/2022/]]}} {{:User:{{PAGENAME}}/Welcome}} <!-- Add comments below --> == Question == Hi I am Wikivuyo I just joined a few days ago and I was wondering if you could help me figure out how to create a new website and to change my account. {{Unsigned|Wikivuyo|8 July 2022‎}} :{{At|Wikivuyo}} Welcome to Wikiversity! You can't really create a new website here. You can edit pages. Start by editing your [[User:Wikivuyo/sandbox|sandbox]]. If you want to create a website, you might try using [[GitHub]] as a free starting point. I'm not sure what you want to change about your account. If you want to rename it, see [[meta:Changing username]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:15, 8 July 2022 (UTC) == text does not wrap == ===text does not wrap=== [[File:0715wrap01.png|thumb|400px|''' 2 screen shots showing where text wraps and where text doesn't wrap.''' </br> ]] Hello Professor Braunschweig, The 2 screenshots are of my sandbox. The left hand shot shows that text when displayed normally doesn't wrap. The right hand shot of preview taken in editing mode shows that text wraps. I prefer the right hand method of text automatically wrapping. How do I ensure this? Thanks. [[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 20:39, 15 July 2022 (UTC) :@[[User:ThaniosAkro|ThaniosAkro]] The right answer is to stop using RoundBoxTop. Those templates are based on tables. Using tables for layout has been discouraged since HTML 4, or since 1999. See [https://www.thoughtco.com/dont-use-tables-for-layout-3468941 ThoughtCo: Don't Use Tables for Layout] for more information. Templates like this should be discouraged in general, and rewritten to use div tags instead of tables where they are necessary. :The only example I can provide offhand is [[Wikiversity:Main Page/Layout]]. I spent (a lot of) time back in 2017 switching the main page to a flexbox layout so that it wraps correctly, both within the boxes and then the boxes themselves. :[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 21:12, 15 July 2022 (UTC) == New user receiving error message during workshop == Hi @[[User:Dave Braunschweig|Dave Braunschweig]]! A new user recently tried to publish a new lesson plan for Eventmath, which included links to Twitter and the Internet Archive. (The lesson plan is based on a tweet, and the user archived the tweet so that it would be permanently available). When attempting to publish, the following error message was generated: <blockquote>This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link</blockquote> The user is brand new, so I offered to help. I checked the [[Special:ListGroupRights|User group rights]] and didn't see anything about this. Can you help? We're running workshops with new users, and I am hoping that this won't continue happening. At least one other user (also new, I think) was able to publish a new page with an external link, without issue. Would it help if they publish the page first, and then add the link as a second edit? Maybe @[[User:Guy vandegrift|Guy vandegrift]] could also help? Thank you so much! -- [[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 02:28, 18 August 2022 (UTC) :{{ping|Greg at Higher Math}} For me the first step is to see the context of the effort. Do you have a reference to the page where this incident occurred? Another issue to be sorted out is any degree of commercialization associated with your project. I am not an expert in such matters, but my guess is that paid workers should be allowed to make such edits under certain circumstances. {{ping|Dave Braunschweig}} I don't know if we are supposed to hold this discussion here or move it to [[Wikiversity:Request_custodian_action]].--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 10:35, 18 August 2022 (UTC) :Wikiversity is bombarded with spamming attempts to create pages with external links. We have filters in place to block this behavior. Please instruct users to create pages without external links and then add links later. Have them start with a clean introductory paragraph, for example. :There are other abuse filters new users may encounter as well. There are work-arounds, but all of them take some time and require some patience. Most importantly, remind users that everything about Wikiversity is volunteer-driven. It may become necessary for you to apply as a [[Wikiversity:Curators|curator]] to fully support these new users. :Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 12:56, 18 August 2022 (UTC) 2fttejxqv2ycbxlkv0fapsrzcyrtmpu 2416146 2416145 2022-08-18T12:57:24Z Dave Braunschweig 426084 Ping wikitext text/x-wiki {{Archive box|[[/2013/]] · [[/2014/]] · [[/2015/]] · [[/2016/]] · [[/2017/]] · [[/2018/]] · [[/2019/]] · [[/2020/]] · [[/2021/]] · [[/2022/]]}} {{:User:{{PAGENAME}}/Welcome}} <!-- Add comments below --> == Question == Hi I am Wikivuyo I just joined a few days ago and I was wondering if you could help me figure out how to create a new website and to change my account. {{Unsigned|Wikivuyo|8 July 2022‎}} :{{At|Wikivuyo}} Welcome to Wikiversity! You can't really create a new website here. You can edit pages. Start by editing your [[User:Wikivuyo/sandbox|sandbox]]. If you want to create a website, you might try using [[GitHub]] as a free starting point. I'm not sure what you want to change about your account. If you want to rename it, see [[meta:Changing username]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:15, 8 July 2022 (UTC) == text does not wrap == ===text does not wrap=== [[File:0715wrap01.png|thumb|400px|''' 2 screen shots showing where text wraps and where text doesn't wrap.''' </br> ]] Hello Professor Braunschweig, The 2 screenshots are of my sandbox. The left hand shot shows that text when displayed normally doesn't wrap. The right hand shot of preview taken in editing mode shows that text wraps. I prefer the right hand method of text automatically wrapping. How do I ensure this? Thanks. [[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 20:39, 15 July 2022 (UTC) :@[[User:ThaniosAkro|ThaniosAkro]] The right answer is to stop using RoundBoxTop. Those templates are based on tables. Using tables for layout has been discouraged since HTML 4, or since 1999. See [https://www.thoughtco.com/dont-use-tables-for-layout-3468941 ThoughtCo: Don't Use Tables for Layout] for more information. Templates like this should be discouraged in general, and rewritten to use div tags instead of tables where they are necessary. :The only example I can provide offhand is [[Wikiversity:Main Page/Layout]]. I spent (a lot of) time back in 2017 switching the main page to a flexbox layout so that it wraps correctly, both within the boxes and then the boxes themselves. :[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 21:12, 15 July 2022 (UTC) == New user receiving error message during workshop == Hi @[[User:Dave Braunschweig|Dave Braunschweig]]! A new user recently tried to publish a new lesson plan for Eventmath, which included links to Twitter and the Internet Archive. (The lesson plan is based on a tweet, and the user archived the tweet so that it would be permanently available). When attempting to publish, the following error message was generated: <blockquote>This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link</blockquote> The user is brand new, so I offered to help. I checked the [[Special:ListGroupRights|User group rights]] and didn't see anything about this. Can you help? We're running workshops with new users, and I am hoping that this won't continue happening. At least one other user (also new, I think) was able to publish a new page with an external link, without issue. Would it help if they publish the page first, and then add the link as a second edit? Maybe @[[User:Guy vandegrift|Guy vandegrift]] could also help? Thank you so much! -- [[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 02:28, 18 August 2022 (UTC) :{{ping|Greg at Higher Math}} For me the first step is to see the context of the effort. Do you have a reference to the page where this incident occurred? Another issue to be sorted out is any degree of commercialization associated with your project. I am not an expert in such matters, but my guess is that paid workers should be allowed to make such edits under certain circumstances. {{ping|Dave Braunschweig}} I don't know if we are supposed to hold this discussion here or move it to [[Wikiversity:Request_custodian_action]].--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 10:35, 18 August 2022 (UTC) :{{At|Greg at Higher Math}} Wikiversity is bombarded with spamming attempts to create pages with external links. We have filters in place to block this behavior. Please instruct users to create pages without external links and then add links later. Have them start with a clean introductory paragraph, for example. :There are other abuse filters new users may encounter as well. There are work-arounds, but all of them take some time and require some patience. Most importantly, remind users that everything about Wikiversity is volunteer-driven. It may become necessary for you to apply as a [[Wikiversity:Curators|curator]] to fully support these new users. :Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 12:56, 18 August 2022 (UTC) 0pdp69pi2ieyct2eqnfimbe1uad1zhq 2416198 2416146 2022-08-18T16:51:28Z Greg at Higher Math Help 2911873 /* New user receiving error message during workshop */ Reply wikitext text/x-wiki {{Archive box|[[/2013/]] · [[/2014/]] · [[/2015/]] · [[/2016/]] · [[/2017/]] · [[/2018/]] · [[/2019/]] · [[/2020/]] · [[/2021/]] · [[/2022/]]}} {{:User:{{PAGENAME}}/Welcome}} <!-- Add comments below --> == Question == Hi I am Wikivuyo I just joined a few days ago and I was wondering if you could help me figure out how to create a new website and to change my account. {{Unsigned|Wikivuyo|8 July 2022‎}} :{{At|Wikivuyo}} Welcome to Wikiversity! You can't really create a new website here. You can edit pages. Start by editing your [[User:Wikivuyo/sandbox|sandbox]]. If you want to create a website, you might try using [[GitHub]] as a free starting point. I'm not sure what you want to change about your account. If you want to rename it, see [[meta:Changing username]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:15, 8 July 2022 (UTC) == text does not wrap == ===text does not wrap=== [[File:0715wrap01.png|thumb|400px|''' 2 screen shots showing where text wraps and where text doesn't wrap.''' </br> ]] Hello Professor Braunschweig, The 2 screenshots are of my sandbox. The left hand shot shows that text when displayed normally doesn't wrap. The right hand shot of preview taken in editing mode shows that text wraps. I prefer the right hand method of text automatically wrapping. How do I ensure this? Thanks. [[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 20:39, 15 July 2022 (UTC) :@[[User:ThaniosAkro|ThaniosAkro]] The right answer is to stop using RoundBoxTop. Those templates are based on tables. Using tables for layout has been discouraged since HTML 4, or since 1999. See [https://www.thoughtco.com/dont-use-tables-for-layout-3468941 ThoughtCo: Don't Use Tables for Layout] for more information. Templates like this should be discouraged in general, and rewritten to use div tags instead of tables where they are necessary. :The only example I can provide offhand is [[Wikiversity:Main Page/Layout]]. I spent (a lot of) time back in 2017 switching the main page to a flexbox layout so that it wraps correctly, both within the boxes and then the boxes themselves. :[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 21:12, 15 July 2022 (UTC) == New user receiving error message during workshop == Hi @[[User:Dave Braunschweig|Dave Braunschweig]]! A new user recently tried to publish a new lesson plan for Eventmath, which included links to Twitter and the Internet Archive. (The lesson plan is based on a tweet, and the user archived the tweet so that it would be permanently available). When attempting to publish, the following error message was generated: <blockquote>This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link</blockquote> The user is brand new, so I offered to help. I checked the [[Special:ListGroupRights|User group rights]] and didn't see anything about this. Can you help? We're running workshops with new users, and I am hoping that this won't continue happening. At least one other user (also new, I think) was able to publish a new page with an external link, without issue. Would it help if they publish the page first, and then add the link as a second edit? Maybe @[[User:Guy vandegrift|Guy vandegrift]] could also help? Thank you so much! -- [[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 02:28, 18 August 2022 (UTC) :{{ping|Greg at Higher Math}} For me the first step is to see the context of the effort. Do you have a reference to the page where this incident occurred? Another issue to be sorted out is any degree of commercialization associated with your project. I am not an expert in such matters, but my guess is that paid workers should be allowed to make such edits under certain circumstances. {{ping|Dave Braunschweig}} I don't know if we are supposed to hold this discussion here or move it to [[Wikiversity:Request_custodian_action]].--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 10:35, 18 August 2022 (UTC) :{{At|Greg at Higher Math}} Wikiversity is bombarded with spamming attempts to create pages with external links. We have filters in place to block this behavior. Please instruct users to create pages without external links and then add links later. Have them start with a clean introductory paragraph, for example. :There are other abuse filters new users may encounter as well. There are work-arounds, but all of them take some time and require some patience. Most importantly, remind users that everything about Wikiversity is volunteer-driven. It may become necessary for you to apply as a [[Wikiversity:Curators|curator]] to fully support these new users. :Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 12:56, 18 August 2022 (UTC) ::@[[User:Dave Braunschweig|Dave Braunschweig]] and @[[User:Guy vandegrift|Guy vandegrift]]: Thank you both for your rapid replies! I'll reply about each point in turn. ::* Eventmath is not commercial in any way. Everything we produce is free and open access, and all content is provided by volunteers. The only exception is [[metawiki:Grants:Project/Eventmath|grant funding from the Wikimedia Foundation]], for project creation and leadership. We are in the early stages of discussing leadership roles for other contributors. ::* Here's the context. To create a new Eventmath lesson plan, contributors follow the guidance on our [[Eventmath/Contributing|Contributing page]]. The key step is to enter a title into an input box and click "Create lesson plan." This opens an editor on a new page with the entered title, and the editor is preloaded with our [[Template:Eventmath lesson plan|lesson plan template]]. Unfortunately, I can't link to the specific page in question, since the user was unable to publish it. ::* I've advised the user to publish the page first, and then add the external link afterward. I'll revise the on-wiki instructions accordingly once we confirm that there are no other problems for the next set of new users. ::* Applying to become a curator sounds like a good idea. For example, we've had a user accidentally make [[Eventmath/Lesson plans/Impact|a page with a title unrelated to their topic]], and then create [[Eventmath/Lesson plans/Using inclusion-exclusion to understand COVID reinfection|a new page with an accurate title]]. My understanding is that a redirect would be inappropriate, since the title of the first page is not related in any way to the second page; also, there is no content on the first page that needs to be retained (the only custom content in the lesson plan is a link, which is also in the second page). ::** I read about curatorship at the link you provided; '''are there any expectations regarding the time commitment'''? I want to be sure I can follow through on the responsibilities. ::** Since I already mentioned it, '''would you be able to delete the page with the inaccurate title'''? I'd be happy to submit a separate request if that is better. ::* I appreciate the heads-up about other abuse filters, and it's always inspiring to me that Wikiversity is run by volunteers, so I'll be sure to remind everyone of this. ::Thanks again! ::-- [[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 16:51, 18 August 2022 (UTC) qu7ldrggpb07s9ocedtk42fvqt8hbkm 2416278 2416198 2022-08-18T19:57:50Z Dave Braunschweig 426084 /* New user receiving error message during workshop */ Reply wikitext text/x-wiki {{Archive box|[[/2013/]] · [[/2014/]] · [[/2015/]] · [[/2016/]] · [[/2017/]] · [[/2018/]] · [[/2019/]] · [[/2020/]] · [[/2021/]] · [[/2022/]]}} {{:User:{{PAGENAME}}/Welcome}} <!-- Add comments below --> == Question == Hi I am Wikivuyo I just joined a few days ago and I was wondering if you could help me figure out how to create a new website and to change my account. {{Unsigned|Wikivuyo|8 July 2022‎}} :{{At|Wikivuyo}} Welcome to Wikiversity! You can't really create a new website here. You can edit pages. Start by editing your [[User:Wikivuyo/sandbox|sandbox]]. If you want to create a website, you might try using [[GitHub]] as a free starting point. I'm not sure what you want to change about your account. If you want to rename it, see [[meta:Changing username]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:15, 8 July 2022 (UTC) == text does not wrap == ===text does not wrap=== [[File:0715wrap01.png|thumb|400px|''' 2 screen shots showing where text wraps and where text doesn't wrap.''' </br> ]] Hello Professor Braunschweig, The 2 screenshots are of my sandbox. The left hand shot shows that text when displayed normally doesn't wrap. The right hand shot of preview taken in editing mode shows that text wraps. I prefer the right hand method of text automatically wrapping. How do I ensure this? Thanks. [[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 20:39, 15 July 2022 (UTC) :@[[User:ThaniosAkro|ThaniosAkro]] The right answer is to stop using RoundBoxTop. Those templates are based on tables. Using tables for layout has been discouraged since HTML 4, or since 1999. See [https://www.thoughtco.com/dont-use-tables-for-layout-3468941 ThoughtCo: Don't Use Tables for Layout] for more information. Templates like this should be discouraged in general, and rewritten to use div tags instead of tables where they are necessary. :The only example I can provide offhand is [[Wikiversity:Main Page/Layout]]. I spent (a lot of) time back in 2017 switching the main page to a flexbox layout so that it wraps correctly, both within the boxes and then the boxes themselves. :[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 21:12, 15 July 2022 (UTC) == New user receiving error message during workshop == Hi @[[User:Dave Braunschweig|Dave Braunschweig]]! A new user recently tried to publish a new lesson plan for Eventmath, which included links to Twitter and the Internet Archive. (The lesson plan is based on a tweet, and the user archived the tweet so that it would be permanently available). When attempting to publish, the following error message was generated: <blockquote>This action has been automatically identified as potentially harmful, and therefore disallowed. If you believe your action was constructive, please [[Wikiversity:Request custodian action|inform an administrator]] of what you were trying to do. A brief description of the abuse rule which your action matched is: New User Created Page with External Link</blockquote> The user is brand new, so I offered to help. I checked the [[Special:ListGroupRights|User group rights]] and didn't see anything about this. Can you help? We're running workshops with new users, and I am hoping that this won't continue happening. At least one other user (also new, I think) was able to publish a new page with an external link, without issue. Would it help if they publish the page first, and then add the link as a second edit? Maybe @[[User:Guy vandegrift|Guy vandegrift]] could also help? Thank you so much! -- [[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 02:28, 18 August 2022 (UTC) :{{ping|Greg at Higher Math}} For me the first step is to see the context of the effort. Do you have a reference to the page where this incident occurred? Another issue to be sorted out is any degree of commercialization associated with your project. I am not an expert in such matters, but my guess is that paid workers should be allowed to make such edits under certain circumstances. {{ping|Dave Braunschweig}} I don't know if we are supposed to hold this discussion here or move it to [[Wikiversity:Request_custodian_action]].--[[User:Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|discuss]] • [[Special:Contributions/Guy vandegrift|contribs]]) 10:35, 18 August 2022 (UTC) :{{At|Greg at Higher Math}} Wikiversity is bombarded with spamming attempts to create pages with external links. We have filters in place to block this behavior. Please instruct users to create pages without external links and then add links later. Have them start with a clean introductory paragraph, for example. :There are other abuse filters new users may encounter as well. There are work-arounds, but all of them take some time and require some patience. Most importantly, remind users that everything about Wikiversity is volunteer-driven. It may become necessary for you to apply as a [[Wikiversity:Curators|curator]] to fully support these new users. :Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 12:56, 18 August 2022 (UTC) ::@[[User:Dave Braunschweig|Dave Braunschweig]] and @[[User:Guy vandegrift|Guy vandegrift]]: Thank you both for your rapid replies! I'll reply about each point in turn. ::* Eventmath is not commercial in any way. Everything we produce is free and open access, and all content is provided by volunteers. The only exception is [[metawiki:Grants:Project/Eventmath|grant funding from the Wikimedia Foundation]], for project creation and leadership. We are in the early stages of discussing leadership roles for other contributors. ::* Here's the context. To create a new Eventmath lesson plan, contributors follow the guidance on our [[Eventmath/Contributing|Contributing page]]. The key step is to enter a title into an input box and click "Create lesson plan." This opens an editor on a new page with the entered title, and the editor is preloaded with our [[Template:Eventmath lesson plan|lesson plan template]]. Unfortunately, I can't link to the specific page in question, since the user was unable to publish it. ::* I've advised the user to publish the page first, and then add the external link afterward. I'll revise the on-wiki instructions accordingly once we confirm that there are no other problems for the next set of new users. ::* Applying to become a curator sounds like a good idea. For example, we've had a user accidentally make [[Eventmath/Lesson plans/Impact|a page with a title unrelated to their topic]], and then create [[Eventmath/Lesson plans/Using inclusion-exclusion to understand COVID reinfection|a new page with an accurate title]]. My understanding is that a redirect would be inappropriate, since the title of the first page is not related in any way to the second page; also, there is no content on the first page that needs to be retained (the only custom content in the lesson plan is a link, which is also in the second page). ::** I read about curatorship at the link you provided; '''are there any expectations regarding the time commitment'''? I want to be sure I can follow through on the responsibilities. ::** Since I already mentioned it, '''would you be able to delete the page with the inaccurate title'''? I'd be happy to submit a separate request if that is better. ::* I appreciate the heads-up about other abuse filters, and it's always inspiring to me that Wikiversity is run by volunteers, so I'll be sure to remind everyone of this. ::Thanks again! ::-- [[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 16:51, 18 August 2022 (UTC) :::@[[User:Greg at Higher Math Help|Greg at Higher Math Help]]: :::Curators are volunteers. They contribute whatever time they can / wish to. :::The easiest way to get a page deleted is to add {{tlx|Delete|reason}} at the top of the page. I deleted the page you had linked. :::Thanks! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 19:57, 18 August 2022 (UTC) e2erc8qpm9m7vra43p24przjghgmowk 0 139384 2416156 2415842 2022-08-18T13:25:04Z Young1lim 21186 /* Adder */ wikitext text/x-wiki {{nocat}} == Adder == * Binary Adder Architecture Exploration ( [[Media:adder.20131113.pdf |pdf]] ) {| class="wikitable" |- ! Adder type !! Overview !! Analysis !! VHDL Level Design !! CMOS Level Design |- | '''1. Ripple Carry Adder''' || [[Media:VLSI.Arith.1A.RCA.20211108.pdf |pdf]] || || [[Media:adder.rca.20140313.pdf |pdf]] || [[Media:VLSI.Arith.1D.RCA.CMOS.20211108.pdf |pdf]] |- | '''2. Carry Lookahead Adder''' || [[Media:VLSI.Arith.1.A.CLA.20211106.pdf |pdf]] || || [[Media:adder.cla.20140313.pdf |pdf]] || |- | '''3. Carry Save Adder''' || [[Media:VLSI.Arith.1.A.CSave.20151209.pdf |pdf]] || || || |- || '''4. Carry Select Adder''' || [[Media:VLSI.Arith.1.A.CSelA.20191002.pdf |pdf]] || || || |- || '''5. Carry Skip Adder''' || [[Media:VLSI.Arith.5A.CSkip.20211111.pdf |pdf]] || || || [[Media:VLSI.Arith.5D.CSkip.CMOS.20211108.pdf |pdf]] |- || '''6. Carry Chain Adder''' || [[Media:VLSI.Arith.6A.CCA.20211109.pdf |pdf]] || || [[Media:VLSI.Arith.6C.CCA.VHDL.20211109.pdf |pdf]], [[Media:adder.cca.20140313.pdf |pdf]] || [[Media:VLSI.Arith.6D.CCA.CMOS.20211109.pdf |pdf]] |- || '''7. Kogge-Stone Adder''' || [[Media:VLSI.Arith.1.A.KSA.20140315.pdf |pdf]] || || [[Media:adder.ksa.20140409.pdf |pdf]] || |- || '''8. Prefix Adder''' || [[Media:VLSI.Arith.1.A.PFA.20140314.pdf |pdf]] || || || |- || '''9. Variable Block Adder''' || [[Media:VLSI.Arith.1.A.VBA.20220818.pdf |pdf]] || || || |} </br> === Adder Architectures Suitable for FPGA === * FPGA Carry-Chain Adder ([[Media:VLSI.Arith.1.A.FPGA-CCA.20210421.pdf |pdf]]) * FPGA Carry Select Adder ([[Media:VLSI.Arith.1.B.FPGA-CarrySelect.20210522.pdf |pdf]]) * FPGA Variable Block Adder ([[Media:VLSI.Arith.1.C.FPGA-VariableBlock.20220125.pdf |pdf]]) * FPGA Carry Lookahead Adder ([[Media:VLSI.Arith.1.D.FPGA-CLookahead.20210304.pdf |pdf]]) * Carry-Skip Adder </br> == Barrel Shifter == * Barrel Shifter Architecture Exploration ([[Media:bshift.20131105.pdf |bshfit.vhdl]], [[Media:bshift.makefile.20131109.pdf |bshfit.makefile]]) </br> '''Mux Based Barrel Shifter''' * Analysis ([[Media:Arith.BShfiter.20151207.pdf |pdf]]) * Implementation </br> == Multiplier == === Array Multipliers === * Analysis ([[Media:VLSI.Arith.1.A.Mult.20151209.pdf |pdf]]) </br> === Tree Mulltipliers === * Lattice Multiplication ([[Media:VLSI.Arith.LatticeMult.20170204.pdf |pdf]]) * Wallace Tree ([[Media:VLSI.Arith.WallaceTree.20170204.pdf |pdf]]) * Dadda Tree ([[Media:VLSI.Arith.DaddaTree.20170701.pdf |pdf]]) </br> === Booth Multipliers === * [[Media:RNS4.BoothEncode.20161005.pdf |Booth Encoding Note]] * Booth Multiplier Note ([[Media:BoothMult.20160929.pdf |H1.pdf]]) </br> == Divider == * Binary Divider ([[Media:VLSI.Arith.1.A.Divider.20131217.pdf |pdf]])</br> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Computer architecture]] 1yi95re7ww1uvbk2va7cihfsg3rl174 0 156113 2416446 2366978 2022-08-19T06:31:03Z Marshallsumter 311529 wikitext text/x-wiki [[Image:NGC 2788A.jpg|thumb|right|250px|This is a visual image of NGC 2788A near the edge of the constellation Volans. Credit: Aladin at SIMBAD.{{tlx|fairuse}}]] The '''first neutron source in Volans''' is unknown. The field of [[Radiation/Neutrons|neutron astronomy]] is the result of observations and theories about neutron sources detected in the sky above. The first astronomical neutron source discovered may have been the [[Stars/Sun|Sun]]. But, neutrons from the Sun are intermingled with other radiation so that the Sun may appear as other than a primary source for neutrons. The early use of sounding rockets and balloons to carry neutron detectors high enough may have detected neutrons from the Sun as early as the 1940s. This is a lesson in map reading, coordinate matching, and researching. It is also a research project in the history of neutron astronomy looking for the first astronomical neutron source discovered in the constellation of Volans. Nearly all the background you need to participate and learn by doing you've probably already been introduced to at a secondary level and perhaps even a primary education level. Some of the material and information is at the college or university level, and as you progress in finding neutron sources, you'll run into concepts and experimental tests that are actual research. ==First step== The first step is to succeed in finding a neutron source in Volans. Next, you'll need to determine the time stamp of its discovery and compare it with any that have already been discovered. Over the history of neutron astronomy a number of sources have been found, many as point sources in the night sky. These points are located on the celestial sphere using coordinate systems. Familiarity with these coordinate systems is not a prerequisite. Here the challenge is geometrical, astrophysical, and historical. The coordinates are usually supplied by the neutron source observers. ==Neutron sources== {{main|Radiation/Neutrons}} '''Def.''' a natural [[wikt:source|source]] usually of [[radiation]] in the sky especially at night is called an '''astronomical source'''. A source of astronomical information on older detections of neutron sources is included in the '''Science''' section of the lecture/article [[Radiation/Neutrons|neutron astronomy]]. ==Traveling neutron sources== Many neutron sources do not remain in a constellation for lengthy periods. Some of these are the Sun and sources apparently in orbit around the Sun. The Sun travels through the 13 constellations along the ecliptic (the plane of the Earth's orbit around the Sun): the 12 of the Zodiac and the constellation Ophiuchus. These are described in [[source astronomy]]. ==Backgrounds== {{main|Radiation astronomy/Backgrounds}} To introduce yourself to some aspects of the challenge may I suggest reading the highlighted links mentioned above, and if you're curious, those listed under the section [[First neutron source in Volans#See also|'''See also''']] below. Neutrons are a form of radiation that is currently part of subatomic particle radiation intersecting the Earth. More information about radiation is in [[Radiation/Astronomy|radiation astronomy]]. ==Constellations== [[Image:Volans IAU.svg|thumb|right|250px|This is an image of the International Astronomical Union (IAU) sky map of the constellation Volans. Credit: IAU and Sky & Telescope magazine (Roger Sinnott & Rick Fienberg).{{tlx|free media}}]] The Wikipedia article about the constellation [[w:Volans|Volans]] contains a high school level description. The figure at right shows the sky map of Volans. Around the edges of the map are coordinates related to longitude and latitude, but with the Earth rotating on its axis every 24 hours the celestial coordinates must remain fixed relative to the background light sources in the sky. Volans is one of the 12 constellations that were introduced by the Dutch navigators Pieter Dirkszoon Keyser and Frederick de Houtman in the late 16th century. It was first depicted on Petrus Plancius’ globe in 1598. Plancius called the constellation ''Vliegendenvis'' (flying fish).<ref name=Constellations>{{ cite book |title=Constellations: A Guide to the Night Sky |year=2013 |url=http://www.constellation-guide.com/constellation-list/volans-constellation/ }}</ref> Also, in the Wikipedia article is a [[w:List of stars in Volans|list of stars in Volans]]. {{clear}} ==Searching catalogs== In the lecture/article [[Radiation/Neutrons|neutron astronomy]] in its science section is a list of older catalogs of neutron sources. Using the constellation description in the previous section and the range of coordinates for the constellation in [[source astronomy]], scan through the coordinates for these neutron sources to see if any may be within Volans. If you find any that are, skip down to the section '''Neutron sources''' in Volans and make an entry. Be sure to check the coordinate era, most B1950 coordinates have changed slightly to the new J2000 set. Try the catalog designation at either SIMBAD website. ==Testing a source== There are many web sites that may have an neutron source listed for the constellation Volans. Some that you may wish to try are in the External links section near the bottom of this lesson. ===Wikipedia sources=== '''A. Constellation article''' Under "Notable features" in the Wikipedia article on the constellation [[w:Volans (constellation)|Volans]] is the [[w:List of stars in Volans|list of stars in Volans]]. Click on this link. In the table of this Wikipedia article is α Vol. To the right are coordinates: Right ascension (RA): 09^h 02^m 26.79592^s and Declination (Dec): -66° 23' 45.8727". Find these coordinates on the Volans map at the right. To evaluate the star as a neutron source, skip ahead to section "Neutron sources". '''B. Wikipedia search''' Another way to look for neutron sources in the constellation is to perform a search on Wikipedia. Try "Volans neutron" without the quotes. This yields only 2 returns which include an [[w:Outline of astronomy|outline of astronomy]] and a Sun computer. The outline of astronomy names no neutron sources. ===SIMBAD sources=== Another way to find possible neutron sources in Volans is to use search queries on [[w:SIMBAD|SIMBAD]]. Click on either SIMBAD link under "External links" below, then click on "Criteria query", or "by criteria". In the tan box, type in "region(09 02 26.79592 -66 23 45.8727,10m)", without the quotes. This tells the SIMBAD computer you are interested in a circular region of the celestial sphere centered on the coordinates for alpha Volantis, with a radius of 10 arcminutes (m), or try 10d for 10 degrees. Notice on the page over at the right from the tan colored box: "'''Return'''". The default is "object count". Click on "submit query". In a few moments a result something like "'''Number of objects: 9'''" should appear. Click "Back" to see the tan box again. Adding an object type such as & otype='IR' to the region request reduces the returned number to those that are infrared sources. But, all of the otypes listed at [http://simbad.u-strasbg.fr/simbad/sim-display?data=otypes Object classification in SIMBAD] contain only neutron stars, not neutron sources. The SIMBAD criteria search allows you to specify spectral types for possible stars. The criteria "sptype" (the exact spectral type): returns only the objects having the requested spectral type (i.e. sptype = 'k0' does not return 'K0III',...). And, "sptypes" should be used to retrive all objects having a spectral type containing the one specified; i.e., sptypes = 'K0' will return all objects having 'K0' as a spectral type, but also 'K0III' or 'K0IIIp', ...). But, here again no information about possible neutron sources is listed. If a flare star is a likely source of neutrons, then entering otype='Fl*' should locate neutron sources. Using only otype='Fl*' on SIMBAD yields 2582 in all of SIMBAD. ===SAO/NASA Astrophysics Data System=== In the naming of sources per constellation, the genitive is in common use. For Volans, the genitive is Volantis. Click on the SAO/NASA Astrophysics Data System link below in the "External links". Try "Volantis neutron" without the quotes, or "alpha volantis" with quotes, followed by neutron. Click on a link below '''# Bibcode Authors'''. If the '''Abstract''' describes the detection of neutrons from a source in the constellation Volans, go to the next section under "SAO/NASA Astrophysics Data System". If it does not try another bibcode link. ==Evaluating neutron sources== There are several ways to evaluate a neutron source for the constellation Volans. ===Wikipedia sources=== Click on the link to the Wikipedia article. After you've enjoyed reading about the source, use the 'find' command of your browser to see if this Wikipedia page mentions anything about neutrons. Does the article mention whether or not the source is a neutron source? What is the current time stamp for the Wikipedia article on the source? [Hint]: look for something like "This page was last modified on 25 December 2011 at 20:12." very near the bottom of the page. For now this is an adequate time stamp. From reading the Wikipedia article on the source, if you believe the text demonstrates that the source is not a neutron source in Volans edit the "Non-neutron sources in Volans" section near the bottom of the page with an entry similar to "# Alpha Volantis 25 December 2011 at 20:12 Wikipedia article "Alpha Volantis", without the quotes, and finish the entry with four "~"s without the quotes after the period. The date included with your designation or username is a time stamp for the entry. The last portion of the entry is the source of your information. On the other hand, if there are one or more sentences in the article that you believe demonstrates that the source is a neutron source in Volans edit the section below "Neutron sources in Volans" with a similar entry. Go to the section entitled, "Challenging an entry". The Wikipedia article on Alpha Volantis mentions: Alpha Volantis is an Am star, 'm' for metal lines, with a stellar classification of kA3hA5mA5&nbsp;V<ref name=Gray>{{ cite journal | author=R. O. Gray, C. J. Corbally, R. F. Garrison, M. T. McFadden, E. J. Bubar, C. E. McGahee, A. A. O'Donoghue, E. R. Knox | title=Contributions to the Nearby Stars (NStars) Project: spectroscopy of stars earlier than M0 within 40 pc-The Southern Sample | journal=The Astronomical Journal | volume=132 | issue=1 | pages=161–70 |month=July |year=2006 | doi=10.1086/504637 | bibcode=2006AJ....132..161G | arxiv=astro-ph/0603770 }}</ref> From the lecture/article on [[Radiation/Neutrons|neutron astronomy]], the neutron radiation band may not have a wavelength temperature pair. Is the primary star of alpha Volantis a neutron star, or a neutron source? ===Wikimedia commons=== Another possible website for neutron sources is Wikimedia Commons, url=https://commons.wikimedia.org/wiki/Main_Page. Try entering "neutron Volans". This returns no images of any neutron sources in Volans. ===SIMBAD sources=== To check any source (even one from Wikipedia) on SIMBAD, click of the "External link" to the "SIMBAD Astronomical Database". At the lower right side of the SIMBAD Astronomical Database page is a "Basic search" box. There are several ways to try your target: # source name: without the quotes or # source coordinates: without the quotes, for example, "12 26 35.94 -63 05 56.6". If you are looking at a SIMBAD generated table which lists possible targets, click on one. Having SIMBAD list all of its 2582 flare stars produces an apparently formidable task. Try searching with your browser using "Vol". While using "Mon" for Monoceros locates 45 in the lost, "Vol" finds none. Many of the flare stars listed do not include a constellation designation. Letting SIMBAD plot all of these flare stars and comparing the plot with the constellation sky chart shows that no flare stars listed in SIMBAD occur within the right ascension and declination ranges of Volans. Another possibility is to use otype='X' for X-ray sources or otype='gam' or otype='gB' for gamma-ray sources. Either of these may indicate a neutron source. There are 3509 otype='gam' listed in SIMBAD. Plotting them may be helpful or scanning them using either -6 or -7 with your browser will eventually reveal six gamma-ray sources that may also be neutron sources, such as AGN AT20G J081103-75302. If you have already found a neutron source (or a table of them) using SIMBAD, click on the blue link identifier for the first. ===SIMBAD time stamp=== Peruse the SIMBAD page for a time stamp or date of last revision. [Hint: it may look something like "2012.01.09CET20:10:02" and be in the upper right.] If the entry at SIMBAD convinces you that the source is not a neutron source, edit the "Non-neutron sources in Volans" section near the bottom of this page and type in an entry similar to "# Source Name 2012.01.09CET20:10:02 SIMBAD article "SIMBAD source name".", without the first set of quotes, followed by four ~s. If your SIMBAD analysis convinces you that you have found a neutron source in Volans (did you check the coordinates vs. the map of Volans?), make an entry something like the ones in the section "Neutron sources in Volans". ===SAO/NASA Astrophysics Data System sources=== If the abstract states that an neutron source in Volans is detected or studied, consider entering it in the section "Neutron sources in Volans" below. ===Abstract time stamp=== On the abstract page is a '''Publication Date''':. This may serve as a time stamp for establishing that the source is detected as a neutron source on or before the date of publication. The time stamp followed by four ~s for your verification as determiner in the section "Neutron sources in Volans" completes your entry. ==Challenging an entry== Any entry in either the section "Neutron sources in Volans" or "Non-neutron sources in Volans" can be challenged. The time stamp can be challenged to see if there is an earlier one. The source can be challenged by an earlier source. ===Wikipedia challenges=== Is Wikipedia a '[[w:Primary source|primary source]]', or does the Wikipedia article cite a source? Even though Wikipedia has an article on the source, is it a good place to stop in testing whether the source has been detected as an astronomical neutron source? If the Wikipedia article cites a primary source, skip down to the section on "Primary sources". ===SIMBAD challenges=== Is SIMBAD a 'primary source'? SIMBAD is an astronomical database provided by the Centre de Données astronomiques de Strasbourg. It is an authoritative source, but they do occasionally make a mistake. If you find a neutron source within the constellation on SIMBAD, the next step is to find the earliest time stamp of discovery. ===SAO/NASA Astrophysics Data System challenge=== Is the SAO/NASA Astrophysics Data System abstract entry a primary source? The SAO/NASA Astrophysics Data System is an astronomical database provided by the High Energy Astrophysics Division at the Harvard-Smithsonian Center for Astrophysics of Harvard University. The abstract has been copied from the actual article in a scientific journal or other publication. Mistakes can be made and the article may record within its text exact dates when the observation or detection of neutrons actually occurred. Such a record may provide an earlier time stamp. ==Primary sources== Primary sources may be searched for possible additional information perhaps not yet evaluated by SIMBAD or not presented in a Wikipedia article about a source. ===Wikipedia test sources=== For a Wikipedia article that cites a primary source, scroll down to the reference and open the reference. Read through the article looking for where the source mentioned in the Wikipedia article occurs. Some primary source authors may use source designations that are not mentioned in the Wikipedia article. To look for other designations, click on the link to SIMBAD in the "External links" on this page, enter the source name from the Wikipedia article, and see if other names are mentioned in the article. When none of the names are mentioned, click on the link for "Google Advanced Search" in the list of "External links", enter the source name or designation(s) such as "Gliese 866", with neutrons to see if the source has a reference indicating it is a neutron source. And, look for the earliest one. Compose an entry using the primary source. ===SIMBAD test sources=== Further down the SIMBAD page is a list of "Identifiers". Click on the blue bold portion. On the page that appears should be a primary source listed after '''Ref:'''. Click on the blue link with the oldest year. This yields an earlier time stamp and entry citation like the current one in the section "Neutron sources in Volans". If you find another source or an earlier time stamp, compose a similar entry and edit the section. Additional information to add into the reference can be found by clicking on "ADS services" from the SIMBAD page. ===SAO/NASA Astrophysics Data System sources test=== Click on either the "Electronic Refereed Journal Article (HTML)" or "Full Refereed Journal Article (PDF/Postscript)", if available. Depending on the article display, if the abstract is repeated and the article is listed as '''FREE''', click on either the PDF or HTML version. While scanning or reading the article look for "Observations" (or use the Find function of your browser) and the possible inclusion of dates for these. If more than one neutron source in Volans are detected, which one(s) would you list in the section "Neutron sources in Volans" below? An example of an article reference is provided in that section. ==Changing an entry== From your analysis of the source so far, is it a neutron source? If you have found an earlier time stamp for the source than the one listed in the section below "Non-neutron sources in Volans" and the answer to the above question is "no", you can edit the section with your result. Or, you can leave the entries as is and try another star. If you have found an earlier time stamp for the source than the one listed in the section below "Neutron sources in Volans", edit the section with your result. Or, if you found another neutron source with a comparable or earlier time stamp, edit the section with your result. ==Neutron sources in Volans== # 1FGL J0849.0-6754 SIMBAD: "C.D.S. - SIMBAD4 rel 1.218 - 2014.02.02CET06:35:51", "otype='gam'", found by searching the list of gamma-ray sources using "-6". --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 05:44, 2 February 2014 (UTC) ==Non-neutron sources in Volans== ==Oldest records== ==SIMBAD annotations== For any particular source, the SIMBAD record may indicate that it is not a neutron source yet above you may have found at least two refereed journal articles to indicate that it is. Use the second SIMBAD External links to directly display the SIMBAD database in France. Enter the name source you have found into the search box. Scroll down to the '''Annotations :'''. Look for the link "add an annotation to this object". With browser open to the literature citations available, click on this link. You may need to register as a user. It's free. Post your annotation containing the literature references. ==Hypotheses== {{main|Hypotheses}} # Neutron sources may be quite common. The "possibility has been discussed by various authors that solar neutrons and 2.2-MeV gamma rays are produced by the nuclear interactions of flare-accelerated particles with solar material."<ref name=Ito>{{ cite journal |author=K. Ito, H. Okazoe, M. Yoshimori |title=Production of neutrons and gamma rays by nuclear interactions in solar flares |journal=Canadian Journal of Physics |month=May |year=1968 |volume=46 |issue=10 |pages=S780-3 |url=http://www.nrcresearchpress.com/doi/abs/10.1139/p68-350?journalCode=cjp#.Uu29ya79UmU |arxiv= |bibcode= |doi=10.1139/p68-350 |pmid= |accessdate=2014-02-01 }}</ref> The "ratio of neutrons to 2.2-MeV gamma rays depends significantly on the direction of motion of the flare-accelerated particles with respect to the solar atmosphere. For the direction of motion of flare-accelerated particles, we shall consider three cases: (a) isotropic motion, (b) upward-directed motion towards the corona, (c) downward-directed motion towards the photosphere. These flare-accelerated particles would interact with the solar atmosphere to produce neutrons with energies above several MeV. Some of the neutrons will escape into space through the corona, while some will be slowed down in the denser atmosphere and captured by the atmospheric protons with the emission of 2.2-MeV gamma rays."<ref name=Ito/> The "ratio of neutrons to 2.2-MeV gamma rays for case (b) is 10–20 times that for (a) or (c). The value for case (c) is twice that for case (a). For the expected solar neutron intensity, the values for (b) and (c) are ~10⁻⁶ and 4 × 10⁻² of that for (a), respectively."<ref name=Ito/> If stellar flares have origins similar to solar flares, then flare stars produce neutrons. Only relativistic neutrons would be able to reach Earth before decaying. ==See also== {{div col|colwidth=20em}} * [[First astronomical sources]] * [[First blue source in Boötes]] * [[First cyan source in Caelum]] * [[Sources/First gamma-ray source in Triangulum Australe|First gamma-ray source in Triangulum Australe]] * [[Sources/First green source in Tucana|First green source in Tucana]] * [[Sources/First infrared source in Crux|First infrared source in Crux]] * [[First neutron source in Volans]] * [[First orange source in Cancer]] * [[First positron source in Phoenix]] * [[First radio source in Pisces]] * [[First red source in Canis Major]] * [[First submillimeter source in Carina]] * [[First superluminal source in Indus]] * [[Sources/First ultraviolet source in Sagittarius|First ultraviolet source in Sagittarius]] * [[First violet source in Leo]] * [[First X-ray source in Apus]] * [[First X-ray source in Andromeda]] * [[First yellow source in Aquila]] {{Div col end}} ==References== {{reflist|2}} ==External links== * [http://www.iau.org/ International Astronomical Union] * [http://nedwww.ipac.caltech.edu/ NASA/IPAC Extragalactic Database - NED] * [http://nssdc.gsfc.nasa.gov/ NASA's National Space Science Data Center.] * [http://www.osti.gov/ Office of Scientific & Technical Information] * [http://www.adsabs.harvard.edu/ The SAO/NASA Astrophysics Data System] * [http://www.scirus.com/srsapp/advanced/index.jsp?q1= Scirus for scientific information only advanced search] * [http://cas.sdss.org/astrodr6/en/tools/quicklook/quickobj.asp SDSS Quick Look tool: SkyServer] * [http://simbad.u-strasbg.fr/simbad/ SIMBAD Astronomical Database] * [http://simbad.harvard.edu/simbad/ SIMBAD Web interface, Harvard alternate] * [http://nssdc.gsfc.nasa.gov/nmc/SpacecraftQuery.jsp Spacecraft Query at NASA.] * [http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/convcoord/convcoord.pl Universal coordinate converter] <!-- footer templates --> {{Principles of radiation astronomy}}{{tlx|Radiation astronomy resources}}{{Sisterlinks|Volans}}{{Sisterlinks|Astronomical neutron sources}} <!-- categories --> [[Category:Radiation astronomy/Lessons]] [[Category:Radiation/Lessons]] [[Category:Sources/Lessons]] pyol4pih900mao6ts6hvsuw6gwz5iwg 0 156144 2416448 2308906 2022-08-19T06:47:24Z Marshallsumter 311529 wikitext text/x-wiki [[Image:Aldebaran.jpg|thumb|right|250px|This is an optical or visual image of Aldebaran. Credit: NASA, ESA and STScI.{{tlx|free media}}]] At present, the geographic coordinates on Earth of latitude and longitude translate to right ascension and declination from the Greenwich meridian and the equator as projected on the celestial sphere. But, other records may exist even here on Earth that use specific stars or bright lights in the sky especially at night that appear fixed relative to others. This problem set is an attempt to acquaint you with alternative coordinate systems that use ''standard stars'' such as those in constellations to preserve a record of celestial events. From your understanding of the unusual or unknown coordinate system try to determine the target star or the likely RA and Dec intended. Angles between known RA and Dec for two stars or other objects may be found using [http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/convcoord/convcoord.pl NASA's Object Position Finder, Coordinate Converter, and Separation Calculator.] {{clear}} ==Problem 1== [[Image:Reseaux 3D aP.png|thumb|right|200px]] A triclinic coordinate system has coordinates of different lengths (a ≠ b ≠ c) along x, y, and z axes, respectively, with interaxial angles that are not 90°. The interaxial angles α, β, and γ vary such that (α ≠ β ≠ γ). These interaxial angles are α = y⋀z, β = z⋀x, and γ = x⋀y, where the symbol "⋀" means "angle between". An observer has spotted an unknown object in the heavens. In the location of the observer and perhaps in time, the now common RA and Dec system is not available. In order to record this object, the observer constructs a device to indicate the objects position in the heavens relative to three well-known and easily seen stars. Once three rods are in place, each pointing to a specific star the observer marks off lengths on each rod until the combination of the three lengths locates the unknown object. With the three lengths along each axis as a_x ~ 5, a_y ~ 6, and a_z ~ 8 marked, the observer measures the three interaxial angles at α = 26°, β = 82°, and γ = 64°. The x-axis is pointing to Achernar. The y-axis is pointing to Rigel. And, the z-axis is pointing to Aldebaran. Using pointers attached to each rod at ninety degrees swinging into the space between each rod, which star or stars are closest to the mutual point or points. What constellation(s) are they in. Use diagrams to justify your answer. What is the RA Dec of the target? {{clear}} ==Problem 2== [[Image:Monoclinic.png|thumb|right|200px]] In a monoclinic coordinate system, a ≠ b ≠ c, and depending on setting α = β = 90° ≠ γ, α = γ = 90° ≠ β, α = 90° ≠ β ≠ γ, or α = β ≠ γ ≠ 90°. As in problem 1, an observer has spotted a luminous object and is attempting to fix its location in the sky using known stars. The x-rod is pointing at Beta Librae (Zubeneschamali). The y-rod points at epsilon Ursa Major (eps UMa). And, the z-rod points at Altair. The angle between y and z is approximately 90°. Between z and x, β = 78°. Between x and z, γ = 76°. The angled pointers are 6 units along the x-axis and 8 units along the y and z-axes. What constellation is the target object in? What star or stars may be the target? What is the RA Dec of the target? {{clear}} ==Problem 3== An observer has located an object of interest using Betelgeuse, Arcturus, and Spica at equal units along each rod. What are the angles between the axes and the RA and Dec of the object? ==Problem 4== An observer has spotted the Large Magellanic Cloud, find three stars to set up a coordinate system. What are the angles between your three stars and the approximate units to pinpoint the LMC? What is the RA Dec of your three stars? ==Problem 5== [[Image:Reseaux 3D oP.png|thumb|right|200px]] In an orthorhombic coordinate system α = β = γ = 90° and a ≠ b ≠ c. An observer has located an object of interest at proportionately a = 6, b = 5, and c = 7 units along axes pointed at Altair, Alioth (epsilon Ursa Major, eps UMa), and Betelgeuse, where the angles between these stars are all approximately 90°. What is the possible object of interest? What are the errors on the angles between these stars? What constellations are each star in? What is the RA Dec of the target? Locate three other bright stars that have interaxial angles of 90°. {{clear}} ==Problem 6== [[Image:Reseaux 3D tP-2011-03-12.png|thumb|right|200px]] A tetragonal coordinate system has α = β = γ = 90°, and a = b ≠ c. Using Betelgeuse, Altair, an epsilon Ursa Major (Alioth), an observer has decided an object of interest has axes proportional units of 6 = a = b, c = 2. What is the likely object and its RA and Dec? What constellation is it in? {{clear}} ==Problem 7== [[Image:Rhombohedral.svg|thumb|right|200px]] A rhombohedral system has a = b = c and α = β = γ < 120°, ≠ 90°. Find three bright stars in which α = β = γ < 120°, ≠ 90°. Find an object relative to these three stars where a = b = c, approximately. {{clear}} ==Problem 8== [[Image:Reseaux 3D hP.png|thumb|right|200px]] A hexagonal system has a = b ≠ c and α = β = 90°, γ = 120°. Find three stars where α = β = 90°, γ = 120°. For example, Polaris and Delta Orionis (Mintaka) are 90° apart. Locate an object along these three rods where proportionately a = b ≠ c. {{clear}} ==See also== {{div col|colwidth=20em}} * [[Angular momentum and energy]] * [[Column densities]] * [[Cosmic circuits]] * [[Energy phantoms]] * [[Furlongs per fortnight]] * [[Planck's equation]] * [[Radiation astronomy/Problem set]] * [[Radiation dosage]] * [[Radiation astromathematics problems]] * [[Spectrographs]] * [[Star jumping]] * [[Synchrotron radiation/Problem set|Synchrotron radiation]] * [[Telescopes and cameras]] * [[Unknown coordinate systems]] * [[Unusual units]] * [[Vectors and coordinates]] {{Div col end}} ==External links== * [http://www.iau.org/ International Astronomical Union] * [http://nedwww.ipac.caltech.edu/ NASA/IPAC Extragalactic Database - NED] * [http://nssdc.gsfc.nasa.gov/ NASA's National Space Science Data Center] * [http://www.osti.gov/ Office of Scientific & Technical Information] * [http://www.adsabs.harvard.edu/ The SAO/NASA Astrophysics Data System] * [http://www.scirus.com/srsapp/advanced/index.jsp?q1= Scirus for scientific information only advanced search] * [http://cas.sdss.org/astrodr6/en/tools/quicklook/quickobj.asp SDSS Quick Look tool: SkyServer] * [http://simbad.u-strasbg.fr/simbad/ SIMBAD Astronomical Database] * [http://simbad.harvard.edu/simbad/ SIMBAD Web interface, Harvard alternate] * [http://nssdc.gsfc.nasa.gov/nmc/SpacecraftQuery.jsp Spacecraft Query at NASA.] * [http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/convcoord/convcoord.pl Universal coordinate converter] <!-- footer templates --> {{tlx|Mathematics resources}}{{Principles of radiation astronomy}}{{tlx|Radiation astronomy resources}}{{tlx|Reasoning resources}}{{Sisterlinks|Coordinate systems}} <!-- categories --> [[Category:Abstractions/Problems]] [[Category:Astrophysics/Problems]] [[Category:Distances/Problems]] [[Category:Mathematics/Problems]] [[Category:Radiation astronomy/Problems]] sxlbao7dyp31q4msz9040vv22dtyiul 0 171005 2416154 2415840 2022-08-18T13:24:28Z Young1lim 21186 /* Geometric Series Examples */ wikitext text/x-wiki Many of the functions that arise naturally in mathematics and real world applications can be extended to and regarded as complex functions, meaning the input, as well as the output, can be complex numbers <math>x+iy</math>, where <math>i=\sqrt{-1}</math>, in such a way that it is a more natural object to study. '''Complex analysis''', which used to be known as '''function theory''' or '''theory of functions of a single complex variable''', is a sub-field of analysis that studies such functions (more specifically, '''holomorphic''' functions) on the complex plane, or part (domain) or extension (Riemann surface) thereof. It notably has great importance in number theory, e.g. the [[Riemann zeta function]] (for the distribution of primes) and other <math>L</math>-functions, modular forms, elliptic functions, etc. <blockquote>The shortest path between two truths in the real domain passes through the complex domain. — [[wikipedia:Jacques_Hadamard|Jacques Hadamard]]</blockquote>In a certain sense, the essence of complex functions is captured by the principle of [[analytic continuation]].{{mathematics}} ==''' Complex Functions '''== * Complex Functions ([[Media:CAnal.1.A.CFunction.20140222.Basic.pdf|1.A.pdf]], [[Media:CAnal.1.B.CFunction.20140111.Octave.pdf|1.B.pdf]], [[Media:CAnal.1.C.CFunction.20140111.Extend.pdf|1.C.pdf]]) * Complex Exponential and Logarithm ([[Media:CAnal.5.A.CLog.20131017.pdf|5.A.pdf]], [[Media:CAnal.5.A.Octave.pdf|5.B.pdf]]) * Complex Trigonometric and Hyperbolic ([[Media:CAnal.7.A.CTrigHyper..pdf|7.A.pdf]], [[Media:CAnal.7.A.Octave..pdf|7.B.pdf]]) '''Complex Function Note''' : 1. Exp and Log Function Note ([[Media:ComplexExp.29160721.pdf|H1.pdf]]) : 2. Trig and TrigH Function Note ([[Media:CAnal.Trig-H.29160901.pdf|H1.pdf]]) : 3. Inverse Trig and TrigH Functions Note ([[Media:CAnal.Hyper.29160829.pdf|H1.pdf]]) ==''' Complex Integrals '''== * Complex Integrals ([[Media:CAnal.2.A.CIntegral.20140224.Basic.pdf|2.A.pdf]], [[Media:CAnal.2.B.CIntegral.20140117.Octave.pdf|2.B.pdf]], [[Media:CAnal.2.C.CIntegral.20140117.Extend.pdf|2.C.pdf]]) ==''' Complex Series '''== * Complex Series ([[Media:CPX.Series.20150226.2.Basic.pdf|3.A.pdf]], [[Media:CAnal.3.B.CSeries.20140121.Octave.pdf|3.B.pdf]], [[Media:CAnal.3.C.CSeries.20140303.Extend.pdf|3.C.pdf]]) ==''' Residue Integrals '''== * Residue Integrals ([[Media:CAnal.4.A.Residue.20140227.Basic.pdf|4.A.pdf]], [[Media:CAnal.4.B.pdf|4.B.pdf]], [[Media:CAnal.4.C.Residue.20140423.Extend.pdf|4.C.pdf]]) ==='''Residue Integrals Note'''=== * Laurent Series with the Residue Theorem Note ([[Media:Laurent.1.Residue.20170713.pdf|H1.pdf]]) * Laurent Series with Applications Note ([[Media:Laurent.2.Applications.20170327.pdf|H1.pdf]]) * Laurent Series and the z-Transform Note ([[Media:Laurent.3.z-Trans.20170831.pdf|H1.pdf]]) * Laurent Series as a Geometric Series Note ([[Media:Laurent.4.GSeries.20170802.pdf|H1.pdf]]) === Laurent Series and the z-Transform Example Note === * Overview ([[Media:Laurent.4.z-Example.20170926.pdf|H1.pdf]]) ====Geometric Series Examples==== * Causality ([[Media:Laurent.5.Causality.1.A.20191026n.pdf|A.pdf]], [[Media:Laurent.5.Causality.1.B.20191026.pdf|B.pdf]]) * Time Shift ([[Media:Laurent.5.TimeShift.2.A.20191028.pdf|A.pdf]], [[Media:Laurent.5.TimeShift.2.B.20191029.pdf|B.pdf]]) * Reciprocity ([[Media:Laurent.5.Reciprocity.3A.20191030.pdf|A.pdf]], [[Media:Laurent.5.Reciprocity.3B.20191031.pdf|B.pdf]]) * Combinations ([[Media:Laurent.5.Combination.4A.20200702.pdf|A.pdf]], [[Media:Laurent.5.Combination.4B.20201002.pdf|B.pdf]]) * Properties ([[Media:Laurent.5.Property.5A.20220105.pdf|A.pdf]], [[Media:Laurent.5.Property.5B.20220126.pdf|B.pdf]]) * Applications ([[Media:Laurent.6.Application.6A.20220818.pdf|A.pdf]], [[Media:Laurent.5.Application.6B.20220723.pdf|B.pdf]]) * Double Pole Case :- Examples ([[Media:Laurent.5.DPoleEx.7A.20220722.pdf|A.pdf]], [[Media:Laurent.5.DPoleEx.7B.20220720.pdf|B.pdf]]) :- Properties ([[Media:Laurent.5.DPoleProp.5A.20190226.pdf|A.pdf]], [[Media:Laurent.5.DPoleProp.5B.20190228.pdf|B.pdf]]) ====The Case Examples==== * Example Overview : ([[Media:Laurent.4.Example.0.A.20171208.pdf|0A.pdf]], [[Media:Laurent.6.CaseExample.0.B.20180205.pdf|0B.pdf]]) * Example Case 1 : ([[Media:Laurent.4.Example.1.A.20171107.pdf|1A.pdf]], [[Media:Laurent.4.Example.1.B.20171227.pdf|1B.pdf]]) * Example Case 2 : ([[Media:Laurent.4.Example.2.A.20171107.pdf|2A.pdf]], [[Media:Laurent.4.Example.2.B.20171227.pdf|2B.pdf]]) * Example Case 3 : ([[Media:Laurent.4.Example.3.A.20171017.pdf|3A.pdf]], [[Media:Laurent.4.Example.3.B.20171226.pdf|3B.pdf]]) * Example Case 4 : ([[Media:Laurent.4.Example.4.A.20171017.pdf|4A.pdf]], [[Media:Laurent.4.Example.4.B.20171228.pdf|4B.pdf]]) * Example Summary : ([[Media:Laurent.4.Example.5.A.20171212.pdf|5A.pdf]], [[Media:Laurent.4.Example.5.B.20171230.pdf|5B.pdf]]) ==''' Conformal Mapping '''== * Conformal Mapping ([[Media:CAnal.6.A.Conformal.20131224.pdf|6.A.pdf]], [[Media:CAnal.6.A.Octave..pdf|6.B.pdf]]) go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] [[Category:Complex analysis]] r5v8pgpeni7cyv46pj3scya56zky8k5 0 171008 2416460 2415787 2022-08-19T09:01:34Z Young1lim 21186 /* The Temporal Characteristics of Random Processes */ wikitext text/x-wiki ==''' Random Variables '''== === Single Random Variables === * Random Variables ([[Media:RV1.RVariable.1.A.20200427.pdf |A.pdf]]) * Distribution Function ([[Media:RV1.Distribution.2.A.201200428.pdf |A.pdf]]) * Density Function ([[Media:RV1.Density.3.A.20200429.pdf |A.pdf]]) * Functions of Random Variables ([[Media:RV1.RVFunction.4.A.20220317.pdf |A.pdf]]) * Gaussian Random Variables ([[Media:RV1.4.Gaussian.20200430.pdf |A.pdf]], [[Media:RV1.4B.Gaussian.20180314.pdf |B.pdf]]) * Other Distribution and Density Functions ([[Media:RV1.5.Other.20200501.pdf |A.pdf]]) * Conditional Distribution and Density Functions ([[Media:RV1.6.Conditional.20200506.pdf |A.pdf]]) </br> === The Characteristics of a Single Random Variable === * Expected Value ([[Media:RV2.Expectation.1.A.20200506.pdf |A.pdf]]) * Moments ([[Media:RV2.Moment.2.B.20200507.pdf |A.pdf]], [[Media:RV2.Moment.2.B.20180320.pdf |B.pdf]]) * Moment Generating Functions ([[Media:RV2.MFunctions.3.A.20200508.pdf |A.pdf]]) * Transformations of Random Variables ([[Media:RV2.Transform.4.A.20200514.pdf |A.pdf]]) </br> === Multiple Random Variables === * Vector Random Variables ([[Media:3MRV.1A.VectorRV.20200515.pdf |A.pdf]]) * Joint Distribution ([[Media:3MRV.2A.JointDist.20200518.pdf |A.pdf]]) * Joint Density ([[Media:3MRV.3A.JointDensity.20200521.pdf |A.pdf]]) * Conditional Joint Distribution and Density ([[Media:3MRV.4A.CondDistrib.20200527.pdf |A.pdf]]) * Statistical Independence ([[Media:3MRV.5A.StatIndep.20200423.pdf |A.pdf]]) * Sums of Random Variables ([[Media:3MRV.6A.RVSum.20200528.pdf |A.pdf]]) * Central Limit Theorem ([[Media:3MRV.7A.CLimit.20200612.pdf |A.pdf]]) </br> === The Characteristics of Multiple Random Variables === * Expected Values ([[Media:4MRV.1A.Expect.20200617.pdf |A.pdf]]) * Joint Characteristic Functions ([[Media:4MRV.2A.JChar.20200618.pdf |A.pdf]]) * Jointly Gaussian Random Variables ([[Media:4MRV.3A.JGauss.20200619.pdf |A.pdf]]) * Transformations of Multiple Random Variables ([[Media:4MRV.4A.Transform.20200620.pdf |A.pdf]]) * Linear Transformation of Gaussian Random Variables ([[Media:4MRV.5A.LinearTrans.20200623.pdf |A.pdf]]) * Simulating Multiple Random Variables ([[Media:4MRV.6A.Simulation.20200624.pdf |A.pdf]]) * Sampling and Some Limit Theorem ([[Media:4MRV.7A.LimitTheorem.20200625.pdf |A.pdf]]) * Complex Random Variables ([[Media:4MRV.8A.ComplexRV.20200626.pdf |A.pdf]]) </br> ==''' Random Processes '''== === The Temporal Characteristics of Random Processes === * Random Processes ([[Media:5MRV.1A.RandomProcess.20210216.pdf |A.pdf]]) * Joint Distribution, Independence, Processes ([[Media:5MRV.2A.JointDistribution.20210220.pdf |A.pdf]]) * Stationary Random Processes ([[Media:5MRV.3A.Stationary.20220402.pdf |A.pdf]], [[Media:5MRV.3B.Stationary.20220817.pdf |B.pdf]]) * Covariance & Correlation of Random Variables ([[Media:5MRV.4A.CovCorrRV.20210910.pdf |A.pdf]]) * Covariance & Correlation of Random Processses ([[Media:5MRV.5A.CovCorrRP.20210911.pdf |A.pdf]]) * Example Random Processes ([[Media:5MRV.4A.Example.20210227.pdf |A.pdf]]) * Ergodic Random Processes ([[Media:5MRV.5A.Ergodicity.20211022.pdf |A.pdf]], [[Media:5MRV.7B.Ergodicity.20211215.pdf |B.pdf]]) * Measurement of Correlation Functions ([[Media:5MRV.6A.Measure.20201013.pdf |A.pdf]]) * Complex Random Processes ([[Media:5MRV.7A.Complex.20201022.pdf |A.pdf]]) </br> === The Spectral Characteristics of Random Processes === * Power Density Spectrum - Continuous Time ([[Media:6MRV.1A.PSpecCT.20210204.pdf |A.pdf]]) * Auto Correlation Function ([[Media:6MRV.2A.AutoCor.20201218.pdf |A.pdf]]) * Power Density Spectrum - Discrete Time ([[Media:6MRV.3A.PSpecDT.20201203.pdf |A.pdf]]) * Cross Power Density Spectrum ([[Media:6MRV.4A.CPSpec.20191108.pdf |A.pdf]]) * Cross Correlation Function ([[Media:6MRV.5A.CCorrel.20191114.pdf |A.pdf]]) * Noise Definitions ([[Media:6MRV.6A.Noise.20191121.pdf |A.pdf]]) * Power Spectrum of Complex Random Processes ([[Media:6MRV.7A.ComplexProc.20191125.pdf |A.pdf]]) </br> === Linear System with Random Inputs === * Continuous Time LTI System ([[Media:7LTI.1A.CTime.20191203.pdf |A.pdf]]) * Discrete Time LTI System ([[Media:7LTI.2A.DTime.20191211.pdf |A.pdf]]) * System Response ([[Media:7LTI.3A.Response.20191224.pdf |A.pdf]]) * Spectral Characteristics ([[Media:7LTI.4A.Spectral.20200104.pdf |A.pdf]]) * Noise Modeling ([[Media:7LTI.5A.Noise.20200122.pdf |A.pdf]]) <br> === Optimum Linear System === * Maximum SNR ([[Media:8OPT.1A.MaxSNR.20200128.pdf |A.pdf]]) * Minimum Squared Error ([[Media:8OPT.2A.MinSE.20200207.pdf |A.pdf]]) <br> === Noise in Some Application Systems === * AM Communication Systems ([[Media:9APP.1A.AM.20200212.pdf |A.pdf]]) * FM Communication Systems ([[Media:9APP.2A.FM.20200221.pdf |A.pdf]]) * Control Systems ([[Media:9APP.3A.Control.20200225.pdf |A.pdf]]) * PLL Systems ([[Media:9APP.4A.PLL.20200305.pdf |A.pdf]]) * Random Waveforms ([[Media:9APP.5A.RandWave.20200311.pdf |A.pdf]]) * Radar Systems ([[Media:9APP.6A.Radar.20200313.pdf |A.pdf]]) <br> ==''' Correlation and Power Spectra '''== # Correlation Functions of Random Signals ([[Media:RAND.1.A.Correlation.20121106.pdf |pdf]]) # Spectra of Random Signals ([[Media:RAND.2.A.Spectra.20121108.pdf |pdf]]) </br> ==''' Ergodicity, Statistics, Estimation '''== </br> ==''' Random Processes and Linear Systems '''== </br> # Time Domain Techniques for Noisy Signals ([[Media:RAND.3.A.Time.20130205.pdf |A.pdf]], [[Media:Dirichlet.pdf |B.pdf]]) # Frequency Domain Techniques for Noisy Signals # Correlation v.s. Convolution for Noisy Signals </br> # System Identification ([[Media:RP.SysId.1.A.pdf |pdf]]) # Matched Filter <ref>[[Understanding Digital Communications]], See Baseband Mod/Demod Section</ref> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ky951h287sp1qxtol8wi08b83zqpk73 2416462 2416460 2022-08-19T09:03:05Z Young1lim 21186 /* The Temporal Characteristics of Random Processes */ wikitext text/x-wiki ==''' Random Variables '''== === Single Random Variables === * Random Variables ([[Media:RV1.RVariable.1.A.20200427.pdf |A.pdf]]) * Distribution Function ([[Media:RV1.Distribution.2.A.201200428.pdf |A.pdf]]) * Density Function ([[Media:RV1.Density.3.A.20200429.pdf |A.pdf]]) * Functions of Random Variables ([[Media:RV1.RVFunction.4.A.20220317.pdf |A.pdf]]) * Gaussian Random Variables ([[Media:RV1.4.Gaussian.20200430.pdf |A.pdf]], [[Media:RV1.4B.Gaussian.20180314.pdf |B.pdf]]) * Other Distribution and Density Functions ([[Media:RV1.5.Other.20200501.pdf |A.pdf]]) * Conditional Distribution and Density Functions ([[Media:RV1.6.Conditional.20200506.pdf |A.pdf]]) </br> === The Characteristics of a Single Random Variable === * Expected Value ([[Media:RV2.Expectation.1.A.20200506.pdf |A.pdf]]) * Moments ([[Media:RV2.Moment.2.B.20200507.pdf |A.pdf]], [[Media:RV2.Moment.2.B.20180320.pdf |B.pdf]]) * Moment Generating Functions ([[Media:RV2.MFunctions.3.A.20200508.pdf |A.pdf]]) * Transformations of Random Variables ([[Media:RV2.Transform.4.A.20200514.pdf |A.pdf]]) </br> === Multiple Random Variables === * Vector Random Variables ([[Media:3MRV.1A.VectorRV.20200515.pdf |A.pdf]]) * Joint Distribution ([[Media:3MRV.2A.JointDist.20200518.pdf |A.pdf]]) * Joint Density ([[Media:3MRV.3A.JointDensity.20200521.pdf |A.pdf]]) * Conditional Joint Distribution and Density ([[Media:3MRV.4A.CondDistrib.20200527.pdf |A.pdf]]) * Statistical Independence ([[Media:3MRV.5A.StatIndep.20200423.pdf |A.pdf]]) * Sums of Random Variables ([[Media:3MRV.6A.RVSum.20200528.pdf |A.pdf]]) * Central Limit Theorem ([[Media:3MRV.7A.CLimit.20200612.pdf |A.pdf]]) </br> === The Characteristics of Multiple Random Variables === * Expected Values ([[Media:4MRV.1A.Expect.20200617.pdf |A.pdf]]) * Joint Characteristic Functions ([[Media:4MRV.2A.JChar.20200618.pdf |A.pdf]]) * Jointly Gaussian Random Variables ([[Media:4MRV.3A.JGauss.20200619.pdf |A.pdf]]) * Transformations of Multiple Random Variables ([[Media:4MRV.4A.Transform.20200620.pdf |A.pdf]]) * Linear Transformation of Gaussian Random Variables ([[Media:4MRV.5A.LinearTrans.20200623.pdf |A.pdf]]) * Simulating Multiple Random Variables ([[Media:4MRV.6A.Simulation.20200624.pdf |A.pdf]]) * Sampling and Some Limit Theorem ([[Media:4MRV.7A.LimitTheorem.20200625.pdf |A.pdf]]) * Complex Random Variables ([[Media:4MRV.8A.ComplexRV.20200626.pdf |A.pdf]]) </br> ==''' Random Processes '''== === The Temporal Characteristics of Random Processes === * Random Processes ([[Media:5MRV.1A.RandomProcess.20210216.pdf |A.pdf]]) * Joint Distribution, Independence, Processes ([[Media:5MRV.2A.JointDistribution.20210220.pdf |A.pdf]]) * Stationary Random Processes ([[Media:5MRV.3A.Stationary.20220402.pdf |A.pdf]], [[Media:5MRV.3B.Stationary.20220818.pdf |B.pdf]]) * Covariance & Correlation of Random Variables ([[Media:5MRV.4A.CovCorrRV.20210910.pdf |A.pdf]]) * Covariance & Correlation of Random Processses ([[Media:5MRV.5A.CovCorrRP.20210911.pdf |A.pdf]]) * Example Random Processes ([[Media:5MRV.4A.Example.20210227.pdf |A.pdf]]) * Ergodic Random Processes ([[Media:5MRV.5A.Ergodicity.20211022.pdf |A.pdf]], [[Media:5MRV.7B.Ergodicity.20211215.pdf |B.pdf]]) * Measurement of Correlation Functions ([[Media:5MRV.6A.Measure.20201013.pdf |A.pdf]]) * Complex Random Processes ([[Media:5MRV.7A.Complex.20201022.pdf |A.pdf]]) </br> === The Spectral Characteristics of Random Processes === * Power Density Spectrum - Continuous Time ([[Media:6MRV.1A.PSpecCT.20210204.pdf |A.pdf]]) * Auto Correlation Function ([[Media:6MRV.2A.AutoCor.20201218.pdf |A.pdf]]) * Power Density Spectrum - Discrete Time ([[Media:6MRV.3A.PSpecDT.20201203.pdf |A.pdf]]) * Cross Power Density Spectrum ([[Media:6MRV.4A.CPSpec.20191108.pdf |A.pdf]]) * Cross Correlation Function ([[Media:6MRV.5A.CCorrel.20191114.pdf |A.pdf]]) * Noise Definitions ([[Media:6MRV.6A.Noise.20191121.pdf |A.pdf]]) * Power Spectrum of Complex Random Processes ([[Media:6MRV.7A.ComplexProc.20191125.pdf |A.pdf]]) </br> === Linear System with Random Inputs === * Continuous Time LTI System ([[Media:7LTI.1A.CTime.20191203.pdf |A.pdf]]) * Discrete Time LTI System ([[Media:7LTI.2A.DTime.20191211.pdf |A.pdf]]) * System Response ([[Media:7LTI.3A.Response.20191224.pdf |A.pdf]]) * Spectral Characteristics ([[Media:7LTI.4A.Spectral.20200104.pdf |A.pdf]]) * Noise Modeling ([[Media:7LTI.5A.Noise.20200122.pdf |A.pdf]]) <br> === Optimum Linear System === * Maximum SNR ([[Media:8OPT.1A.MaxSNR.20200128.pdf |A.pdf]]) * Minimum Squared Error ([[Media:8OPT.2A.MinSE.20200207.pdf |A.pdf]]) <br> === Noise in Some Application Systems === * AM Communication Systems ([[Media:9APP.1A.AM.20200212.pdf |A.pdf]]) * FM Communication Systems ([[Media:9APP.2A.FM.20200221.pdf |A.pdf]]) * Control Systems ([[Media:9APP.3A.Control.20200225.pdf |A.pdf]]) * PLL Systems ([[Media:9APP.4A.PLL.20200305.pdf |A.pdf]]) * Random Waveforms ([[Media:9APP.5A.RandWave.20200311.pdf |A.pdf]]) * Radar Systems ([[Media:9APP.6A.Radar.20200313.pdf |A.pdf]]) <br> ==''' Correlation and Power Spectra '''== # Correlation Functions of Random Signals ([[Media:RAND.1.A.Correlation.20121106.pdf |pdf]]) # Spectra of Random Signals ([[Media:RAND.2.A.Spectra.20121108.pdf |pdf]]) </br> ==''' Ergodicity, Statistics, Estimation '''== </br> ==''' Random Processes and Linear Systems '''== </br> # Time Domain Techniques for Noisy Signals ([[Media:RAND.3.A.Time.20130205.pdf |A.pdf]], [[Media:Dirichlet.pdf |B.pdf]]) # Frequency Domain Techniques for Noisy Signals # Correlation v.s. Convolution for Noisy Signals </br> # System Identification ([[Media:RP.SysId.1.A.pdf |pdf]]) # Matched Filter <ref>[[Understanding Digital Communications]], See Baseband Mod/Demod Section</ref> </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] j75e92mfmwnindtzpngplo0m3v09241 1 201463 2416443 2364684 2022-08-19T05:13:05Z U3216256 2942574 /* Resources */ new section wikitext text/x-wiki ==Comments== Hey I just wanted to suggest you look at contrasting cultures like america and islam and the self esteem presented by individuals. Very interesting chapter and looking forward to reading it. All the best [[User:Iga.leszczynska|Iga.leszczynska]] ([[User talk:Iga.leszczynska|discuss]] • [[Special:Contributions/Iga.leszczynska|contribs]]) 02:19, 9 October 2015 (UTC) ==Heading casing== {| style="float: center; background:transparent;" |- | [[File:Crystal Clear app ktip.svg|48px|left]] | FYI, the convention on [[Wikiversity]] is for <u>lower-cased headings</u>. For example, use: <nowiki>==Cats and dogs==</nowiki><br><br> rather than<br><br> <nowiki>==Cats and Dogs==</nowiki><br><br> -- [[User:Jtneill|Jtneill]] - <small>[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]</small> 03:46, 19 October 2015 (UTC) |} == Resources == Hi These resources might be useful Hamamura, T., & Septarini, B. G. (2017). Culture and Self-Esteem Over Time: A Cross-Temporal Meta-Analysis Among Australians, 1978–2014. ''Social Psychological & Personality Science, 8''(8), 904–909. https://doi.org/10.1177/1948550617698205 Schmitt, D. P., & Allik, J. (2005). Simultaneous Administration of the Rosenberg Self-Esteem Scale in 53 Nations: Exploring the Universal and Culture-Specific Features of Global Self-Esteem. ''Journal of Personality and Social Psychology, 89''(4), 623–642. https://doi.org/10.1037/0022-3514.89.4.623 Bleidorn, W., Arslan, R. C., Denissen, J. J. A., Rentfrow, P. J., Gebauer, J. E., Potter, J., & Gosling, S. D. (2016). Age and Gender Differences in Self-Esteem-A Cross-Cultural Window. ''Journal of Personality and Social Psychology, 111''(3), 396–410. https://doi.org/10.1037/pspp0000078 [[User:U3216256|U3216256]] ([[User talk:U3216256|discuss]] • [[Special:Contributions/U3216256|contribs]]) 05:13, 19 August 2022 (UTC) ic6r03djaleaokqskj6ejdyla9vi1kf 0 203942 2416314 2415683 2022-08-18T22:00:58Z Young1lim 21186 /* Monads III : Mutable State Monads */ wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Function Oriented Typeclasses== === Background === * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Expressions ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) : - Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) : - Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) : - Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) : - Expressions ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) : - Lambda Calculus ([[Media:MP3.1F.Mut.LambdaCal.20220817.pdf |pdf]]) : - Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Computer programming]] pgbn52aw2zwij267glc71ulefreb2kx 2416318 2416314 2022-08-18T22:05:47Z Young1lim 21186 /* Monads III : Mutable State Monads */ wikitext text/x-wiki ==Introduction== * Overview I ([[Media:HSKL.Overview.1.A.20160806.pdf |pdf]]) * Overview II ([[Media:HSKL.Overview.2.A.20160926.pdf |pdf]]) * Overview III ([[Media:HSKL.Overview.3.A.20161011.pdf |pdf]]) * Overview IV ([[Media:HSKL.Overview.4.A.20161104.pdf |pdf]]) * Overview V ([[Media:HSKL.Overview.5.A.20161108.pdf |pdf]]) </br> ==Applications== * Sudoku Background ([[Media:Sudoku.Background.0.A.20161108.pdf |pdf]]) * Bird's Implementation :- Specification ([[Media:Sudoku.1Bird.1.A.Spec.20170425.pdf |pdf]]) :- Rules ([[Media:Sudoku.1Bird.2.A.Rule.20170201.pdf |pdf]]) :- Pruning ([[Media:Sudoku.1Bird.3.A.Pruning.20170211.pdf |pdf]]) :- Expanding ([[Media:Sudoku.1Bird.4.A.Expand.20170506.pdf |pdf]]) </br> ==Using GHCi== * Getting started ([[Media:GHCi.Start.1.A.20170605.pdf |pdf]]) </br> ==Using Libraries== * Library ([[Media:Library.1.A.20170605.pdf |pdf]]) </br> </br> ==Function Oriented Typeclasses== === Background === * Constructors ([[Media:Background.1.A.Constructor.20180904.pdf |pdf]]) * TypeClasses ([[Media:Background.1.B.TypeClass.20180904.pdf |pdf]]) * Functions ([[Media:Background.1.C.Function.20180712.pdf |pdf]]) * Expressions ([[Media:Background.1.D.Expression.20180707.pdf |pdf]]) * Operators ([[Media:Background.1.E.Operator.20180707.pdf |pdf]]) === Functors === * Functor Overview ([[Media:Functor.1.A.Overview.20180802.pdf |pdf]]) * Function Functor ([[Media:Functor.2.A.Function.20180804.pdf |pdf]]) * Functor Lifting ([[Media:Functor.2.B.Lifting.20180721.pdf |pdf]]) === Applicatives === * Applicatives Overview ([[Media:Applicative.3.A.Overview.20180606.pdf |pdf]]) * Applicatives Methods ([[Media:Applicative.3.B.Method.20180519.pdf |pdf]]) * Function Applicative ([[Media:Applicative.3.A.Function.20180804.pdf |pdf]]) * Applicatives Sequencing ([[Media:Applicative.3.C.Sequencing.20180606.pdf |pdf]]) === Monads I : Background === * Side Effects ([[Media:Monad.P1.1A.SideEffect.20190316.pdf |pdf]]) * Monad Overview ([[Media:Monad.P1.2A.Overview.20190308.pdf |pdf]]) * Monadic Operations ([[Media:Monad.P1.3A.Operations.20190308.pdf |pdf]]) * Maybe Monad ([[Media:Monad.P1.4A.Maybe.201900606.pdf |pdf]]) * IO Actions ([[Media:Monad.P1.5A.IOAction.20190606.pdf |pdf]]) * Several Monad Types ([[Media:Monad.P1.6A.Types.20191016.pdf |pdf]]) === Monads II : State Transformer Monads === * State Transformer : - State Transformer Basics ([[Media:MP2.1A.STrans.Basic.20191002.pdf |pdf]]) : - State Transformer Generic Monad ([[Media:MP2.1B.STrans.Generic.20191002.pdf |pdf]]) : - State Transformer Monads ([[Media:MP2.1C.STrans.Monad.20191022.pdf |pdf]]) * State Monad : - State Monad Basics ([[Media:MP2.2A.State.Basic.20190706.pdf |pdf]]) : - State Monad Methods ([[Media:MP2.2B.State.Method.20190706.pdf |pdf]]) : - State Monad Examples ([[Media:MP2.2C.State.Example.20190706.pdf |pdf]]) === Monads III : Mutable State Monads === * Mutability Background : - Types ([[Media:MP3.1A.Mut.Type.20200721.pdf |pdf]]) : - Primitive Types ([[Media:MP3.1B.Mut.PrimType.20200611.pdf |pdf]]) : - Polymorphic Types ([[Media:MP3.1C.Mut.Polymorphic.20201212.pdf |pdf]]) : - Continuation Passing Style ([[Media:MP3.1D.Mut.Continuation.20220110.pdf |pdf]]) : - Expressions ([[Media:MP3.1E.Mut.Expression.20220628.pdf |pdf]]) : - Lambda Calculus ([[Media:MP3.1F.Mut.LambdaCal.20220818.pdf |pdf]]) : - Non-terminating Expressions ([[Media:MP3.1F.Mut.Non-terminating.20220616.pdf |pdf]]) : - Inhabitedness ([[Media:MP3.1F.Mut.Inhabited.20220319.pdf |pdf]]) : - Existential Types ([[Media:MP3.1E.Mut.Existential.20220128.pdf |pdf]]) : - forall Keyword ([[Media:MP3.1E.Mut.forall.20210316.pdf |pdf]]) : - Mutability and Strictness ([[Media:MP3.1C.Mut.Strictness.20200613.pdf |pdf]]) : - Strict and Lazy Packages ([[Media:MP3.1D.Mut.Package.20200620.pdf |pdf]]) * Mutable Objects : - Mutable Variables ([[Media:MP3.1B.Mut.Variable.20200224.pdf |pdf]]) : - Mutable Data Structures ([[Media:MP3.1D.Mut.DataStruct.20191226.pdf |pdf]]) * IO Monad : - IO Monad Basics ([[Media:MP3.2A.IO.Basic.20191019.pdf |pdf]]) : - IO Monad Methods ([[Media:MP3.2B.IO.Method.20191022.pdf |pdf]]) : - IORef Mutable Variable ([[Media:MP3.2C.IO.IORef.20191019.pdf |pdf]]) * ST Monad : - ST Monad Basics ([[Media:MP3.3A.ST.Basic.20191031.pdf |pdf]]) : - ST Monad Methods ([[Media:MP3.3B.ST.Method.20191023.pdf |pdf]]) : - STRef Mutable Variable ([[Media:MP3.3C.ST.STRef.20191023.pdf |pdf]]) === Monads IV : Reader and Writer Monads === * Function Monad ([[Media:Monad.10.A.Function.20180806.pdf |pdf]]) * Monad Transformer ([[Media:Monad.3.I.Transformer.20180727.pdf |pdf]]) * MonadState Class :: - State & StateT Monads ([[Media:Monad.9.A.MonadState.Monad.20180920.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.9.B.MonadState.Class.20180920.pdf |pdf]]) * MonadReader Class :: - Reader & ReaderT Monads ([[Media:Monad.11.A.Reader.20180821.pdf |pdf]]) :: - MonadReader Class ([[Media:Monad.12.A.MonadReader.20180821.pdf |pdf]]) * Control Monad ([[Media:Monad.9.A.Control.20180908.pdf |pdf]]) === Monoid === * Monoids ([[Media:Monoid.4.A.20180508.pdf |pdf]]) === Arrow === * Arrows ([[Media:Arrow.1.A.20190504.pdf |pdf]]) </br> ==Polymorphism== * Polymorphism Overview ([[Media:Poly.1.A.20180220.pdf |pdf]]) </br> ==Concurrent Haskell == </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ==External links== * [http://learnyouahaskell.com/introduction Learn you Haskell] * [http://book.realworldhaskell.org/read/ Real World Haskell] * [http://www.scs.stanford.edu/14sp-cs240h/slides/ Standford Class Material] [[Category:Computer programming]] 54nbx6us2fe2q1zf6423507nasmv6um 0 213047 2416163 2415619 2022-08-18T13:51:52Z 2409:4061:2E9B:B5B0:6296:C3DA:E022:6602 Add objection wikitext text/x-wiki {{Wikidebate}} {{Religion}} [[File:Doré, Gustave - Paradiso Canto 31.jpg|thumb|alt=A man staring into a large swirl of angels and light|God as a point of light in {{W|Paradiso (Dante)|Dante's ''Paradiso''}}, engraving by {{W|Gustave Doré}}.]] [[File:Does God exist%3F.png|thumb|Visual snapshot of this debate sometime in the past.]] Questions about the nature of ultimate reality have been asked as long as humans have been conscious. For thousands of years, across thousands of cultures, belief in a supreme being has been more or less common but some have always called into question whether or not God exists or can even be known. By "God" we mean the metaphysically ultimate being, all-knowing, all-powerful, all-good, timeless, simple and devoid of any anthropomorphic qualities. We do not necessarily mean the Abrahamic God, although these ideas may share some overlap. So is there a God? == God exists == === Arguments for === * {{Argument for}} The laws of logic are necessary true propositions. Propositions are real entities, but cannot be physical entities; they are essentially thoughts.<ref>"Propositions are not on paper, in your brain, or anywhere else" [https://www.academia.edu/630293/Propositions_Are_Not_on_Paper_In_Your_Brain_or_Anywhere_Else Direct Paper link]</ref>{{Clarify}} So the laws of logic are necessarily true thoughts. Since they are true in every possible world, they must exist in every possible world.<ref>"From Necessary Truth to Necessary Existence" [http://www.joshualrasmussen.com/articles/existence.pdf Direct paper link (PDF)]</ref> But if there are necessarily existent thoughts, there must be a necessarily existent mind; and if there is a necessarily existent mind, there must be a necessarily existent person. A necessarily existent person must be spiritual in nature, because no physical entity exists necessarily. Thus, if there are laws of logic, there must also be a necessarily existent, personal, spiritual being.<ref>{{Cite journal|last=Anderson|first=James N.|last2=Welty|first2=Greg|date=2011|title=The Lord of Noncontradiction: An Argument for God from Logic|url=https://www.semanticscholar.org/paper/The-Lord-of-Noncontradiction%3A-An-Argument-for-God-Anderson-Welty/e0f49c35c70e54174bf201c59f4abfadd223c583?p2df|doi=10.5840/PC201113229}}</ref><ref>"A Defense of Theistic Argument from the Law of Non-contradiction", [https://oaktrust.library.tamu.edu/bitstream/handle/1969.1/157717/NAVARRO-THESIS-2016.pdf?sequence=1&isAllowed=y Direct paper link (PDF)]</ref><ref>{{Cite journal|last=Weaver|first=Christopher G.|title=Why is There Anything?|url=https://www.academia.edu/20392976/Why_is_There_Anything|journal=Two Dozen (or so) Arguments for God: The Plantinga Project}}</ref><ref>{{Cite journal|last=Rasmussen|first=Joshua|date=2009-01-01|title=From a necessary being to god|url=https://www.academia.edu/630287/From_a_necessary_being_to_god|journal=International journal for philosophy of religion}}</ref> ** {{Objection}} This proposition is not correct: "Since they are true in very possible world, they must exist in every possible world". Why? Because the expression "possible world" has two different meanings. Is an amphibology. In first case, its meaning is: "possible world that we imagine". In the second case, its meaning is: "possible world that can exist". * {{Argument for}} Existence of God has been proven ''a priori'' using higher-order logic and reasonable axioms.<ref>Axioms:[https://arxiv.org/pdf/2001.04701]<br>A1 Self-identity is a positive property, self-difference is not.<br>A2 A property entailed or necessarily entailed by a positive property is positive.<br>A3 The conjunction of any collection of positive properties is positive.</ref><ref>"A (Simplified) Supreme Being Necessarily Exists -- Says the Computer!", [https://arxiv.org/pdf/2001.04701 Direct Paper link (PDF)]</ref>{{Clarify}} * {{Argument for}} Various phenomena in the Universe appear to be designed and suggest a designer, God.{{Example}} ** {{Objection}} Consider Auroras, these phenomena occurs due to scattering of charged particles in atmosphere which exists as of evolutionary phases of Earth. Not designed by anyone. * {{Argument for}} Language plays an integral role in the laws of nature and of DNA. As encoded meaning, language is non-material in its ultimate essence. Apart from something akin to the human mind, there are no serious candidates for explaining how linguistic phenomena might otherwise arise. The only reasonable way to account for the linguistic aspects of the laws of nature and of DNA is an intellect with capacities so vast that most people would immediately identify this entity as God.<ref>"A Linguistic Argument for God's existence", [http://www.etsjets.org/files/JETS-PDFs/58/58-4/JETS_58-4_771-86_Baumgardner&Lyon.pdf Direct Paper link (PDF)]</ref> * {{Argument for}} Under a naturalistic worldview, the coincidence of all the cognitive faculties required for knowledge is highly unlikely, since evolution favors survival, not truth, and any overlap of survival and truth is highly implausible.<ref>"IF KNOWLEDGE THEN GOD: THE EPISTEMOLOGICAL THEISTIC ARGUMENTS OF PLANTINGA AND VAN TIL", [http://www.proginosko.com/docs/If_Knowledge_Then_God.pdf Direct paper link (PDF)]</ref><ref>{{Cite web|url=https://www.spectator.co.uk/2019/10/donald-hoffmans-the-case-against-reality-is-hard-to-get-your-head-around/|title=Donald Hoffman's The Case Against Reality is hard to get your head around|date=2019-10-12|website=The Spectator|language=en-US}}</ref><ref>{{Cite web|url=https://www.quantamagazine.org/the-evolutionary-argument-against-reality-20160421/|title=The Evolutionary Argument Against Reality|last=Amanda Gefter|website=Quanta Magazine|language=en}}</ref><ref>{{Cite web|url=https://iai.tv/articles/the-evolution-of-reality-auid-1274|title=The Evolution of Reality {{!}} Why the world is not how we see it|date=2019-11-29|website=IAI TV - Changing how the world thinks|language=en-GB}}</ref> If the reliability of cognitive faculties is low, any belief is not warranted including metaphysical naturalism, which becomes self-defeating.<ref>"Probability And Defeaters", [http://www.proginosko.com/docs/If_Knowledge_Then_God.pdf Direct Paper link (PDF)]</ref> ** {{Objection}} This doesn't imply that God exists, only that naturalism is false. * {{Argument for}} Chanting of names of God (like Hare Krishna) gives immense happiness to devotees. One can't expect such happiness from a non-theistic worldview, but God existence could explain such happiness. ** {{Objection}} Chanting in unison in a crowd during a sports game can cause similar feelings of happiness. Happy feelings from chanting is not evidence for God. * {{Argument for}} For the Universe to exist, there must be an uncaused cause, God, or the Universe is eternal. So either there's no explanation for God, or there's no explanation for the Universe. The Big Bang is not an explanation, it's a description with no explanation for why it came to be. We then have to rely on chance and happenstance. God fits the picture better. ** {{Objection}} If there must be an uncaused cause, then why can't the Universe be the uncaused cause? Adding God to the chain only adds unnecessary complexity making it a less likely explanation. Just start with a natural, unintelligent and minimally powerful uncaused cause. *** {{Objection}} Everything that we know in the Universe has a cause external in space and previous in time. Why should the Universe itself be any different? But God is timeless, so the same rules don't necessarily apply. **** {{Objection}} The Big Bang was the beginning of spacetime. As such, we cannot say that the particles that cause the Big Bang follow the same rules. ** {{Objection}} Not knowing how the Big Bang came about is not proof or evidence that God exists. ** {{Objection}} This assumes the need for an explanation, which is just a [[Wikipedia:False equivalence|false equivalence]]. Things within the Universe require an explanation, but the Universe itself does not require an explanation, because we explain things inside the Universe based on the assumption that there is an external factor already explained. This does not apply to the Universe itself because there is nothing external to the Universe by definition. Either the Universe caused itself or simply has no cause. This might strike many as nonsensical but that is simply because they are unconsciously and inappropriately extending the logic of parts to the whole. *** {{Objection}} Everything we know has a cause different from itself. Thus, we must pursue this principle to its natural end, and conclude that the Universe has a cause different from itself, unless contrary evidence can be provided. You can't merely assume that the Universe is all there is and write in an exception into the definition of the Universe. **** {{Objection}} If you question the origin of the Universe, it's like questioning the water cycle. Where does evaporated water come from? From condensed water. Where does condensed water come from? From precipitated water. Where does precipitated water come from? From ran off water. Where does ran off water come from? From evaporated water. And the cycle goes on and on and on. Similarly, if you question the origin of the Universe, I would say from the Big Bang. If you question the origin of the Big Bang, I would say from the Singularity. If you question the origin of the Singularity, I would say from the Big Crunch. If you question the origin of the Big Crunch, I would say from the Universe. If God does exist, why did he make principles that lead to his non-existence? If God created science, why we can't connect science to him? Even principles are self-looping, so there's also indeed no creator. Energy loops, matter loops, the burnt paper turns into carbon and gas, it just turns into something else to become paper again. The Universe exists in the first place because it loops so there's no beginning or end. The [[Wikipedia:Conservation of energy|law of conservation of energy]] states that energy can neither be created nor destroyed, only be transferred or transformed from one form to another. This implies that energy and the Universe existed in the first place. There's no creator. Just accept the fact that there is no zero in the Universe, there's a fixed number of materials and energy. The Universe is just a continuous loop of energy. It inflates through the Big Bang, reaches the maximum inflation, deflates through the Big Crunch, and reaches the maximum deflation, and the cycle goes on. * {{Argument for}} God may not have provided evidence to everyone because not all people want God, so God may be fulfilling their desire by not giving them sufficient evidence. ** {{Objection}} Absence of evidence is not evidence of presence. Lack of evidence doesn't imply that God is fulfilling the desire of people who don't want God, nor that God exists. *** {{Objection}} Subjective evidence suggest that either God exists who wants to known to people as God may choose or either all people deluded, lier or anything else. But principle of incredulity say that we should believe in experience unless there is good reason to not otherwise. Now if there is positive evidence for non existence of God, it can't counted. But as far as we know, there is no positive evidence of non existence of God. Therefore subjective evidence can't be rejected. Now question one can ask is why would God don't provide subjective evidence to everyone? So, one explanation is not all people want God, and God may not provides evidence to those who doesn't want God or who doesn't qualify. * {{Argument for}} If the Universe is a simulation, then there's a creator beyond spacetime who designed and fine-tuned the simulation. Now either the Universe is a simulation or the Universe is base reality. If the Universe is a simulation, it can be a simulation level 1, level 2, level 1000 or any level. If the Universe is base reality, it can exist in only one way. Therefore, there're more possibilities of the Universe being a simulation than base reality. ** {{Objection}} Even if our Universe is a simulation and has a creator, that doesn't imply a creator of base reality. Our Universe could be a simulation one level below base reality, which implies a creator outside of our local spacetime that exists in base reality, but that does not imply a creator outside spacetime for that base reality since the argument only applies to simulations. * {{Argument for}} Difference between natural and supernatural is artificial. It depends on definition of Supernatural, and gives reason to reject God on artificial grounds. If Supernatural is defined as something which science can't explain, then most of phenomenon can't explainable by science because it depends on our observations, and it doesn't necessarily that observations shows real content of reality out there. If Supernatural is defined as something done that violates laws of physics, there is contradiction in definition. Because laws of physics are not fixed set of laws, but we try to find laws by observations and whatever comes, we describes it as laws of physics. If Supernatural is defined as some actions by agents, it is unclear whether to consider ghost as Supernatural because if some explanation found for it, it becomes natural. And it isn't necessary that nature behaves on it's own. If physical reality doesn't know what it does, or how it does, then it doesn't have to continue to exist. All physical reality can suddenly appears and disappears. It isn't necessarily for reality to such as it should change, it can be such as it remains static for infinity, and nothing happens. It behaves in such a way that is not expected if it run on it's own. However if material nature works under directions of God, it is exactly what we normally expect. And evidence from spontaneous emergent, chaotic unpredictability further suggest material nature can be works under supervision of God, as scripture like Bhagavad Gita also suggest.<ref>{{Cite web|url=https://asitis.com/|title=Bhagavad Gita As It Is Original by Prabhupada|website=asitis.com|access-date=2022-08-16|quote=This material nature is working under My direction, O son of Kunti, and it is producing all moving and unmoving beings. By its rule this manifestation is created and annihilated again and again.}}</ref> * {{Argument for}} The Universe is fine-tuned to support life. This fine-tuning is less surprising and even probable if God exists, but highly unlikely in a godless Universe.<ref>{{Cite journal|last=Chan|first=Man Ho|date=2017-05-24|title=The fine-tuned universe and the existence of God|url=https://repository.hkbu.edu.hk/etd_oa/447|journal=Open Access Theses and Dissertations|quote=To conclude, after a comprehensive study of the fine-tuning arguments, the fine-tuning phenomena strongly support the theistic worldview.}}</ref> ** {{Objection}} This argument is biased to carbon-based lifeforms. Life could exist in ways that are not based on carbon meaning the fine-tuning of the universe is not as necessary for life to exist as a carbon-based life form might think. ** {{Objection}} Considering multiverse theory, there may be many universes with their own cosmological constants. In this scenario, of course a life form in a "fine tuned" universe sees fine tuning even though that life form's existence can be relegated to chance, not God. This is an example of selection bias. The multitude of other universes inhospitable to life would never develop life forms capable of posing such questions. * {{Argument for}} If the Universe is ultimately meaningless, devoid of any purpose or design, then all-purpose and meaning one assigns should be imaginary. Because if the Universe doesn't have ultimately any purpose, then life just happened to appear in accidental ways such as it doesn't have any inherent meaning or purpose. However, if God is the reason for the existence of us, there can be the purpose of life. In some religions, it is suggested that God is originally in the spiritual universe with living entities. When a living entity doesn't want God, God makes a material universe for the fulfilment of the desire of them of not wanting God. The material universe is created temporarily such as living entity can realise their connection with God, and when they want God, they can return to God, which may be the purpose of life. ** {{Objection}} This argument assumes its own premise. It relies on the presumption that 'meaning' is an inherent characteristic of life, which is just a rudimentary form of Anthropocentrism. There is no reason why there should be 'meaning' to one's life for life to exist. * {{Argument for}} The Universe follows mathematical laws independently of how humans describe them. So mathematics must exist independently of human minds. But all mathematics needs axioms. How can axioms exist independently of human minds? An axiom generator system is needed, or meta-axioms that create the axioms required for mathematical laws. But how can meta-axioms exist? Meta-meta-axioms are needed, and so on. This makes it implausible or even impossible for any mathematical laws to exist. However, it's not impossible if mathematics exists in the mind of God. Because God can conceptualize mathematics. * {{Argument for}} Objective morality exists and requires an absolute moral authority. Without some absolute authority, all morality is an individual interpretation of morals or shared morality decided by groups of people, which is ultimately subjective. If morality comes just from the survival of fittest, it can be moral to steal or murder, if it results in survival. However, most people regard it as not moral. This absolute authority is equivalent to God who may have created humans and provided some rules or laws which may be inherent in us. ** {{Objection}} The argument assumes its own premise, that objective morality exists, an assumption that isn't necessarily correct but fundamentally necessary for the functionality of the argument. Since, it is from the idea of the existence of objective morality that this argument derives the existence of an absolute moral authority, and from the existence of an absolute moral authority, the existence of God, we necessarily have to conclude that, since no evidence is provided for the positive claim of the existence of objective morality, then no evidence has been provided for the positive claims of the existence of an absolute moral authority or a God. Since the existence of objective morality is provided merely as an assumption lacking supporting evidence, then the existence of an absolute moral authority or the existence of a God is just that, an assumption lacking supporting evidence. * {{Argument for}} Suppose there was no intelligence behind the Universe, no creative mind. In that case, nobody designed my brain for the purpose of thinking. It is merely that when the atoms inside my skull happen, for physical or chemical reasons, to arrange themselves in a certain way, this gives me, as a by-product, the sensation I call thought. But, if so, how can I trust my own thinking to be true? It's like upsetting a milk jug and hoping that the way it splashes itself will give you a map of London. But if I can't trust my own thinking, of course I can't trust the arguments leading to Atheism, and therefore have no reason to be an Atheist, or anything else. Unless I believe in God, I cannot believe in thought: so I can never use thought to disbelieve in God.<ref>The original version of this argument was brought forth by {{W|C. S. Lewis}}.</ref> ** {{Objection}} A similar argument applies to the theist. Suppose an intelligence designed our brains. This could mean that our brains were designed for thinking rationally, or it could mean that our brains were designed to come to the wrong conclusions. How do we know which is true? We can't. If I can't trust my own thinking, I can't trust the arguments leading to theism. Assuming God exists does not lead to knowledge that we think rationally. *** {{Objection}} But problem is more serious in universe which just happened to be like it, which is accidental, mindless, purposeless, arbitrary. In such godless universe, there would millions or perhaps billions of coincidence required for brain to function exact right, and there would much more possibility that brain is unreliable. How can anyone know that brain is anything more than quantum fluctuations? In such, how can one can sure that quantum fluctuations should behaves exactly by which it makes logical decision when it doesn't know anything? How can one have faith in such quantum fluctuations such as it shows real reality than shows illusion of reality? However if God creates physical Universe for living entity for giving chance to those who don't want them, then living entity can realize God and can comes to conclusion of God by realizing that material universe is illusionary, and full of suffering which is not ultimate place for him, because of which they can comes to God. In such state, they can have faith in God and can surrender God. Additionally if God wants, God can make known their existence to someone at absolutely certain such as one can becomes certain that God exist, and can realize purpose of existence. ** {{Objection}} Nobody designed the stomach for digesting food either, yet with modern biology, we know through a process of evolution over millions of years that the digestive system evolved naturally through an accumulation of beneficial steps. Just like the eye, just like the mind. In short, evolution provides a better explanation than God. *** {{Objection}} Not really. How can physical reality behaves so perfectly? If everything is ultimately stochastic thermodynamics progress and quantum fluctuations, why such great coincidence occurs by which it works so well, such as it seems design? If it is not really design but apparent design, it can also be that there is no evolution but apparent evolution, and all entropic thermodynamics progression which doesn't know anything, just happened to behaves like that. But it's seems highly implausible, however it becomes likely if God direct nature or physical reality. ** {{Objection}} If God designed our minds, then why is our reasoning ability so imperfect? Why do people confuse correlation with causation? Why do people believe in astrology and other obvious nonsense? *** {{Objection}} Intelligence is a gift endowed to humans that cannot be explained through evolution. It's a gift, but humans are not perfect. If so, we would be divine. How you choose to believe this happened is a matter of faith.{{Clarify}} * {{Argument for}} If you trust your own thinking, then you must have an absolute perfection, a highest logic, against which to measure your thoughts. ** {{Objection}} Why must that vision of perfection actually exist? Is it not merely a measure in our minds against an imaginary height of perfection? Thought does not imply existence. * {{Argument for}} A huge number of humans, throughout centuries, have reported all sorts of encounters with God, from the personal internal type to shared apparitions and public miracles. Experiences differ in many ways, but they all support a common cause: the existence of God. It is highly implausible that many reports all be false or misled, and as we trust our experience unless we have good reason to think otherwise, it is reasonable to think that God is reason behind such experiences. ** {{Objection}} Testimonies are generally inconsistent unless they are sufficiently connected by cultural myths or sufficiently vague. Many cultures have the concept of magic, not because magic is real, but because magic is a sufficiently vague concept to hold many different conceptions of it. Unsurprisingly, the reports about God are more similar to the more closely connected the cultures of the witnesses are, which indicates they are cultural phenomena rather than independent observations that corroborate each other. *** {{Objection}} How testimony is inconsistent? In general experience of God is very difficult to describe or explain. Think how do you describe the experience of the color red to someone who is blind from birth? And cultural explanation applies to everything we experienced. Whatever we experienced is shaped by culture and our background. Like if someone who knows about tree interprets the experience of light for seeing something different than someone who doesn't know about the tree, but it doesn't mean such experience of light is not real. And when we claim that the experience of God must be the same for all, it is not reasonable. Suppose someone prays to God in the form of Lord Krishna, now if God appears as Jesus, he probably requests God to appears in form in which he remembers. Same as if someone remembers God in form of Jesus, and if they experience God in form of other, they may request God that they like God in form of Jesus, therefore, please give me appearance in the form in which I remember. Therefore, it is reasonable to think that God may not experience the same as for every devotee because devotees want God in a specific form they love, and God may fulfill a desire of devotees. Also, it is reasonable to think that God has unlimited forms, and also formless because they are absolute. Also if we reject experience because just they are different, should we also reject the experience of reality if it experienced inconsistently by different people? Most people may answer no. So why should we reject the experience of few billions of people who think they have experienced God? Unless we have positive evidence for the non-existence of God, it shouldn't be rejected. ** {{Objection}} To know that all these testimonies are testimonies about the same thing, we should know their object (God) independently from these testimonies. We should first know the object we are talking about (God) in order to be able to recognize that all reports deal with the same object. *** {{Objection}} All knowledge ultimately reduces to the testimony of one or more people. Objects cannot be known independently of all testimony. **** {{Objection}} This claim attempts to homogenize testimony. It fails to recognize not all testimony is created equal; some testimony is backed by evidence while some is not. ** {{Objection}} Eyewitness testimony is notoriously unreliable. Many people have testified seeing thousands of people celebrate the 9/11 attacks in New Jersey, even though this event never took place. If you put an idea in people's minds, some people will believe that they personally saw whatever that idea is, even if the idea turns out to be false. *** {{Objection}} But people who have experienced God includes medical professionals who know about different psychological phenomenon, and also they have experienced God in which they may differentiate between psychological condition and the real one. Numbers of Medical professionals, physicists, cosmologists experienced God, which makes it less likely to be a purely psychological phenomenon. Additionally, many people go through psychological tests after such experience, in which many are shown to be normal and healthy. If the experience of God is purely delusional, they should show signs of delusion, but instead, people show no such phenomenon. And if God wants, God can make someone know about their existence with absolute certainly. ** {{Objection}} Although there're millions of believers, there're not that many of eyewitnesses, relative to the number of believers. *** {{Objection}} It can be because the connection with God may be difficult. Some people who love God so much, God may give him direct realization than someone who is not of that level. However many people are spiritually shallow, but they have experienced God slightly like newborn babies experience light slightly and as time passes, he can more clearly experience light. But it doesn't mean most newborn doesn't have any experience of light, and maybe the same as most people who are spiritually shallow, may not experience God deeper. ** {{Objection}} [[Wikipedia:Argumentum ad populum|''Argumentum ad populum'']]''.'' *** {{Objection}} It can argumentum ad populum only if it claims like "People believe in God therefore God exists". But this is not the same. This argument offers that because millions of people have experienced God, and because we have no positive evidence of the non-existence of God, the experience of God can't be rejected as delusion, hallucinations or lies because else one can reject all experience of all people by same without any positive evidence for its non-existence. ** {{Objection}} Personal experiences can't be accounted for as evidence because there's no evidence to support these reports. How would one prove that these encounters were not a trick of the mind, such as mirages or sleep paralysis, or completely fictional? Experiences of God's existence could easily all be hallucinations, delusions, or attributions of a supernatural cause to natural phenomena which, them being as theists, leading to them being caused by God. I can hallucinate too when I am a theist and I can refer to it as a supernatural occurrence caused by God. It leads to the assumption that it is God-caused because you are a theist. *** {{Objection}} Personal experience can be accounted for as evidence. Otherwise, it would not be reasonable to believe (unless you personally experience it yourself) that all humans are conscious, certain drugs induce hallucinations or certain psychological phenomena exist, such as dreams, sleep paralysis, Alice in Wonderland syndrome, phantom limb, etc. *** {{Objection}} When corroborated by so many people, they cannot be so handily dismissed, though. ** {{Objection}} Attributing some encounter with nature or with some unusual phenomenon to the existence of God is a speculative conclusion based on a subjective assessment of the available information. Is contemplation of the beauty of a flower an encounter with God, or simply an appreciation of the fractal nature of the cellular structure that has evolved over millions of years? Are reports of virgin birth evidence of a miracle, or simply a translation error, a misunderstanding of the mechanisms of conception, or marketing hype? *** {{Objection}} Trying to prove the existence of miracles scientifically is like trying to prove that Gandhi was Indian ''linguistically''. It is the wrong outlet, as we can never re-experience what they did, along with millions of others. These miracles are a matter of faith to them. * {{Argument for}} Science is built on materialistic assumptions, so it already excludes the existence of God. ** {{Objection}} Current science accepts the existence of many immaterial entities, such as light, energy, spacetime and mathematical entities, so it's not built on materialistic assumptions.<ref>{{Cite journal|last=Moulines|first=C. Ulises|date=1977|title=Por qué no soy materialista|url=http://www.jstor.org/stable/40104059|journal=Crítica: Revista Hispanoamericana de Filosofía|volume=9|issue=26|pages=25–37|issn=0011-1503}}</ref> ** {{Objection}} Even if science were built on materialistic assumptions and excluded the existence of God, that doesn't imply that God exists. === Arguments against === * {{Argument against}} Since there are many religions in the world, all of which have their own idea of God and their own ideas of an afterlife (i.e. heaven/hell, reincarnation, etc.), then which God is real, and which afterlife is real? ** {{Objection}} To claim there is no rational basis for belief in any one religion if at most one can be correct out of thousands is incorrect. The mistake here is assuming all religions have equal probability of being correct, which is not the case: some religions are internally inconsistent, for example, and so have a far lower probability of correctness than more consistent religions. To claim that the credibility of a religion is dictated by its followers' backgrounds is also incorrect; the origin of someone's belief has little bearing on whether the belief is true. ** {{Objection}} God may have appeared in different parts of the world in different ways so that people of that place and time can understand God, according to circumstances of that time. Therefore, even though God appears different in different religions, it can be same. *** {{Objection}} If God provides different versions of Himself to different people at different times, then the definition of God is strained. What are we talking about if the definition of God depends on the time and place? Suggesting that God only reveals himself in ways that are not equivalent to His true nature suggests that God hasn't really revealed Himself at all. It suggests he has only revealed caricatures of Himself dependent on the cultures to which he displays these caricatures to. ** {{Objection}} This does not exclude the possibility that one of those religions might turn out to be the correct one, with the correct idea of God and an afterlife, even if that religion contradicts all other religions and this means all religions except for one of them turn out to be wrong. *** {{Objection}} If at most one religion can be correct, out of the many thousands that exist, and it is possible that they are all wrong, there is no rational basis to believe in any one religion over all the others. Most people who believe in a religion do so because of social reasons, for instance being raised in that religion, or falling in love with and marrying a follower of that religion, rather than any rational basis to believe that their particular religion is any more likely to be true than all the other religions it contradicts. No one religion is obviously superior to all of the others enough to persuade all the followers of the other religions to convert. **** {{Objection}} We aren't debating over the merits of Christianity or Islam for example--these are matters of faith. We are debating whether a God of some sort exists as a starting ground ***** {{Objection}} It is not about the merits of a religion. This is about a lack of communication (or effective communication) which is weak evidence that there is no God. **** {{Objection}} All monotheistic religions basically have the same main idea, which is worship of a higher power, a God. The religion itself is a set of values or ideas that one certain group "binds" to a higher power. This may be meant in a way of pleasing or satisfying the higher power, which we as humans often feel the need to do. Take away these values and traditions, which is most likely human made. What we have left is the acknowledgement of a higher power. Monotheistic religions are just different ways of saying the same thing. ***** {{Objection}} Well, you're cherry picking monotheistic religion which already is a set that can include the existence only of a single god, which is then worshiped. If you take a larger set, such as all religion, you get differing beliefs even regarding the nature and even the existence of a higher power. It is not just the traditions that are most likely human made, it is the very notion of god(s), having a clear progression from more utilitarian deities to more abstract ideas which are more resistant to empirical disproof. * {{Argument against}} God's existence would imply that he can change the past. This would imply that some things happened and didn't happen at the same time and in the same sense. But contradictions are impossible, so not only God doesn't exist: his existence is impossible. ** {{Objection}} This is not a contradiction. Only the edited version of history actually happened. If God, an omnipotent being, changed the path of history, history is changed. As odd as it seemed, the previous scenario never happened. *** {{Objection}} However you are still presenting a contradiction: a scenario which existed and never existed. ** {{Objection}} That wouldn't be a contradiction, because God would know that He changed the past (per being all-knowing). So things that God changed would have happened at the same time, but not in the same sense. God would be able to distinguish them. *** {{Objection}} Doing one impossible thing is no more difficult than doing two impossible things.<ref>"For why should God not be able to perform the task in question? To be sure, it is a task—the task of lifting a stone which He cannot lift—whose description is self-contradictory. But if God is supposed capable of performing one task whose description is self-contradictory—that of creating the problematic stone in the first place—why should He not be supposed capable of performing another—that of lifting the stone? After all, is there any greater trick in performing two logically impossible tasks than there is in performing one?" Frankfurt, Harry. "The Logic of Omnipotence" first published in 1964 in ''Philosophical Review'' and now in ''Necessity, Volition, and Love''. Cambridge University Press November 28, 1998 pp.1–2</ref> ** {{Objection}} God's omnipotence is often described as "can do anything that is not logically impossible", or similarly defined so as to rule out paradoxes deriving from His omnipotence. *** {{Objection}} If God's omnipotence is limited by "can do anything that is not logically impossible", then the fact that God is also defined as omniscient and knowing everything would mean God possesses complete knowledge ahead of time of all things that He will do, and is bound by logic to do what He predicted He would do. This would reduce his omnipotence to complete powerlessness since he would never have any choice at all other than to do what he predicted he would do, and thus he would not have any power at all. **** {{Objection}} By omnipotence and omniscience, which theists usually refer to, we are speaking of something vastly beyond our understanding and quantifying it in human terms. Omnipotence implies that he has the choice to exercise his omnipotence in any particular way, which he knows what his choice will be. Knowing what you plan to do in a circumstance beforehand by no means makes one powerless ** {{Objection}} If God can change the past, it does not mean he would; this is dictated by his character and motivations, which are particular to different religions. This is similar to the argument that if God can create an immovable object he is not all powerful, but if he cannot then he is not all powerful either: it is wordplay, since such a definition is nonsensical. God is unbounded by the universe's laws. * {{Argument against}} If we are talking about a God which affects the physical world in some way, then saying that God exists is an empirical statement. But there is no hard evidence supporting that statement, and evidence is necessary to prove an empirical statement. ** {{Objection}} God is outside of the confines of our physical world, and whether or not God is actively involved in this world is not being debated. Therefore, saying that God exists is not an empirical statement, it's a metaphysical statement, and empirical proof for metaphysical statements is not necessary or even possible. *** {{Objection}} By this logic, there is no such thing as a false statement. Anything can be inferred. I have a 3 headed donkey in my backyard except you can't see it, it doesn't smell, and you can't touch it; there is no physical evidence for its existence whatsoever, but it exists metaphysically, above the plane of our existence. At this point, you have to ask, what do I mean when I say the donkey "exists"? If physical evidence isn't required to say that something exists, then I'm correct when I say that a 3 headed, 5 headed, 50,000 headed, and 50,001 headed donkey all exist in my backyard. ** {{Objection}} God may have perfect reason for not giving hard evidence to everyone. There are some people who don't like God, don't want God or if they know that God exist, they may becomes envy of God. God may not hides their existence from someone who doesn't want God as fulfilling their desires, and which may be better for them. As scriptures like Bhagavad Gita says, it can be that God manifest in proportion to one surrender to God.<ref>{{Cite web|url=https://asitis.com/|title=Bhagavad Gita As It Is Original by Prabhupada|website=asitis.com|access-date=2022-08-16|quote=Always think of Me and become My devotee. Worship Me and offer your homage unto Me. Thus you will come to Me without fail. I promise you this because you are My very dear friend.}}</ref><ref>{{Cite web|url=https://asitis.com/|title=Bhagavad Gita As It Is Original by Prabhupada|website=asitis.com|access-date=2022-08-16|quote=Abandon all varieties of religion and just surrender unto Me. I shall deliver you from all sinful reaction. Do not fear.}}</ref> * {{Argument against}} God is conceived as all-good, all-knowing and all-powerful. So, if God exists, then under any ordinary definition of evil, evil shouldn't exist. But evil clearly exists. Therefore, God does not exist. ** {{Objection}} There are several problem with it. First we don't know what God exactly do if they have perfections. We think God should do this or other, but we can't imagine it from way God would think. We can think if I am photon, I would do this or other, but we don't know what exactly happens when one be a photon. God who is outside of material realism, may have perfect reason to produce seemingly imperfect universe. We don't know God's reason for something, and if we know it we also think that everything is like it should be as best. Secondly, we define ourselves as moral authority and decides what is good or not, but God may beyond good and evil, the absolute one. Because God is free, if living entity is part and parcel of God, living entity should be free, and because of it they have minutes of free will. Now if living entity not like God or doesn't want God, God may fulfill his desire to becomes independent of God, and creates material world where living entity can be independent of God, and enjoy without God. Now as living entity has free Will, they can choose actions for which they are responsible, to which God doesn't intervene as they respect free will. But if they choose to do something bad, it is entirely on them because they have chosen it from their will. God may created suffering in material world to let someone know that it is not permanent place and living entity can again realize God, and can return to them. ** {{Objection}} God could have given humans the power to do evil. If humans can do evil, then evil can exist despite there being an all-powerful, all-knowing and all-good God. Moral responsibility is not hereditary. If my (grownup) child commits a crime, no society will (or should) blame me for it. Similarly, if a human does evil, we shouldn't blame God for it. Giving the power to do evil is not the same as doing it. God may even be the (metaphysical) cause of evil, while not being morally responsible for it. But no one contends that God raised us that way. He merely gave us the freedom to choose. In fact, because we turned away from his benevolence, evil arose, according to theists. Should we not live with the consequences of our actions and disobedience, then? *** {{Objection}} This would still make God responsible for evil, albeit indirectly. Moral responsibility is partially hereditary. If I knowingly raise my child in a way that makes it highly likely they will commit a crime when grown up that makes me responsible for that crime and subject to blame. An omniscient God should have known that if he gave humans the freedom to choose, we would do evil. Therefore, he allowed evil to exist, which contradicts the benevolent nature of God. *** {{Objection}} Even if God gave humans free will and the power to do evil, this doesn't imply that there should be evil. A world where free will exists but evil does not is logically possible. God can create any world that is logically possible, so God chose a world where there is unnecessary evil. This contradicts the all-powerful and all-good nature of God. *** {{Objection}} We haven't dealt with the problem of natural evil. Is cancer also the consequence of human disobedience? ** {{Objection}} Assuming God exists, since he created everything, he also would have created the moral standard. God gave humans free will to commit evil, therefore, if God exists, free will is the moral choice. Humans do not get to say that God is immoral, when God is the one who determines what is and is not moral. * {{Argument against}} In order to exist, an entity must exist as something. To exist as something, the entity must have positive primary attributes (i.e. I'm a material entity, made up of atoms). All of God's attributes are either negatively defined (ex. omniscience can be reduced to 'without limits of knowledge'), secondary (i.e. good) or relational (i.e. creator). If a god is Creator, then it must be immaterial, as nothing can cause itself. But "immaterial" is a negatively defined term. Therefore a god's substance is undefined. All of this is to say that the god concept is incoherent. If this indeed turns out to be the case, then positive belief in such a concept is not possible. ** {{Objection}} Positive and negative properties are vague notions, often interchangeable. 'Closed' can be reduced to 'not open', just as 'open' can be reduced to 'not closed'. Similarly, 'omniscience' can be reduced to positive terms, like 'with total knowledge' just as it can be reduced to negative ones, like 'without limits of knowledge' or 'without ignorance'. Other properties of God, such as 'all-powerful', can also be thought as either positive or negative: 'with complete power' or 'without limits to its power'. *** {{Objection}} Even so, saying that something is omniscient is a secondary characteristic - it's telling us what something can do, NOT what it is. If I said humans were an IQ of 120, that doesn't really tell me much of anything about what a human IS (as opposed to saying something like an entity in space/time made up of matter, etc). **** {{Objection}} Yes, it does. Intelligence is an attribute of humans, is it not? ** {{Objection}} {{W|Dark matter}} and {{W|dark energy}} are entities whose existence is generally accepted by the scientific community, despite the fact that we don't know what they are made of. The fact that we don't know what something is made of doesn't imply that it's made of nothing, or that it doesn't exist. *** {{Objection}} Dark matter and energy are theoretical. **** {{Objection}} Global warming, evolution, the Theory of Relativity, and even gravity are also theoretical. This does not mean they are wrong. The scientific community can have almost complete certainty in something but still classify something as "just a theory". ***** {{Objection}} This is irrelevant to the primary point of the original argument: "All of God's attributes are either negatively defined (ex. omniscience can be reduced to 'without limits of knowledge'), secondary (ex. good) or relational (ex. creator)." The part of the original argument stating that God's substance is undefined is not necessary for the original argument to be correct... if the unnecessary sentence "Therefore a god's substance is undefined." were left out of the original argument it would be a perfectly valid argument and this objection against it would not work. The main point of the original argument is that in order to exist, that entity must has positive primary attributes, of which there still are none for God. This is a red herring, if we remove that unnecessary sentence from the original argument. ** {{Objection}} How about the fact that the Universe exists in the first place. The fact that the necessary things exist in the first place that leads to the big bang theory and the creation of the Universe. Then the Universe shrinks again into a big matter, crushing everything in its collision, bringing back to the theory of big bang, and the cycle goes on infinite time. What if the Universe is not zero in the first place? The Universe exists without the creation of anything. We can't think of anything that might have created the Universe, because it's just there. * {{Argument against}} God doesn't exist because of Theophagus, the god-eater. Since Theophagus is god-eating by definition, he has no choice but to eat God. So if God exists, He would immediately cease to exist as a result of being eaten. Unless it's proven that Theophagus doesn't exist, then God doesn't exist. ** {{Objection}} Without any evidence or logical argument for the existence of such a being, there's no reason to believe Theophagus exists. *** {{Objection}} The same argument against Theophagus works on God: Without any evidence or logical argument for the existence of such a being, there's no reason to believe God exists. So either the argument you raised against Theophagus is valid, in which case it is also valid against God, and thus there is no reason to believe God exists, or the argument you raised against Theophagus is invalid, in which case Theophagus has eaten God and God no longer exists. **** {{Objection}} If you've read the entire "Arguments for" section, one would see that there are arguments for God's existence ***** {{Objection}} Some Of the arguments you're pointing to are all unsound or plainly invalid. There actually are not better arguments for God's existence than for Theophagus. ** {{Objection}} God is omnipotent and omnipresent, so even if Theophagus exists, God can't be eaten by him. *** {{Objection}} By its definition, Theophagus eats omnipotent and omnipresent beings. *** {{Objection}} Nothing about omniscience and omnipotence precludes being eaten. *** {{Objection}} If God is the most powerful being, and Theophagus can eat God, then Theophagus is more powerful than God, so Theophagus is God, therefore Theophagus/God eats itself and Theophagus/God cease to exist. ** {{Objection}} If Theophagus can eat God, who cannot be eaten, his existence creates a contradiction. Therefore, Theophagus cannot exist *** {{Objection}} If Theophagus can eat God, who cannot be eaten, then God's existence creates a contradiction. Therefore, God cannot exist. To refute this argument, we must prove that Theophagus does not exist independent of the existence of God. ** {{Objection}} We have two options. Either Theophagus is God or Theophagus is not God. Now attribute of Theophagus is God-like, who need to be omnipotent to eat God, who also need to be omniscient to know everything which requires to eat omnipotent God. If Theophagus is god, he has to eat himself before he eat Actual God. So, if he has eaten himself before eating Actual God, he can't eat God and this all create recursion loop. If Theophagus is not god, he needs to have less than omnipotent and omniscient, by which he can't he eat Actual God, because God is perfect omnipotent who knows everything about Theophagus. * {{Argument against}} Particles don't have a position until their wave function collapses, and wave functions collapse when observed. From experiments such as the double-slit experiment, we infer that there are uncollapsed wave functions. Therefore, there is no being observing all particles, no omniscient being, no God. ** {{Objection}} We don't really understand ''how'' observation causes superposition to collapse nor ''how'' a being who is outside of spacetime (or alternately who exists in all of spacetime) would even affect superposition. As {{W|George Berkeley}} argued in his version of {{W|Idealism}}, all of the physical Universe exists ''because'' God is perceiving it. *** {{Objection}} God of the gaps fallacy. **** {{Objection}} He was merely pointing out that the original statement need not be always true, nor has any true weight because God exists outside of the physical world. *** {{Objection}} What we do know about it is that observation causes collapse of the wave function (if you want to claim that being outside space time is somehow different in that regard you'll have to substantiate that claim). ** {{Objection}} The wave function collapse does not happen because of observation per se, but when a wave function interacts with a classical environment. If God is all-powerful, he can observe a wave function without interacting with it. *** {{Objection}} Being all-powerful is self-contradictory. This is because an all-powerful God would be able to predict the future, but also be able to take actions which would contradict His predictions of the future. Since knowing things is a power, being all-powerful implies being all-knowing. And then, since an all-knowing God would know all of the actions He would take ahead of time, an all-knowing God would know in advance all actions He would ever take, and, in order to prevent any paradoxes and allow God to exist, God's "all-powerful" powers would have to be reduced to just doing what God predicted He would do ahead of time. Thus the whole idea of being all-powerful is nonsense. **** {{Objection}} This has already been addressed in above arguments, but I'll reiterate--knowing what you will do beforehand does not take away from your freedom of choice. His knowledge is his choice. ***** {{Objection}} Knowing that something will happen effectively means that nothing different will happen. If nothing else will happen then God cannot do something else other than what he foresaw, being then effectively limited in what he can do. *** {{Objection}} In quantum systems observation is intrinsically linked to the behavior of the system. Your assumption that God is omnipotent is therefore in contradiction which a known feature of the actual world (and being God defined as being omnipotent its very existence is inconsistent with observation of properties of the actual world). ** {{Objection}} If God is all-knowing, he does not need to observe particles to know their position. *** {{Objection}} When a particle exists in a quantum superposition that can be described using a wave function, prior to wave function collapse, that particle does not actually have any definite position, but just probabilities of being in different locations. This has been experimentally verified in the aforementioned double-slit experiment. So talking about the position of a particle whose wavefunction has not yet collapsed as if it is something definite makes no sense, since such particles can and do exist in multiple locations at the same time, which is what produces the interference fringes in the double-slit experiment, from a particle in different locations interfering with itself in other locations, meaning, it really has no one location. So if God were to perceive such a particle as being at one specific location, God would be incorrect. * {{Argument against}} Some infinite traits, such as "omniscience", have a computational complexity equal to infinity, thus the {{W|Kolmogorov complexity}} of a God defined with these attributes is infinite, the prior probability for his existence is epsilon, and "P(X exists) is epsilon" is the statistically literate way of saying "X does not exist". * {{Argument against}} Non-theism is the parsimonious worldview.<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Beyond_Theism#Non-Theism_is_the_Null_Hypothesis|title=Beyond Theism - Wikiversity|website=en.wikiversity.org|language=en|access-date=2022-08-16}}</ref> == See also == * [[Do humans have free will?]] == Notes and references == {{Reflist}} [[Category:Religion]] liefhvsqn96t3izeebjvm0r1z0p6qgt 2416164 2416163 2022-08-18T14:23:45Z Sophivorus 85690 Add objection wikitext text/x-wiki {{Wikidebate}} {{Religion}} [[File:Doré, Gustave - Paradiso Canto 31.jpg|thumb|alt=A man staring into a large swirl of angels and light|God as a point of light in {{W|Paradiso (Dante)|Dante's ''Paradiso''}}, engraving by {{W|Gustave Doré}}.]] [[File:Does God exist%3F.png|thumb|Visual snapshot of this debate sometime in the past.]] Questions about the nature of ultimate reality have been asked as long as humans have been conscious. For thousands of years, across thousands of cultures, belief in a supreme being has been more or less common but some have always called into question whether or not God exists or can even be known. By "God" we mean the metaphysically ultimate being, all-knowing, all-powerful, all-good, timeless, simple and devoid of any anthropomorphic qualities. We do not necessarily mean the Abrahamic God, although these ideas may share some overlap. So is there a God? == God exists == === Arguments for === * {{Argument for}} The laws of logic are necessary true propositions. Propositions are real entities, but cannot be physical entities; they are essentially thoughts.<ref>"Propositions are not on paper, in your brain, or anywhere else" [https://www.academia.edu/630293/Propositions_Are_Not_on_Paper_In_Your_Brain_or_Anywhere_Else Direct Paper link]</ref>{{Clarify}} So the laws of logic are necessarily true thoughts. Since they are true in every possible world, they must exist in every possible world.<ref>"From Necessary Truth to Necessary Existence" [http://www.joshualrasmussen.com/articles/existence.pdf Direct paper link (PDF)]</ref> But if there are necessarily existent thoughts, there must be a necessarily existent mind; and if there is a necessarily existent mind, there must be a necessarily existent person. A necessarily existent person must be spiritual in nature, because no physical entity exists necessarily. Thus, if there are laws of logic, there must also be a necessarily existent, personal, spiritual being.<ref>{{Cite journal|last=Anderson|first=James N.|last2=Welty|first2=Greg|date=2011|title=The Lord of Noncontradiction: An Argument for God from Logic|url=https://www.semanticscholar.org/paper/The-Lord-of-Noncontradiction%3A-An-Argument-for-God-Anderson-Welty/e0f49c35c70e54174bf201c59f4abfadd223c583?p2df|doi=10.5840/PC201113229}}</ref><ref>"A Defense of Theistic Argument from the Law of Non-contradiction", [https://oaktrust.library.tamu.edu/bitstream/handle/1969.1/157717/NAVARRO-THESIS-2016.pdf?sequence=1&isAllowed=y Direct paper link (PDF)]</ref><ref>{{Cite journal|last=Weaver|first=Christopher G.|title=Why is There Anything?|url=https://www.academia.edu/20392976/Why_is_There_Anything|journal=Two Dozen (or so) Arguments for God: The Plantinga Project}}</ref><ref>{{Cite journal|last=Rasmussen|first=Joshua|date=2009-01-01|title=From a necessary being to god|url=https://www.academia.edu/630287/From_a_necessary_being_to_god|journal=International journal for philosophy of religion}}</ref> ** {{Objection}} This proposition is not correct: "Since they are true in very possible world, they must exist in every possible world". Why? Because the expression "possible world" has two different meanings. Is an amphibology. In first case, its meaning is: "possible world that we imagine". In the second case, its meaning is: "possible world that can exist". * {{Argument for}} Existence of God has been proven ''a priori'' using higher-order logic and reasonable axioms.<ref>Axioms:[https://arxiv.org/pdf/2001.04701]<br>A1 Self-identity is a positive property, self-difference is not.<br>A2 A property entailed or necessarily entailed by a positive property is positive.<br>A3 The conjunction of any collection of positive properties is positive.</ref><ref>"A (Simplified) Supreme Being Necessarily Exists -- Says the Computer!", [https://arxiv.org/pdf/2001.04701 Direct Paper link (PDF)]</ref>{{Clarify}} * {{Argument for}} Various phenomena in the Universe appear to be designed and suggest a designer, God.{{Example}} ** {{Objection}} Consider Auroras, these phenomena occurs due to scattering of charged particles in atmosphere which exists as of evolutionary phases of Earth. Not designed by anyone. *** {{Objection}} The fact that some phenomena were not designed by God deos not imply that no phenomena were designed by God. * {{Argument for}} Language plays an integral role in the laws of nature and of DNA. As encoded meaning, language is non-material in its ultimate essence. Apart from something akin to the human mind, there are no serious candidates for explaining how linguistic phenomena might otherwise arise. The only reasonable way to account for the linguistic aspects of the laws of nature and of DNA is an intellect with capacities so vast that most people would immediately identify this entity as God.<ref>"A Linguistic Argument for God's existence", [http://www.etsjets.org/files/JETS-PDFs/58/58-4/JETS_58-4_771-86_Baumgardner&Lyon.pdf Direct Paper link (PDF)]</ref> * {{Argument for}} Under a naturalistic worldview, the coincidence of all the cognitive faculties required for knowledge is highly unlikely, since evolution favors survival, not truth, and any overlap of survival and truth is highly implausible.<ref>"IF KNOWLEDGE THEN GOD: THE EPISTEMOLOGICAL THEISTIC ARGUMENTS OF PLANTINGA AND VAN TIL", [http://www.proginosko.com/docs/If_Knowledge_Then_God.pdf Direct paper link (PDF)]</ref><ref>{{Cite web|url=https://www.spectator.co.uk/2019/10/donald-hoffmans-the-case-against-reality-is-hard-to-get-your-head-around/|title=Donald Hoffman's The Case Against Reality is hard to get your head around|date=2019-10-12|website=The Spectator|language=en-US}}</ref><ref>{{Cite web|url=https://www.quantamagazine.org/the-evolutionary-argument-against-reality-20160421/|title=The Evolutionary Argument Against Reality|last=Amanda Gefter|website=Quanta Magazine|language=en}}</ref><ref>{{Cite web|url=https://iai.tv/articles/the-evolution-of-reality-auid-1274|title=The Evolution of Reality {{!}} Why the world is not how we see it|date=2019-11-29|website=IAI TV - Changing how the world thinks|language=en-GB}}</ref> If the reliability of cognitive faculties is low, any belief is not warranted including metaphysical naturalism, which becomes self-defeating.<ref>"Probability And Defeaters", [http://www.proginosko.com/docs/If_Knowledge_Then_God.pdf Direct Paper link (PDF)]</ref> ** {{Objection}} This doesn't imply that God exists, only that naturalism is false. * {{Argument for}} Chanting of names of God (like Hare Krishna) gives immense happiness to devotees. One can't expect such happiness from a non-theistic worldview, but God existence could explain such happiness. ** {{Objection}} Chanting in unison in a crowd during a sports game can cause similar feelings of happiness. Happy feelings from chanting is not evidence for God. * {{Argument for}} For the Universe to exist, there must be an uncaused cause, God, or the Universe is eternal. So either there's no explanation for God, or there's no explanation for the Universe. The Big Bang is not an explanation, it's a description with no explanation for why it came to be. We then have to rely on chance and happenstance. God fits the picture better. ** {{Objection}} If there must be an uncaused cause, then why can't the Universe be the uncaused cause? Adding God to the chain only adds unnecessary complexity making it a less likely explanation. Just start with a natural, unintelligent and minimally powerful uncaused cause. *** {{Objection}} Everything that we know in the Universe has a cause external in space and previous in time. Why should the Universe itself be any different? But God is timeless, so the same rules don't necessarily apply. **** {{Objection}} The Big Bang was the beginning of spacetime. As such, we cannot say that the particles that cause the Big Bang follow the same rules. ** {{Objection}} Not knowing how the Big Bang came about is not proof or evidence that God exists. ** {{Objection}} This assumes the need for an explanation, which is just a [[Wikipedia:False equivalence|false equivalence]]. Things within the Universe require an explanation, but the Universe itself does not require an explanation, because we explain things inside the Universe based on the assumption that there is an external factor already explained. This does not apply to the Universe itself because there is nothing external to the Universe by definition. Either the Universe caused itself or simply has no cause. This might strike many as nonsensical but that is simply because they are unconsciously and inappropriately extending the logic of parts to the whole. *** {{Objection}} Everything we know has a cause different from itself. Thus, we must pursue this principle to its natural end, and conclude that the Universe has a cause different from itself, unless contrary evidence can be provided. You can't merely assume that the Universe is all there is and write in an exception into the definition of the Universe. **** {{Objection}} If you question the origin of the Universe, it's like questioning the water cycle. Where does evaporated water come from? From condensed water. Where does condensed water come from? From precipitated water. Where does precipitated water come from? From ran off water. Where does ran off water come from? From evaporated water. And the cycle goes on and on and on. Similarly, if you question the origin of the Universe, I would say from the Big Bang. If you question the origin of the Big Bang, I would say from the Singularity. If you question the origin of the Singularity, I would say from the Big Crunch. If you question the origin of the Big Crunch, I would say from the Universe. If God does exist, why did he make principles that lead to his non-existence? If God created science, why we can't connect science to him? Even principles are self-looping, so there's also indeed no creator. Energy loops, matter loops, the burnt paper turns into carbon and gas, it just turns into something else to become paper again. The Universe exists in the first place because it loops so there's no beginning or end. The [[Wikipedia:Conservation of energy|law of conservation of energy]] states that energy can neither be created nor destroyed, only be transferred or transformed from one form to another. This implies that energy and the Universe existed in the first place. There's no creator. Just accept the fact that there is no zero in the Universe, there's a fixed number of materials and energy. The Universe is just a continuous loop of energy. It inflates through the Big Bang, reaches the maximum inflation, deflates through the Big Crunch, and reaches the maximum deflation, and the cycle goes on. * {{Argument for}} God may not have provided evidence to everyone because not all people want God, so God may be fulfilling their desire by not giving them sufficient evidence. ** {{Objection}} Absence of evidence is not evidence of presence. Lack of evidence doesn't imply that God is fulfilling the desire of people who don't want God, nor that God exists. *** {{Objection}} Subjective evidence suggest that either God exists who wants to known to people as God may choose or either all people deluded, lier or anything else. But principle of incredulity say that we should believe in experience unless there is good reason to not otherwise. Now if there is positive evidence for non existence of God, it can't counted. But as far as we know, there is no positive evidence of non existence of God. Therefore subjective evidence can't be rejected. Now question one can ask is why would God don't provide subjective evidence to everyone? So, one explanation is not all people want God, and God may not provides evidence to those who doesn't want God or who doesn't qualify. * {{Argument for}} If the Universe is a simulation, then there's a creator beyond spacetime who designed and fine-tuned the simulation. Now either the Universe is a simulation or the Universe is base reality. If the Universe is a simulation, it can be a simulation level 1, level 2, level 1000 or any level. If the Universe is base reality, it can exist in only one way. Therefore, there're more possibilities of the Universe being a simulation than base reality. ** {{Objection}} Even if our Universe is a simulation and has a creator, that doesn't imply a creator of base reality. Our Universe could be a simulation one level below base reality, which implies a creator outside of our local spacetime that exists in base reality, but that does not imply a creator outside spacetime for that base reality since the argument only applies to simulations. * {{Argument for}} Difference between natural and supernatural is artificial. It depends on definition of Supernatural, and gives reason to reject God on artificial grounds. If Supernatural is defined as something which science can't explain, then most of phenomenon can't explainable by science because it depends on our observations, and it doesn't necessarily that observations shows real content of reality out there. If Supernatural is defined as something done that violates laws of physics, there is contradiction in definition. Because laws of physics are not fixed set of laws, but we try to find laws by observations and whatever comes, we describes it as laws of physics. If Supernatural is defined as some actions by agents, it is unclear whether to consider ghost as Supernatural because if some explanation found for it, it becomes natural. And it isn't necessary that nature behaves on it's own. If physical reality doesn't know what it does, or how it does, then it doesn't have to continue to exist. All physical reality can suddenly appears and disappears. It isn't necessarily for reality to such as it should change, it can be such as it remains static for infinity, and nothing happens. It behaves in such a way that is not expected if it run on it's own. However if material nature works under directions of God, it is exactly what we normally expect. And evidence from spontaneous emergent, chaotic unpredictability further suggest material nature can be works under supervision of God, as scripture like Bhagavad Gita also suggest.<ref>{{Cite web|url=https://asitis.com/|title=Bhagavad Gita As It Is Original by Prabhupada|website=asitis.com|access-date=2022-08-16|quote=This material nature is working under My direction, O son of Kunti, and it is producing all moving and unmoving beings. By its rule this manifestation is created and annihilated again and again.}}</ref> * {{Argument for}} The Universe is fine-tuned to support life. This fine-tuning is less surprising and even probable if God exists, but highly unlikely in a godless Universe.<ref>{{Cite journal|last=Chan|first=Man Ho|date=2017-05-24|title=The fine-tuned universe and the existence of God|url=https://repository.hkbu.edu.hk/etd_oa/447|journal=Open Access Theses and Dissertations|quote=To conclude, after a comprehensive study of the fine-tuning arguments, the fine-tuning phenomena strongly support the theistic worldview.}}</ref> ** {{Objection}} This argument is biased to carbon-based lifeforms. Life could exist in ways that are not based on carbon meaning the fine-tuning of the universe is not as necessary for life to exist as a carbon-based life form might think. ** {{Objection}} Considering multiverse theory, there may be many universes with their own cosmological constants. In this scenario, of course a life form in a "fine tuned" universe sees fine tuning even though that life form's existence can be relegated to chance, not God. This is an example of selection bias. The multitude of other universes inhospitable to life would never develop life forms capable of posing such questions. * {{Argument for}} If the Universe is ultimately meaningless, devoid of any purpose or design, then all-purpose and meaning one assigns should be imaginary. Because if the Universe doesn't have ultimately any purpose, then life just happened to appear in accidental ways such as it doesn't have any inherent meaning or purpose. However, if God is the reason for the existence of us, there can be the purpose of life. In some religions, it is suggested that God is originally in the spiritual universe with living entities. When a living entity doesn't want God, God makes a material universe for the fulfilment of the desire of them of not wanting God. The material universe is created temporarily such as living entity can realise their connection with God, and when they want God, they can return to God, which may be the purpose of life. ** {{Objection}} This argument assumes its own premise. It relies on the presumption that 'meaning' is an inherent characteristic of life, which is just a rudimentary form of Anthropocentrism. There is no reason why there should be 'meaning' to one's life for life to exist. * {{Argument for}} The Universe follows mathematical laws independently of how humans describe them. So mathematics must exist independently of human minds. But all mathematics needs axioms. How can axioms exist independently of human minds? An axiom generator system is needed, or meta-axioms that create the axioms required for mathematical laws. But how can meta-axioms exist? Meta-meta-axioms are needed, and so on. This makes it implausible or even impossible for any mathematical laws to exist. However, it's not impossible if mathematics exists in the mind of God. Because God can conceptualize mathematics. * {{Argument for}} Objective morality exists and requires an absolute moral authority. Without some absolute authority, all morality is an individual interpretation of morals or shared morality decided by groups of people, which is ultimately subjective. If morality comes just from the survival of fittest, it can be moral to steal or murder, if it results in survival. However, most people regard it as not moral. This absolute authority is equivalent to God who may have created humans and provided some rules or laws which may be inherent in us. ** {{Objection}} The argument assumes its own premise, that objective morality exists, an assumption that isn't necessarily correct but fundamentally necessary for the functionality of the argument. Since, it is from the idea of the existence of objective morality that this argument derives the existence of an absolute moral authority, and from the existence of an absolute moral authority, the existence of God, we necessarily have to conclude that, since no evidence is provided for the positive claim of the existence of objective morality, then no evidence has been provided for the positive claims of the existence of an absolute moral authority or a God. Since the existence of objective morality is provided merely as an assumption lacking supporting evidence, then the existence of an absolute moral authority or the existence of a God is just that, an assumption lacking supporting evidence. * {{Argument for}} Suppose there was no intelligence behind the Universe, no creative mind. In that case, nobody designed my brain for the purpose of thinking. It is merely that when the atoms inside my skull happen, for physical or chemical reasons, to arrange themselves in a certain way, this gives me, as a by-product, the sensation I call thought. But, if so, how can I trust my own thinking to be true? It's like upsetting a milk jug and hoping that the way it splashes itself will give you a map of London. But if I can't trust my own thinking, of course I can't trust the arguments leading to Atheism, and therefore have no reason to be an Atheist, or anything else. Unless I believe in God, I cannot believe in thought: so I can never use thought to disbelieve in God.<ref>The original version of this argument was brought forth by {{W|C. S. Lewis}}.</ref> ** {{Objection}} A similar argument applies to the theist. Suppose an intelligence designed our brains. This could mean that our brains were designed for thinking rationally, or it could mean that our brains were designed to come to the wrong conclusions. How do we know which is true? We can't. If I can't trust my own thinking, I can't trust the arguments leading to theism. Assuming God exists does not lead to knowledge that we think rationally. *** {{Objection}} But problem is more serious in universe which just happened to be like it, which is accidental, mindless, purposeless, arbitrary. In such godless universe, there would millions or perhaps billions of coincidence required for brain to function exact right, and there would much more possibility that brain is unreliable. How can anyone know that brain is anything more than quantum fluctuations? In such, how can one can sure that quantum fluctuations should behaves exactly by which it makes logical decision when it doesn't know anything? How can one have faith in such quantum fluctuations such as it shows real reality than shows illusion of reality? However if God creates physical Universe for living entity for giving chance to those who don't want them, then living entity can realize God and can comes to conclusion of God by realizing that material universe is illusionary, and full of suffering which is not ultimate place for him, because of which they can comes to God. In such state, they can have faith in God and can surrender God. Additionally if God wants, God can make known their existence to someone at absolutely certain such as one can becomes certain that God exist, and can realize purpose of existence. ** {{Objection}} Nobody designed the stomach for digesting food either, yet with modern biology, we know through a process of evolution over millions of years that the digestive system evolved naturally through an accumulation of beneficial steps. Just like the eye, just like the mind. In short, evolution provides a better explanation than God. *** {{Objection}} Not really. How can physical reality behaves so perfectly? If everything is ultimately stochastic thermodynamics progress and quantum fluctuations, why such great coincidence occurs by which it works so well, such as it seems design? If it is not really design but apparent design, it can also be that there is no evolution but apparent evolution, and all entropic thermodynamics progression which doesn't know anything, just happened to behaves like that. But it's seems highly implausible, however it becomes likely if God direct nature or physical reality. ** {{Objection}} If God designed our minds, then why is our reasoning ability so imperfect? Why do people confuse correlation with causation? Why do people believe in astrology and other obvious nonsense? *** {{Objection}} Intelligence is a gift endowed to humans that cannot be explained through evolution. It's a gift, but humans are not perfect. If so, we would be divine. How you choose to believe this happened is a matter of faith.{{Clarify}} * {{Argument for}} If you trust your own thinking, then you must have an absolute perfection, a highest logic, against which to measure your thoughts. ** {{Objection}} Why must that vision of perfection actually exist? Is it not merely a measure in our minds against an imaginary height of perfection? Thought does not imply existence. * {{Argument for}} A huge number of humans, throughout centuries, have reported all sorts of encounters with God, from the personal internal type to shared apparitions and public miracles. Experiences differ in many ways, but they all support a common cause: the existence of God. It is highly implausible that many reports all be false or misled, and as we trust our experience unless we have good reason to think otherwise, it is reasonable to think that God is reason behind such experiences. ** {{Objection}} Testimonies are generally inconsistent unless they are sufficiently connected by cultural myths or sufficiently vague. Many cultures have the concept of magic, not because magic is real, but because magic is a sufficiently vague concept to hold many different conceptions of it. Unsurprisingly, the reports about God are more similar to the more closely connected the cultures of the witnesses are, which indicates they are cultural phenomena rather than independent observations that corroborate each other. *** {{Objection}} How testimony is inconsistent? In general experience of God is very difficult to describe or explain. Think how do you describe the experience of the color red to someone who is blind from birth? And cultural explanation applies to everything we experienced. Whatever we experienced is shaped by culture and our background. Like if someone who knows about tree interprets the experience of light for seeing something different than someone who doesn't know about the tree, but it doesn't mean such experience of light is not real. And when we claim that the experience of God must be the same for all, it is not reasonable. Suppose someone prays to God in the form of Lord Krishna, now if God appears as Jesus, he probably requests God to appears in form in which he remembers. Same as if someone remembers God in form of Jesus, and if they experience God in form of other, they may request God that they like God in form of Jesus, therefore, please give me appearance in the form in which I remember. Therefore, it is reasonable to think that God may not experience the same as for every devotee because devotees want God in a specific form they love, and God may fulfill a desire of devotees. Also, it is reasonable to think that God has unlimited forms, and also formless because they are absolute. Also if we reject experience because just they are different, should we also reject the experience of reality if it experienced inconsistently by different people? Most people may answer no. So why should we reject the experience of few billions of people who think they have experienced God? Unless we have positive evidence for the non-existence of God, it shouldn't be rejected. ** {{Objection}} To know that all these testimonies are testimonies about the same thing, we should know their object (God) independently from these testimonies. We should first know the object we are talking about (God) in order to be able to recognize that all reports deal with the same object. *** {{Objection}} All knowledge ultimately reduces to the testimony of one or more people. Objects cannot be known independently of all testimony. **** {{Objection}} This claim attempts to homogenize testimony. It fails to recognize not all testimony is created equal; some testimony is backed by evidence while some is not. ** {{Objection}} Eyewitness testimony is notoriously unreliable. Many people have testified seeing thousands of people celebrate the 9/11 attacks in New Jersey, even though this event never took place. If you put an idea in people's minds, some people will believe that they personally saw whatever that idea is, even if the idea turns out to be false. *** {{Objection}} But people who have experienced God includes medical professionals who know about different psychological phenomenon, and also they have experienced God in which they may differentiate between psychological condition and the real one. Numbers of Medical professionals, physicists, cosmologists experienced God, which makes it less likely to be a purely psychological phenomenon. Additionally, many people go through psychological tests after such experience, in which many are shown to be normal and healthy. If the experience of God is purely delusional, they should show signs of delusion, but instead, people show no such phenomenon. And if God wants, God can make someone know about their existence with absolute certainly. ** {{Objection}} Although there're millions of believers, there're not that many of eyewitnesses, relative to the number of believers. *** {{Objection}} It can be because the connection with God may be difficult. Some people who love God so much, God may give him direct realization than someone who is not of that level. However many people are spiritually shallow, but they have experienced God slightly like newborn babies experience light slightly and as time passes, he can more clearly experience light. But it doesn't mean most newborn doesn't have any experience of light, and maybe the same as most people who are spiritually shallow, may not experience God deeper. ** {{Objection}} [[Wikipedia:Argumentum ad populum|''Argumentum ad populum'']]''.'' *** {{Objection}} It can argumentum ad populum only if it claims like "People believe in God therefore God exists". But this is not the same. This argument offers that because millions of people have experienced God, and because we have no positive evidence of the non-existence of God, the experience of God can't be rejected as delusion, hallucinations or lies because else one can reject all experience of all people by same without any positive evidence for its non-existence. ** {{Objection}} Personal experiences can't be accounted for as evidence because there's no evidence to support these reports. How would one prove that these encounters were not a trick of the mind, such as mirages or sleep paralysis, or completely fictional? Experiences of God's existence could easily all be hallucinations, delusions, or attributions of a supernatural cause to natural phenomena which, them being as theists, leading to them being caused by God. I can hallucinate too when I am a theist and I can refer to it as a supernatural occurrence caused by God. It leads to the assumption that it is God-caused because you are a theist. *** {{Objection}} Personal experience can be accounted for as evidence. Otherwise, it would not be reasonable to believe (unless you personally experience it yourself) that all humans are conscious, certain drugs induce hallucinations or certain psychological phenomena exist, such as dreams, sleep paralysis, Alice in Wonderland syndrome, phantom limb, etc. *** {{Objection}} When corroborated by so many people, they cannot be so handily dismissed, though. ** {{Objection}} Attributing some encounter with nature or with some unusual phenomenon to the existence of God is a speculative conclusion based on a subjective assessment of the available information. Is contemplation of the beauty of a flower an encounter with God, or simply an appreciation of the fractal nature of the cellular structure that has evolved over millions of years? Are reports of virgin birth evidence of a miracle, or simply a translation error, a misunderstanding of the mechanisms of conception, or marketing hype? *** {{Objection}} Trying to prove the existence of miracles scientifically is like trying to prove that Gandhi was Indian ''linguistically''. It is the wrong outlet, as we can never re-experience what they did, along with millions of others. These miracles are a matter of faith to them. * {{Argument for}} Science is built on materialistic assumptions, so it already excludes the existence of God. ** {{Objection}} Current science accepts the existence of many immaterial entities, such as light, energy, spacetime and mathematical entities, so it's not built on materialistic assumptions.<ref>{{Cite journal|last=Moulines|first=C. Ulises|date=1977|title=Por qué no soy materialista|url=http://www.jstor.org/stable/40104059|journal=Crítica: Revista Hispanoamericana de Filosofía|volume=9|issue=26|pages=25–37|issn=0011-1503}}</ref> ** {{Objection}} Even if science were built on materialistic assumptions and excluded the existence of God, that doesn't imply that God exists. === Arguments against === * {{Argument against}} Since there are many religions in the world, all of which have their own idea of God and their own ideas of an afterlife (i.e. heaven/hell, reincarnation, etc.), then which God is real, and which afterlife is real? ** {{Objection}} To claim there is no rational basis for belief in any one religion if at most one can be correct out of thousands is incorrect. The mistake here is assuming all religions have equal probability of being correct, which is not the case: some religions are internally inconsistent, for example, and so have a far lower probability of correctness than more consistent religions. To claim that the credibility of a religion is dictated by its followers' backgrounds is also incorrect; the origin of someone's belief has little bearing on whether the belief is true. ** {{Objection}} God may have appeared in different parts of the world in different ways so that people of that place and time can understand God, according to circumstances of that time. Therefore, even though God appears different in different religions, it can be same. *** {{Objection}} If God provides different versions of Himself to different people at different times, then the definition of God is strained. What are we talking about if the definition of God depends on the time and place? Suggesting that God only reveals himself in ways that are not equivalent to His true nature suggests that God hasn't really revealed Himself at all. It suggests he has only revealed caricatures of Himself dependent on the cultures to which he displays these caricatures to. ** {{Objection}} This does not exclude the possibility that one of those religions might turn out to be the correct one, with the correct idea of God and an afterlife, even if that religion contradicts all other religions and this means all religions except for one of them turn out to be wrong. *** {{Objection}} If at most one religion can be correct, out of the many thousands that exist, and it is possible that they are all wrong, there is no rational basis to believe in any one religion over all the others. Most people who believe in a religion do so because of social reasons, for instance being raised in that religion, or falling in love with and marrying a follower of that religion, rather than any rational basis to believe that their particular religion is any more likely to be true than all the other religions it contradicts. No one religion is obviously superior to all of the others enough to persuade all the followers of the other religions to convert. **** {{Objection}} We aren't debating over the merits of Christianity or Islam for example--these are matters of faith. We are debating whether a God of some sort exists as a starting ground ***** {{Objection}} It is not about the merits of a religion. This is about a lack of communication (or effective communication) which is weak evidence that there is no God. **** {{Objection}} All monotheistic religions basically have the same main idea, which is worship of a higher power, a God. The religion itself is a set of values or ideas that one certain group "binds" to a higher power. This may be meant in a way of pleasing or satisfying the higher power, which we as humans often feel the need to do. Take away these values and traditions, which is most likely human made. What we have left is the acknowledgement of a higher power. Monotheistic religions are just different ways of saying the same thing. ***** {{Objection}} Well, you're cherry picking monotheistic religion which already is a set that can include the existence only of a single god, which is then worshiped. If you take a larger set, such as all religion, you get differing beliefs even regarding the nature and even the existence of a higher power. It is not just the traditions that are most likely human made, it is the very notion of god(s), having a clear progression from more utilitarian deities to more abstract ideas which are more resistant to empirical disproof. * {{Argument against}} God's existence would imply that he can change the past. This would imply that some things happened and didn't happen at the same time and in the same sense. But contradictions are impossible, so not only God doesn't exist: his existence is impossible. ** {{Objection}} This is not a contradiction. Only the edited version of history actually happened. If God, an omnipotent being, changed the path of history, history is changed. As odd as it seemed, the previous scenario never happened. *** {{Objection}} However you are still presenting a contradiction: a scenario which existed and never existed. ** {{Objection}} That wouldn't be a contradiction, because God would know that He changed the past (per being all-knowing). So things that God changed would have happened at the same time, but not in the same sense. God would be able to distinguish them. *** {{Objection}} Doing one impossible thing is no more difficult than doing two impossible things.<ref>"For why should God not be able to perform the task in question? To be sure, it is a task—the task of lifting a stone which He cannot lift—whose description is self-contradictory. But if God is supposed capable of performing one task whose description is self-contradictory—that of creating the problematic stone in the first place—why should He not be supposed capable of performing another—that of lifting the stone? After all, is there any greater trick in performing two logically impossible tasks than there is in performing one?" Frankfurt, Harry. "The Logic of Omnipotence" first published in 1964 in ''Philosophical Review'' and now in ''Necessity, Volition, and Love''. Cambridge University Press November 28, 1998 pp.1–2</ref> ** {{Objection}} God's omnipotence is often described as "can do anything that is not logically impossible", or similarly defined so as to rule out paradoxes deriving from His omnipotence. *** {{Objection}} If God's omnipotence is limited by "can do anything that is not logically impossible", then the fact that God is also defined as omniscient and knowing everything would mean God possesses complete knowledge ahead of time of all things that He will do, and is bound by logic to do what He predicted He would do. This would reduce his omnipotence to complete powerlessness since he would never have any choice at all other than to do what he predicted he would do, and thus he would not have any power at all. **** {{Objection}} By omnipotence and omniscience, which theists usually refer to, we are speaking of something vastly beyond our understanding and quantifying it in human terms. Omnipotence implies that he has the choice to exercise his omnipotence in any particular way, which he knows what his choice will be. Knowing what you plan to do in a circumstance beforehand by no means makes one powerless ** {{Objection}} If God can change the past, it does not mean he would; this is dictated by his character and motivations, which are particular to different religions. This is similar to the argument that if God can create an immovable object he is not all powerful, but if he cannot then he is not all powerful either: it is wordplay, since such a definition is nonsensical. God is unbounded by the universe's laws. * {{Argument against}} If we are talking about a God which affects the physical world in some way, then saying that God exists is an empirical statement. But there is no hard evidence supporting that statement, and evidence is necessary to prove an empirical statement. ** {{Objection}} God is outside of the confines of our physical world, and whether or not God is actively involved in this world is not being debated. Therefore, saying that God exists is not an empirical statement, it's a metaphysical statement, and empirical proof for metaphysical statements is not necessary or even possible. *** {{Objection}} By this logic, there is no such thing as a false statement. Anything can be inferred. I have a 3 headed donkey in my backyard except you can't see it, it doesn't smell, and you can't touch it; there is no physical evidence for its existence whatsoever, but it exists metaphysically, above the plane of our existence. At this point, you have to ask, what do I mean when I say the donkey "exists"? If physical evidence isn't required to say that something exists, then I'm correct when I say that a 3 headed, 5 headed, 50,000 headed, and 50,001 headed donkey all exist in my backyard. ** {{Objection}} God may have perfect reason for not giving hard evidence to everyone. There are some people who don't like God, don't want God or if they know that God exist, they may becomes envy of God. God may not hides their existence from someone who doesn't want God as fulfilling their desires, and which may be better for them. As scriptures like Bhagavad Gita says, it can be that God manifest in proportion to one surrender to God.<ref>{{Cite web|url=https://asitis.com/|title=Bhagavad Gita As It Is Original by Prabhupada|website=asitis.com|access-date=2022-08-16|quote=Always think of Me and become My devotee. Worship Me and offer your homage unto Me. Thus you will come to Me without fail. I promise you this because you are My very dear friend.}}</ref><ref>{{Cite web|url=https://asitis.com/|title=Bhagavad Gita As It Is Original by Prabhupada|website=asitis.com|access-date=2022-08-16|quote=Abandon all varieties of religion and just surrender unto Me. I shall deliver you from all sinful reaction. Do not fear.}}</ref> * {{Argument against}} God is conceived as all-good, all-knowing and all-powerful. So, if God exists, then under any ordinary definition of evil, evil shouldn't exist. But evil clearly exists. Therefore, God does not exist. ** {{Objection}} There are several problem with it. First we don't know what God exactly do if they have perfections. We think God should do this or other, but we can't imagine it from way God would think. We can think if I am photon, I would do this or other, but we don't know what exactly happens when one be a photon. God who is outside of material realism, may have perfect reason to produce seemingly imperfect universe. We don't know God's reason for something, and if we know it we also think that everything is like it should be as best. Secondly, we define ourselves as moral authority and decides what is good or not, but God may beyond good and evil, the absolute one. Because God is free, if living entity is part and parcel of God, living entity should be free, and because of it they have minutes of free will. Now if living entity not like God or doesn't want God, God may fulfill his desire to becomes independent of God, and creates material world where living entity can be independent of God, and enjoy without God. Now as living entity has free Will, they can choose actions for which they are responsible, to which God doesn't intervene as they respect free will. But if they choose to do something bad, it is entirely on them because they have chosen it from their will. God may created suffering in material world to let someone know that it is not permanent place and living entity can again realize God, and can return to them. ** {{Objection}} God could have given humans the power to do evil. If humans can do evil, then evil can exist despite there being an all-powerful, all-knowing and all-good God. Moral responsibility is not hereditary. If my (grownup) child commits a crime, no society will (or should) blame me for it. Similarly, if a human does evil, we shouldn't blame God for it. Giving the power to do evil is not the same as doing it. God may even be the (metaphysical) cause of evil, while not being morally responsible for it. But no one contends that God raised us that way. He merely gave us the freedom to choose. In fact, because we turned away from his benevolence, evil arose, according to theists. Should we not live with the consequences of our actions and disobedience, then? *** {{Objection}} This would still make God responsible for evil, albeit indirectly. Moral responsibility is partially hereditary. If I knowingly raise my child in a way that makes it highly likely they will commit a crime when grown up that makes me responsible for that crime and subject to blame. An omniscient God should have known that if he gave humans the freedom to choose, we would do evil. Therefore, he allowed evil to exist, which contradicts the benevolent nature of God. *** {{Objection}} Even if God gave humans free will and the power to do evil, this doesn't imply that there should be evil. A world where free will exists but evil does not is logically possible. God can create any world that is logically possible, so God chose a world where there is unnecessary evil. This contradicts the all-powerful and all-good nature of God. *** {{Objection}} We haven't dealt with the problem of natural evil. Is cancer also the consequence of human disobedience? ** {{Objection}} Assuming God exists, since he created everything, he also would have created the moral standard. God gave humans free will to commit evil, therefore, if God exists, free will is the moral choice. Humans do not get to say that God is immoral, when God is the one who determines what is and is not moral. * {{Argument against}} In order to exist, an entity must exist as something. To exist as something, the entity must have positive primary attributes (i.e. I'm a material entity, made up of atoms). All of God's attributes are either negatively defined (ex. omniscience can be reduced to 'without limits of knowledge'), secondary (i.e. good) or relational (i.e. creator). If a god is Creator, then it must be immaterial, as nothing can cause itself. But "immaterial" is a negatively defined term. Therefore a god's substance is undefined. All of this is to say that the god concept is incoherent. If this indeed turns out to be the case, then positive belief in such a concept is not possible. ** {{Objection}} Positive and negative properties are vague notions, often interchangeable. 'Closed' can be reduced to 'not open', just as 'open' can be reduced to 'not closed'. Similarly, 'omniscience' can be reduced to positive terms, like 'with total knowledge' just as it can be reduced to negative ones, like 'without limits of knowledge' or 'without ignorance'. Other properties of God, such as 'all-powerful', can also be thought as either positive or negative: 'with complete power' or 'without limits to its power'. *** {{Objection}} Even so, saying that something is omniscient is a secondary characteristic - it's telling us what something can do, NOT what it is. If I said humans were an IQ of 120, that doesn't really tell me much of anything about what a human IS (as opposed to saying something like an entity in space/time made up of matter, etc). **** {{Objection}} Yes, it does. Intelligence is an attribute of humans, is it not? ** {{Objection}} {{W|Dark matter}} and {{W|dark energy}} are entities whose existence is generally accepted by the scientific community, despite the fact that we don't know what they are made of. The fact that we don't know what something is made of doesn't imply that it's made of nothing, or that it doesn't exist. *** {{Objection}} Dark matter and energy are theoretical. **** {{Objection}} Global warming, evolution, the Theory of Relativity, and even gravity are also theoretical. This does not mean they are wrong. The scientific community can have almost complete certainty in something but still classify something as "just a theory". ***** {{Objection}} This is irrelevant to the primary point of the original argument: "All of God's attributes are either negatively defined (ex. omniscience can be reduced to 'without limits of knowledge'), secondary (ex. good) or relational (ex. creator)." The part of the original argument stating that God's substance is undefined is not necessary for the original argument to be correct... if the unnecessary sentence "Therefore a god's substance is undefined." were left out of the original argument it would be a perfectly valid argument and this objection against it would not work. The main point of the original argument is that in order to exist, that entity must has positive primary attributes, of which there still are none for God. This is a red herring, if we remove that unnecessary sentence from the original argument. ** {{Objection}} How about the fact that the Universe exists in the first place. The fact that the necessary things exist in the first place that leads to the big bang theory and the creation of the Universe. Then the Universe shrinks again into a big matter, crushing everything in its collision, bringing back to the theory of big bang, and the cycle goes on infinite time. What if the Universe is not zero in the first place? The Universe exists without the creation of anything. We can't think of anything that might have created the Universe, because it's just there. * {{Argument against}} God doesn't exist because of Theophagus, the god-eater. Since Theophagus is god-eating by definition, he has no choice but to eat God. So if God exists, He would immediately cease to exist as a result of being eaten. Unless it's proven that Theophagus doesn't exist, then God doesn't exist. ** {{Objection}} Without any evidence or logical argument for the existence of such a being, there's no reason to believe Theophagus exists. *** {{Objection}} The same argument against Theophagus works on God: Without any evidence or logical argument for the existence of such a being, there's no reason to believe God exists. So either the argument you raised against Theophagus is valid, in which case it is also valid against God, and thus there is no reason to believe God exists, or the argument you raised against Theophagus is invalid, in which case Theophagus has eaten God and God no longer exists. **** {{Objection}} If you've read the entire "Arguments for" section, one would see that there are arguments for God's existence ***** {{Objection}} Some Of the arguments you're pointing to are all unsound or plainly invalid. There actually are not better arguments for God's existence than for Theophagus. ** {{Objection}} God is omnipotent and omnipresent, so even if Theophagus exists, God can't be eaten by him. *** {{Objection}} By its definition, Theophagus eats omnipotent and omnipresent beings. *** {{Objection}} Nothing about omniscience and omnipotence precludes being eaten. *** {{Objection}} If God is the most powerful being, and Theophagus can eat God, then Theophagus is more powerful than God, so Theophagus is God, therefore Theophagus/God eats itself and Theophagus/God cease to exist. ** {{Objection}} If Theophagus can eat God, who cannot be eaten, his existence creates a contradiction. Therefore, Theophagus cannot exist *** {{Objection}} If Theophagus can eat God, who cannot be eaten, then God's existence creates a contradiction. Therefore, God cannot exist. To refute this argument, we must prove that Theophagus does not exist independent of the existence of God. ** {{Objection}} We have two options. Either Theophagus is God or Theophagus is not God. Now attribute of Theophagus is God-like, who need to be omnipotent to eat God, who also need to be omniscient to know everything which requires to eat omnipotent God. If Theophagus is god, he has to eat himself before he eat Actual God. So, if he has eaten himself before eating Actual God, he can't eat God and this all create recursion loop. If Theophagus is not god, he needs to have less than omnipotent and omniscient, by which he can't he eat Actual God, because God is perfect omnipotent who knows everything about Theophagus. * {{Argument against}} Particles don't have a position until their wave function collapses, and wave functions collapse when observed. From experiments such as the double-slit experiment, we infer that there are uncollapsed wave functions. Therefore, there is no being observing all particles, no omniscient being, no God. ** {{Objection}} We don't really understand ''how'' observation causes superposition to collapse nor ''how'' a being who is outside of spacetime (or alternately who exists in all of spacetime) would even affect superposition. As {{W|George Berkeley}} argued in his version of {{W|Idealism}}, all of the physical Universe exists ''because'' God is perceiving it. *** {{Objection}} God of the gaps fallacy. **** {{Objection}} He was merely pointing out that the original statement need not be always true, nor has any true weight because God exists outside of the physical world. *** {{Objection}} What we do know about it is that observation causes collapse of the wave function (if you want to claim that being outside space time is somehow different in that regard you'll have to substantiate that claim). ** {{Objection}} The wave function collapse does not happen because of observation per se, but when a wave function interacts with a classical environment. If God is all-powerful, he can observe a wave function without interacting with it. *** {{Objection}} Being all-powerful is self-contradictory. This is because an all-powerful God would be able to predict the future, but also be able to take actions which would contradict His predictions of the future. Since knowing things is a power, being all-powerful implies being all-knowing. And then, since an all-knowing God would know all of the actions He would take ahead of time, an all-knowing God would know in advance all actions He would ever take, and, in order to prevent any paradoxes and allow God to exist, God's "all-powerful" powers would have to be reduced to just doing what God predicted He would do ahead of time. Thus the whole idea of being all-powerful is nonsense. **** {{Objection}} This has already been addressed in above arguments, but I'll reiterate--knowing what you will do beforehand does not take away from your freedom of choice. His knowledge is his choice. ***** {{Objection}} Knowing that something will happen effectively means that nothing different will happen. If nothing else will happen then God cannot do something else other than what he foresaw, being then effectively limited in what he can do. *** {{Objection}} In quantum systems observation is intrinsically linked to the behavior of the system. Your assumption that God is omnipotent is therefore in contradiction which a known feature of the actual world (and being God defined as being omnipotent its very existence is inconsistent with observation of properties of the actual world). ** {{Objection}} If God is all-knowing, he does not need to observe particles to know their position. *** {{Objection}} When a particle exists in a quantum superposition that can be described using a wave function, prior to wave function collapse, that particle does not actually have any definite position, but just probabilities of being in different locations. This has been experimentally verified in the aforementioned double-slit experiment. So talking about the position of a particle whose wavefunction has not yet collapsed as if it is something definite makes no sense, since such particles can and do exist in multiple locations at the same time, which is what produces the interference fringes in the double-slit experiment, from a particle in different locations interfering with itself in other locations, meaning, it really has no one location. So if God were to perceive such a particle as being at one specific location, God would be incorrect. * {{Argument against}} Some infinite traits, such as "omniscience", have a computational complexity equal to infinity, thus the {{W|Kolmogorov complexity}} of a God defined with these attributes is infinite, the prior probability for his existence is epsilon, and "P(X exists) is epsilon" is the statistically literate way of saying "X does not exist". * {{Argument against}} Non-theism is the parsimonious worldview.<ref>{{Cite web|url=https://en.wikiversity.org/wiki/Beyond_Theism#Non-Theism_is_the_Null_Hypothesis|title=Beyond Theism - Wikiversity|website=en.wikiversity.org|language=en|access-date=2022-08-16}}</ref> == See also == * [[Do humans have free will?]] == Notes and references == {{Reflist}} [[Category:Religion]] cjbbdfp9aqc2umh0mikfd1ytxw72pdv 0 215895 2416332 2415637 2022-08-18T22:43:32Z Sophivorus 85690 Add argument against wikitext text/x-wiki {{Wikidebate}} {{Law}} {{Medicine}} '''Voluntary euthanasia''', also known as '''assisted death''', is a medical process which is similar to but distinct from the question [[should suicide be legal?]] In euthanasia, a person facing terminal illness has life-prolonging treatment denied and some actions by medical practitioners can active kill the patient. Does this represent a violation of doctors' commitment to do no harm or is it actually an example of it? Euthanasia is the practice of intentionally ending a life in order to relieve pain and suffering. Voluntary euthanasia is different from non-voluntary euthanasia and involuntary euthanasia in that it has the consent of the patient. Some forms of voluntary euthanasia are legal in Belgium, Luxembourg, the Netherlands, Switzerland, and the US states of Oregon and Washington. Voluntary euthanasia should be legal worldwide. == Voluntary euthanasia should be legal == === Arguments for === * {{Argument for}} Every person has a fundamental right over his or her own body and life. Forbidding voluntary euthanasia would be preventing willing but unable individuals to exercise this right. The practice of euthanasia isn't expensive or complicated (quite the opposite), so society has no reason left to deny it to a willing individual. * {{Argument for}} Another reason is that the parents don't want their child to suffer so they tell them to take euthanasia. === Arguments against === * {{Argument against}} If euthanasia were legalized, some health insurance companies would see it as a profit-enhancing and cost-saving tool. It is well known that many health insurance companies have an economic interest in denying treatment to their patients, and they routinely find excuses to do so. If euthanasia were made an alternative, then it would not be long until they start recommending it instead of more expensive treatments, even if such treatments may lead to recovery. ** {{Objection}} One still has the decision and just because ones decision could be influenced, the possibility to decide shouldn't be removed. As an example, we are always being influenced by advertisement to buy certain products, but just because it would be bad for us if we followed all those suggestions we aren't forbidden to buy the products. * {{Argument against}} If voluntary euthanasia is made legal, then it won't always be voluntary. Emotional and psychological pressures could become overpowering for depressed or dependent people. If the choice of euthanasia is considered as good as a decision as receiving care, many people will feel guilty for not choosing death. Financial considerations, added to the concern about 'being a burden', could serve as powerful forces that would lead people to 'choose' euthanasia or assisted suicide. ** {{Objection}} Dependent people already feel the pressure of 'being a burden'. The legality of the act does not bear upon serious suicidal considerations. Put another way, think about who a dependent person might be concerned with burdening. It might be their family, friends, or care providers, but certainly not the state. * {{Argument against}} Legalizing euthanasia can lead to suicide contagion. When the media portrays assisted suicide as a means of 'taking control' or claims that someone helping another person kill themselves is 'death with dignity', then society (including teenagers) is receiving the dangerous message that suicide is a legitimate answer to life's problems.<ref>{{Cite web|url=https://www.nationalreview.com/human-exceptionalism/suicide-contagion-real-wesley-j-smith/|title=Suicide Contagion is Real|date=2013-05-21|website=National Review|language=en-US|access-date=2022-08-16}}</ref> ** {{Objection}} The argument relies on speculation as to the behavior of media and the behavior of society as a result. One might also speculate that the "contagion" effect is limited to adolescents, since the study also showed that the effect lessens with age.<ref>{{Cite web|url=https://www2.uottawa.ca/about-us/media|title=Media|website=About us|language=en|access-date=2022-08-16}}</ref> ** {{Objection}} The argument fails to show that an increase in suicide rates ought to be addressed through law (or at all). * {{Argument against}} Laws against euthanasia and assisted suicide are in place to prevent abuse and to protect people from unscrupulous medics and others. ** {{Objection}} If euthanasia were legal, abuse may arise in some instances, but your argument does not sufficiently show that it would be the norm. The mere possibility for abuse is not a reason to outlaw such a service. What your argument really suggests is the need for well regulated euthanasia and a thorough record of consent, incorporating practices that are resistant to malicious manipulation. *{{Argument against}}Euthanasia contradicts professional medical ethics where life is sought to be preserved. == See also == * [[Should suicide be legal?]] * [[Should capital punishment be legal?]] == Notes and references == {{Reflist}} [[Category:Politics]] [[Category:Death]] [[Category:Wikiversity Law Review]] [[Category:Legal ethics]] bnbe595z3o2vzkkcyrwxdbhte8sc4jq 2416333 2416332 2022-08-18T22:46:01Z Sophivorus 85690 Generalize argument wikitext text/x-wiki {{Wikidebate}} {{Law}} {{Medicine}} '''Voluntary euthanasia''', also known as '''assisted death''', is a medical process which is similar to but distinct from the question [[should suicide be legal?]] In euthanasia, a person facing terminal illness has life-prolonging treatment denied and some actions by medical practitioners can active kill the patient. Does this represent a violation of doctors' commitment to do no harm or is it actually an example of it? Euthanasia is the practice of intentionally ending a life in order to relieve pain and suffering. Voluntary euthanasia is different from non-voluntary euthanasia and involuntary euthanasia in that it has the consent of the patient. Some forms of voluntary euthanasia are legal in Belgium, Luxembourg, the Netherlands, Switzerland, and the US states of Oregon and Washington. Voluntary euthanasia should be legal worldwide. == Voluntary euthanasia should be legal == === Arguments for === * {{Argument for}} Every person has a fundamental right over his or her own body and life. Forbidding voluntary euthanasia would be preventing willing but unable individuals to exercise this right. The practice of euthanasia isn't expensive or complicated (quite the opposite), so society has no reason left to deny it to a willing individual. * {{Argument for}} By not legalizing euthanasia, many people would continue to suffer, waiting for them to die, and with euthanasia they could stop suffering. === Arguments against === * {{Argument against}} If euthanasia were legalized, some health insurance companies would see it as a profit-enhancing and cost-saving tool. It is well known that many health insurance companies have an economic interest in denying treatment to their patients, and they routinely find excuses to do so. If euthanasia were made an alternative, then it would not be long until they start recommending it instead of more expensive treatments, even if such treatments may lead to recovery. ** {{Objection}} One still has the decision and just because ones decision could be influenced, the possibility to decide shouldn't be removed. As an example, we are always being influenced by advertisement to buy certain products, but just because it would be bad for us if we followed all those suggestions we aren't forbidden to buy the products. * {{Argument against}} If voluntary euthanasia is made legal, then it won't always be voluntary. Emotional and psychological pressures could become overpowering for depressed or dependent people. If the choice of euthanasia is considered as good as a decision as receiving care, many people will feel guilty for not choosing death. Financial considerations, added to the concern about 'being a burden', could serve as powerful forces that would lead people to 'choose' euthanasia or assisted suicide. ** {{Objection}} Dependent people already feel the pressure of 'being a burden'. The legality of the act does not bear upon serious suicidal considerations. Put another way, think about who a dependent person might be concerned with burdening. It might be their family, friends, or care providers, but certainly not the state. * {{Argument against}} Legalizing euthanasia can lead to suicide contagion. When the media portrays assisted suicide as a means of 'taking control' or claims that someone helping another person kill themselves is 'death with dignity', then society (including teenagers) is receiving the dangerous message that suicide is a legitimate answer to life's problems.<ref>{{Cite web|url=https://www.nationalreview.com/human-exceptionalism/suicide-contagion-real-wesley-j-smith/|title=Suicide Contagion is Real|date=2013-05-21|website=National Review|language=en-US|access-date=2022-08-16}}</ref> ** {{Objection}} The argument relies on speculation as to the behavior of media and the behavior of society as a result. One might also speculate that the "contagion" effect is limited to adolescents, since the study also showed that the effect lessens with age.<ref>{{Cite web|url=https://www2.uottawa.ca/about-us/media|title=Media|website=About us|language=en|access-date=2022-08-16}}</ref> ** {{Objection}} The argument fails to show that an increase in suicide rates ought to be addressed through law (or at all). * {{Argument against}} Laws against euthanasia and assisted suicide are in place to prevent abuse and to protect people from unscrupulous medics and others. ** {{Objection}} If euthanasia were legal, abuse may arise in some instances, but your argument does not sufficiently show that it would be the norm. The mere possibility for abuse is not a reason to outlaw such a service. What your argument really suggests is the need for well regulated euthanasia and a thorough record of consent, incorporating practices that are resistant to malicious manipulation. *{{Argument against}}Euthanasia contradicts professional medical ethics where life is sought to be preserved. == See also == * [[Should suicide be legal?]] * [[Should capital punishment be legal?]] == Notes and references == {{Reflist}} [[Category:Politics]] [[Category:Death]] [[Category:Wikiversity Law Review]] [[Category:Legal ethics]] sd4j3qwlmum4fzuzo78dyc5shni9h6w 0 215897 2416334 2394966 2022-08-18T22:55:09Z Sophivorus 85690 Bring objections from the Spanish Wikidebate wikitext text/x-wiki {{Wikidebate}} {{Law}} By marriage we mean the legal union, not the religious ceremony. ==Same-sex marriage should be legal== ===Arguments for=== *{{Argument for}} Forbidding a gay couple from marriage when it's explicitly available to a heterosexual couple is preventing their freedom of expression. *{{Argument for}} Forbidding gay marriage is making their union less valuable than another's by preventing it from reaching an official level. *{{Argument for}} Every human is equal and free to express his or her sexuality. **{{Objection}} There are many ways to live sexuality outside of marriage, such as free union for example. **{{Objection}} This debate isn't about the right to have sex or otherwise express one's sexuality towards someone of the same sex. It's about marriage, a legal figure. ***{{Objection}} Marriage is an expression of sexuality as much as it is a legal statement. ****{{Objection}} What about celibate couples? *{{Argument for}} Marriage is no longer based on complementary, gender-based roles, and therefore the gender of participants no longer matters. **{{Objection}} This is not the case in many jurisdictions. **{{Objection}} There will always be roles, even in same-sex couples there are roles. ===Arguments against=== *{{Argument against}} Marriage has been understood as the union or legal contract between a man and a woman for millennia. If gay people want equal rights, they may have them, but there is no need to force the meaning of such a long-lasting institution. New laws and institutions may be created to accommodate their civil requests, without upsetting traditions and the people who support them. **{{Objection}} Social concepts change, usually not because they are 'forced' or because they are attacks on the traditional, but because society evolves. Traditions are abandoned, changed, or conserved throughout history. No one demands you wear boots with buckles out of respect to those who follow the traditions of Pilgrim dress. Gay marriage is no more an attack on heterosexuality than modern dress is an attack on Pilgrims; or no more than non-Christians marrying are an attack on Christian marriage. **{{Objection}} The objection here seems to be the use of the term "marriage". If we called it "garriage" would that help? In the end if it walks like a duck, quacks like a duck and swims like a duck then call it a duck. How does a gay marriage devalue or affect in anyway a hetero marriage when they are totally de-coupled event? ==No marriage should recognized== *{{Argument for}} Aside from things such as civic unions, government should get out of the marriage business. Let individuals internally validate the quality of their personal relationships, or perhaps also their communities. There are people who are married who might as well not be; and there are people who are married in all but designation. ==See also== *[[Is morality objective?]] *[[Should polygamy be legal?]] *[[Same-sex marriage]] ==Notes and references== {{Reflist}} ==External links== *[[Wikiquote:Same-sex marriage]] [[Category:Politics]] [[Category:Sexuality]] [[Category:Wikiversity Law Review]] [[Category:Legal ethics]] 7q2od4wxk21u55jarvzzjq5ei8b3yhy 2 216921 2416349 2399843 2022-08-18T23:07:14Z Sophivorus 85690 Undo all revisions. Resource is empty, but not [[Wikiversity:Deletions|deleted]]. wikitext text/x-wiki phoiac9h4m842xq45sp7s6u21eteeq1 1 228878 2416398 2415968 2022-08-19T01:38:22Z OhanaUnited 18921 /* Editorial board application of Dr. Raj Kumar Yadav */ Reply wikitext text/x-wiki <noinclude> {{WikiJournal editorial application top |archive box = {{Archive box|[[/Archive 2017]] <br>[[/Archive 2018]] <br>[[/Archive 2019]] <br>[[/Archive 2020]] }} }} </noinclude> ==Editorial board application of Dr. Raj Kumar Yadav== {{WikiJournal editor application submitted | position =Editorial board | name =Dr. Raj Kumar Yadav | qualifications =PhD | link =http://cup.edu.in/rajkumaryadav.php | areas_of_expertise =Criminal Law, Clinical Legal Education | professional_experience =14 Years | publishing_experience =11 Years | open_experience =Wikipedia | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:HariSinghw|HariSinghw]] ([[User talk:HariSinghw|discuss]] • [[Special:Contributions/HariSinghw|contribs]]) 13:29, 19 July 2022 (UTC) }} * {{comment}} Experience editing on wiki [https://guc.toolforge.org/?by=date&user=HariSinghw looks solid]. My concern is that your [[User:HariSinghw|userpage]] states that you are Hari Singh Yadav. But your application and the faculty website that you have provided says your name is Raj Kumar Yadav. Can you clarify this discrepancy? I want to note that similar issue was brought up in July 2022 at [[:w:Wikipedia:Articles for deletion/Raj Kumar Yadav (academic)|en.wp AfD for your page]] over who you really are. Are you Raj Kumar Yadav? Or are you Hari Singh Yadav editing on behalf Raj Kumar Yadav? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 19:47, 17 August 2022 (UTC) *:Sir, previously this account was handled by my father now I Dr. Raj Kumar Yadav operate this account because creation of double account may declare us sock puppet as it happened years back.[[User:HariSinghw|HariSinghw]] ([[User talk:HariSinghw|discuss]] • [[Special:Contributions/HariSinghw|contribs]]) 00:40, 18 August 2022 (UTC) *::{{re|HariSinghw}} Actually, you should be using a different account since each editor is expected to not share their account with other individuals for accountability purpose. I strongly suggest you to read and familiarize yourself with [[w:WP:SOCK|Sockpuppetry policy]]. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 01:38, 19 August 2022 (UTC) j0syvrtix2mbb55t1h8kbuy9ocvo6lv 2416470 2416398 2022-08-19T11:24:41Z HariSinghw 2937420 /* Editorial board application of Dr. Raj Kumar Yadav */ Reply wikitext text/x-wiki <noinclude> {{WikiJournal editorial application top |archive box = {{Archive box|[[/Archive 2017]] <br>[[/Archive 2018]] <br>[[/Archive 2019]] <br>[[/Archive 2020]] }} }} </noinclude> ==Editorial board application of Dr. Raj Kumar Yadav== {{WikiJournal editor application submitted | position =Editorial board | name =Dr. Raj Kumar Yadav | qualifications =PhD | link =http://cup.edu.in/rajkumaryadav.php | areas_of_expertise =Criminal Law, Clinical Legal Education | professional_experience =14 Years | publishing_experience =11 Years | open_experience =Wikipedia | policy_confirm =I confirm that I will act in accordance with the policies of the WikiJournal of Humanities. [[User:HariSinghw|HariSinghw]] ([[User talk:HariSinghw|discuss]] • [[Special:Contributions/HariSinghw|contribs]]) 13:29, 19 July 2022 (UTC) }} * {{comment}} Experience editing on wiki [https://guc.toolforge.org/?by=date&user=HariSinghw looks solid]. My concern is that your [[User:HariSinghw|userpage]] states that you are Hari Singh Yadav. But your application and the faculty website that you have provided says your name is Raj Kumar Yadav. Can you clarify this discrepancy? I want to note that similar issue was brought up in July 2022 at [[:w:Wikipedia:Articles for deletion/Raj Kumar Yadav (academic)|en.wp AfD for your page]] over who you really are. Are you Raj Kumar Yadav? Or are you Hari Singh Yadav editing on behalf Raj Kumar Yadav? [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 19:47, 17 August 2022 (UTC) *:Sir, previously this account was handled by my father now I Dr. Raj Kumar Yadav operate this account because creation of double account may declare us sock puppet as it happened years back.[[User:HariSinghw|HariSinghw]] ([[User talk:HariSinghw|discuss]] • [[Special:Contributions/HariSinghw|contribs]]) 00:40, 18 August 2022 (UTC) *::{{re|HariSinghw}} Actually, you should be using a different account since each editor is expected to not share their account with other individuals for accountability purpose. I strongly suggest you to read and familiarize yourself with [[w:WP:SOCK|Sockpuppetry policy]]. [[User:OhanaUnited|<b><span style="color: #0000FF;">OhanaUnited</span></b>]][[User talk:OhanaUnited|<b><span style="color: green;">^{Talk page}</span></b>]] 01:38, 19 August 2022 (UTC) *:::Thanks [[User:HariSinghw|HariSinghw]] ([[User talk:HariSinghw|discuss]] • [[Special:Contributions/HariSinghw|contribs]]) 11:24, 19 August 2022 (UTC) 99onxnddc9v99leddroga4n7jkz323k 0 232469 2416326 2415714 2022-08-18T22:31:31Z Young1lim 21186 /* ARM Assembly Programming (II) */ wikitext text/x-wiki == '''Background''' == '''Combinational and Sequential Circuits''' * [[Media:DD2.B.4..Adder.20131007.pdf |Adder]] * [[Media:DD3.A.1.LatchFF.20160308.pdf |Latches and Flipflops]] '''FSM''' * [[Media:DD3.A.3.FSM.20131030.pdf |FSM]] * [[Media:CArch.2.A.Bubble.20131021.pdf |FSM Example]] '''Tiny CPU Example''' * [[Media:CDsgn6.TinyCPU.2.A.ISA.20160511.pdf |Instruction Set]] * [[Media:CDsgn6.TinyCPU.2.B.DPath.20160502.pdf |Data Path]] * [[Media:CDsgn6.TinyCPU.2.C.CPath.20160427.pdf |Control Path]] * [[Media:CDsgn6.TinyCPU.2.D.Implement.20160513.pdf |FPGA Implementation]] </br> == '''Microprocessor Architecture''' == * ARM Architecture : - Programmer's Model ([[Media:ARM.1Arch.1A.Model.20180321.pdf |pdf]]) : - Pipelined Architecture ([[Media:ARM.1Arch.2A.Pipeline.20180419.pdf |pdf]]) * ARM Organization * ARM Cortex-M Processor Architecture * ARM Processor Cores </br> == '''Instruction Set Architecture''' == * ARM Instruction Set : - Overview ([[Media:ARM.2ISA.1A.Overview.20190611.pdf |pdf]]) : - Addressing Modes ([[Media:ARM.2ISA.2A.AddrMode.20191108.pdf |pdf]]) : - Multiple Transfer ([[Media:ARM.2ISA.3A.MTransfer.20190903.pdf |pdf]]) : - Assembler Format :: - Data Processing ([[Media:ARM.2ISA.4A.Proc.Format.20200204.pdf |pdf]]) :: - Data Transfer ([[Media:ARM.2ISA.4B.Trans.Format.20200205.pdf |pdf]]) :: - Coprocessor ([[Media:ARM.2ISA.4C.CoProc.Format.20191214.pdf |pdf]]) :: - Summary ([[Media:ARM.2ISA.4D.Summary.Format.20200205.pdf |pdf]]) : - Binary Encoding ([[Media:ARM.2ISA.5A.Encoding.201901105.pdf |pdf]]) * Thumb Instruction Set </br> == '''Assembly Programming''' == === ARM Assembly Programming (I) === * 1. Overview ([[Media:ARM.2ASM.1A.Overview.20200101.pdf |pdf]]) * 2. Example Programs ([[Media:ARM.2ASM.2A.Program.20200108.pdf |pdf]]) * 3. Addressing Modes ([[Media:ARM.2ASM.3A.Address.20200127.pdf |pdf]]) * 4. Data Transfer ([[Media:ARM.2ASM.4A.DTransfer.20200206.pdf |pdf]]) * 5. Data Processing ([[Media:ARM.2ASM.5A.DProcess.20200208.pdf |pdf]]) * 6. Control ([[Media:ARM.2ASM.6A.Control.20200215.pdf |pdf]]) * 7. Arrays ([[Media:ARM.2ASM.7A.Array.20200311.pdf |pdf]]) * 8. Data Structures ([[Media:ARM.2ASM.8A.DataStruct.20200718.pdf |pdf]]) * 9. Finite State Machines ([[Media:ARM.2ASM.9A.FSM.20200417.pdf |pdf]]) * 10. Functions ([[Media:ARM.2ASM.10A.Function.20210115.pdf |pdf]]) * 11. Parameter Passing ([[Media:ARM.2ASM.11A.Parameter.20210106.pdf |pdf]]) * 12. Stack Frames ([[Media:ARM.2ASM.12A.StackFrame.20210611.pdf |pdf]]) :: :: === ARM Assembly Programming (II) === :: * 1. Thumb instruction programming ([[Media:ARM.2ASM.Thumb.20210612.pdf |pdf]]) * 2. Exceptions ([[Media:ARM.2ASM.Exception.20220722.pdf |pdf]]) * 3. Exception Programming ([[Media:ARM.2ASM.ExceptionProg.20220311.pdf |pdf]]) * 4. Exception Handlers ([[Media:ARM.2ASM.ExceptionHandler.20220131.pdf |pdf]]) * 5. Interrupt Programming ([[Media:ARM.2ASM.InterruptProg.20211030.pdf |pdf]]) * 6. Interrupt Handlers ([[Media:ARM.2ASM.InterruptHandler.20211030.pdf |pdf]]) * 7. Vectored Interrupt Programming ([[Media:ARM.2ASM.VectorInt.20220817.pdf |pdf]]) </br> * ARM Assembly Exercises ([[Media:ESys.3.A.ARM-ASM-Exercise.20160608.pdf |A.pdf]], [[Media:ESys.3.B.Assembly.20160716.pdf |B.pdf]]) :: === ARM Assembly Programming (III) === * 1. Fixed point arithmetic (integer division) * 2. Floating point arithmetic * 3. Matrix multiply === ARM Linking === * arm link ([[Media:arm_link.20211208.pdf |pdf]]) </br> === ARM Microcontroller Programming === * 1. Input / Output * 2. Serial / Parallel Port Interfacing * 3. Analog I/O Interfacing * 4. Communication </br> == '''Memory Architecture''' == </br> === '''Memory Hierarchy''' === </br> === '''System and Peripheral Buses''' === </br> === '''Architectural Support''' === * High Level Languages * System Development * Operating Systems </br> == '''Peripheral Architecture''' == </br> === '''Vectored Interrupt Controller ''' === </br> === '''Timers ''' === * Timer / Counter ([[Media:ARM.4ASM.Timer.20220801.pdf |pdf]]) * Real Time Clock * Watchdog Timer </br> === '''Serial Bus''' === * '''UART''' : Universal Asynchronous Receiver/Transmitter ([[Media:ARM.4ASM.UART.20220816.pdf |pdf]]) * '''I2C''' : Inter-Integrated Circuit * '''SPI''' : Serial Peripheral Interface * '''USB''' : Universal Serial Bus Device Controller </br> === '''I/Os ''' === * General Purpose Input/Output ports (GPIO) * Pulse Width Modulator * Analog-to-Digital Converter (ADC) * Digital-to-Analog Converter (DAC) </br> <!-- == '''Interrupts and Exceptions ''' == --> </br> == '''Synchrnoization'''== </br> === H/W and S/W Synchronization === * busy wait synchronization * handshake interface </br> === Interrupt Synchronization === * interrupt synchronization * reentrant programming * buffered IO * periodic interrupt * periodic polling </br> ==''' Interfacing '''== </br> === Time Interfacing === * input capture * output compare </br> === Serial Interfacing === * Programming UART * Programming SPI * Programming I2C * Programming USB </br> === Analog Interfacing === * OP Amp * Filters * ADC * DAC </br> == '''Old materials''' == === '''Instruction Set Architecture''' === * ARM Instruction Set :: - Overview ([[Media:ARM.2ISA.1A.Overview.20180528.pdf |pdf]]) :: - Binary Encoding ([[Media:ARM.2ISA.2A.Encoding.20180528.pdf |pdf]]) :: - Assembler Format ([[Media:ARM.2ISA.3A.Format.20180528.pdf |pdf]]) * Thumb Instruction Set * ARM Assembly Language ([[Media:ESys3.1A.Assembly.20160608.pdf |pdf]]) * ARM Machine Language ([[Media:ESys3.2A.Machine.20160615.pdf |pdf]]) </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] 0x4rxy5557v65tevsy2ksery7isipnj 2416328 2416326 2022-08-18T22:33:31Z Young1lim 21186 /* ARM Assembly Programming (II) */ wikitext text/x-wiki == '''Background''' == '''Combinational and Sequential Circuits''' * [[Media:DD2.B.4..Adder.20131007.pdf |Adder]] * [[Media:DD3.A.1.LatchFF.20160308.pdf |Latches and Flipflops]] '''FSM''' * [[Media:DD3.A.3.FSM.20131030.pdf |FSM]] * [[Media:CArch.2.A.Bubble.20131021.pdf |FSM Example]] '''Tiny CPU Example''' * [[Media:CDsgn6.TinyCPU.2.A.ISA.20160511.pdf |Instruction Set]] * [[Media:CDsgn6.TinyCPU.2.B.DPath.20160502.pdf |Data Path]] * [[Media:CDsgn6.TinyCPU.2.C.CPath.20160427.pdf |Control Path]] * [[Media:CDsgn6.TinyCPU.2.D.Implement.20160513.pdf |FPGA Implementation]] </br> == '''Microprocessor Architecture''' == * ARM Architecture : - Programmer's Model ([[Media:ARM.1Arch.1A.Model.20180321.pdf |pdf]]) : - Pipelined Architecture ([[Media:ARM.1Arch.2A.Pipeline.20180419.pdf |pdf]]) * ARM Organization * ARM Cortex-M Processor Architecture * ARM Processor Cores </br> == '''Instruction Set Architecture''' == * ARM Instruction Set : - Overview ([[Media:ARM.2ISA.1A.Overview.20190611.pdf |pdf]]) : - Addressing Modes ([[Media:ARM.2ISA.2A.AddrMode.20191108.pdf |pdf]]) : - Multiple Transfer ([[Media:ARM.2ISA.3A.MTransfer.20190903.pdf |pdf]]) : - Assembler Format :: - Data Processing ([[Media:ARM.2ISA.4A.Proc.Format.20200204.pdf |pdf]]) :: - Data Transfer ([[Media:ARM.2ISA.4B.Trans.Format.20200205.pdf |pdf]]) :: - Coprocessor ([[Media:ARM.2ISA.4C.CoProc.Format.20191214.pdf |pdf]]) :: - Summary ([[Media:ARM.2ISA.4D.Summary.Format.20200205.pdf |pdf]]) : - Binary Encoding ([[Media:ARM.2ISA.5A.Encoding.201901105.pdf |pdf]]) * Thumb Instruction Set </br> == '''Assembly Programming''' == === ARM Assembly Programming (I) === * 1. Overview ([[Media:ARM.2ASM.1A.Overview.20200101.pdf |pdf]]) * 2. Example Programs ([[Media:ARM.2ASM.2A.Program.20200108.pdf |pdf]]) * 3. Addressing Modes ([[Media:ARM.2ASM.3A.Address.20200127.pdf |pdf]]) * 4. Data Transfer ([[Media:ARM.2ASM.4A.DTransfer.20200206.pdf |pdf]]) * 5. Data Processing ([[Media:ARM.2ASM.5A.DProcess.20200208.pdf |pdf]]) * 6. Control ([[Media:ARM.2ASM.6A.Control.20200215.pdf |pdf]]) * 7. Arrays ([[Media:ARM.2ASM.7A.Array.20200311.pdf |pdf]]) * 8. Data Structures ([[Media:ARM.2ASM.8A.DataStruct.20200718.pdf |pdf]]) * 9. Finite State Machines ([[Media:ARM.2ASM.9A.FSM.20200417.pdf |pdf]]) * 10. Functions ([[Media:ARM.2ASM.10A.Function.20210115.pdf |pdf]]) * 11. Parameter Passing ([[Media:ARM.2ASM.11A.Parameter.20210106.pdf |pdf]]) * 12. Stack Frames ([[Media:ARM.2ASM.12A.StackFrame.20210611.pdf |pdf]]) :: :: === ARM Assembly Programming (II) === :: * 1. Thumb instruction programming ([[Media:ARM.2ASM.Thumb.20210612.pdf |pdf]]) * 2. Exceptions ([[Media:ARM.2ASM.Exception.20220722.pdf |pdf]]) * 3. Exception Programming ([[Media:ARM.2ASM.ExceptionProg.20220311.pdf |pdf]]) * 4. Exception Handlers ([[Media:ARM.2ASM.ExceptionHandler.20220131.pdf |pdf]]) * 5. Interrupt Programming ([[Media:ARM.2ASM.InterruptProg.20211030.pdf |pdf]]) * 6. Interrupt Handlers ([[Media:ARM.2ASM.InterruptHandler.20211030.pdf |pdf]]) * 7. Vectored Interrupt Programming ([[Media:ARM.2ASM.VectorInt.20220818.pdf |pdf]]) </br> * ARM Assembly Exercises ([[Media:ESys.3.A.ARM-ASM-Exercise.20160608.pdf |A.pdf]], [[Media:ESys.3.B.Assembly.20160716.pdf |B.pdf]]) :: === ARM Assembly Programming (III) === * 1. Fixed point arithmetic (integer division) * 2. Floating point arithmetic * 3. Matrix multiply === ARM Linking === * arm link ([[Media:arm_link.20211208.pdf |pdf]]) </br> === ARM Microcontroller Programming === * 1. Input / Output * 2. Serial / Parallel Port Interfacing * 3. Analog I/O Interfacing * 4. Communication </br> == '''Memory Architecture''' == </br> === '''Memory Hierarchy''' === </br> === '''System and Peripheral Buses''' === </br> === '''Architectural Support''' === * High Level Languages * System Development * Operating Systems </br> == '''Peripheral Architecture''' == </br> === '''Vectored Interrupt Controller ''' === </br> === '''Timers ''' === * Timer / Counter ([[Media:ARM.4ASM.Timer.20220801.pdf |pdf]]) * Real Time Clock * Watchdog Timer </br> === '''Serial Bus''' === * '''UART''' : Universal Asynchronous Receiver/Transmitter ([[Media:ARM.4ASM.UART.20220816.pdf |pdf]]) * '''I2C''' : Inter-Integrated Circuit * '''SPI''' : Serial Peripheral Interface * '''USB''' : Universal Serial Bus Device Controller </br> === '''I/Os ''' === * General Purpose Input/Output ports (GPIO) * Pulse Width Modulator * Analog-to-Digital Converter (ADC) * Digital-to-Analog Converter (DAC) </br> <!-- == '''Interrupts and Exceptions ''' == --> </br> == '''Synchrnoization'''== </br> === H/W and S/W Synchronization === * busy wait synchronization * handshake interface </br> === Interrupt Synchronization === * interrupt synchronization * reentrant programming * buffered IO * periodic interrupt * periodic polling </br> ==''' Interfacing '''== </br> === Time Interfacing === * input capture * output compare </br> === Serial Interfacing === * Programming UART * Programming SPI * Programming I2C * Programming USB </br> === Analog Interfacing === * OP Amp * Filters * ADC * DAC </br> == '''Old materials''' == === '''Instruction Set Architecture''' === * ARM Instruction Set :: - Overview ([[Media:ARM.2ISA.1A.Overview.20180528.pdf |pdf]]) :: - Binary Encoding ([[Media:ARM.2ISA.2A.Encoding.20180528.pdf |pdf]]) :: - Assembler Format ([[Media:ARM.2ISA.3A.Format.20180528.pdf |pdf]]) * Thumb Instruction Set * ARM Assembly Language ([[Media:ESys3.1A.Assembly.20160608.pdf |pdf]]) * ARM Machine Language ([[Media:ESys3.2A.Machine.20160615.pdf |pdf]]) </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] kjml4n1n31xjx3r4kzh5jt49jby70g8 2416335 2416328 2022-08-18T22:55:24Z Young1lim 21186 /* Serial Bus */ wikitext text/x-wiki == '''Background''' == '''Combinational and Sequential Circuits''' * [[Media:DD2.B.4..Adder.20131007.pdf |Adder]] * [[Media:DD3.A.1.LatchFF.20160308.pdf |Latches and Flipflops]] '''FSM''' * [[Media:DD3.A.3.FSM.20131030.pdf |FSM]] * [[Media:CArch.2.A.Bubble.20131021.pdf |FSM Example]] '''Tiny CPU Example''' * [[Media:CDsgn6.TinyCPU.2.A.ISA.20160511.pdf |Instruction Set]] * [[Media:CDsgn6.TinyCPU.2.B.DPath.20160502.pdf |Data Path]] * [[Media:CDsgn6.TinyCPU.2.C.CPath.20160427.pdf |Control Path]] * [[Media:CDsgn6.TinyCPU.2.D.Implement.20160513.pdf |FPGA Implementation]] </br> == '''Microprocessor Architecture''' == * ARM Architecture : - Programmer's Model ([[Media:ARM.1Arch.1A.Model.20180321.pdf |pdf]]) : - Pipelined Architecture ([[Media:ARM.1Arch.2A.Pipeline.20180419.pdf |pdf]]) * ARM Organization * ARM Cortex-M Processor Architecture * ARM Processor Cores </br> == '''Instruction Set Architecture''' == * ARM Instruction Set : - Overview ([[Media:ARM.2ISA.1A.Overview.20190611.pdf |pdf]]) : - Addressing Modes ([[Media:ARM.2ISA.2A.AddrMode.20191108.pdf |pdf]]) : - Multiple Transfer ([[Media:ARM.2ISA.3A.MTransfer.20190903.pdf |pdf]]) : - Assembler Format :: - Data Processing ([[Media:ARM.2ISA.4A.Proc.Format.20200204.pdf |pdf]]) :: - Data Transfer ([[Media:ARM.2ISA.4B.Trans.Format.20200205.pdf |pdf]]) :: - Coprocessor ([[Media:ARM.2ISA.4C.CoProc.Format.20191214.pdf |pdf]]) :: - Summary ([[Media:ARM.2ISA.4D.Summary.Format.20200205.pdf |pdf]]) : - Binary Encoding ([[Media:ARM.2ISA.5A.Encoding.201901105.pdf |pdf]]) * Thumb Instruction Set </br> == '''Assembly Programming''' == === ARM Assembly Programming (I) === * 1. Overview ([[Media:ARM.2ASM.1A.Overview.20200101.pdf |pdf]]) * 2. Example Programs ([[Media:ARM.2ASM.2A.Program.20200108.pdf |pdf]]) * 3. Addressing Modes ([[Media:ARM.2ASM.3A.Address.20200127.pdf |pdf]]) * 4. Data Transfer ([[Media:ARM.2ASM.4A.DTransfer.20200206.pdf |pdf]]) * 5. Data Processing ([[Media:ARM.2ASM.5A.DProcess.20200208.pdf |pdf]]) * 6. Control ([[Media:ARM.2ASM.6A.Control.20200215.pdf |pdf]]) * 7. Arrays ([[Media:ARM.2ASM.7A.Array.20200311.pdf |pdf]]) * 8. Data Structures ([[Media:ARM.2ASM.8A.DataStruct.20200718.pdf |pdf]]) * 9. Finite State Machines ([[Media:ARM.2ASM.9A.FSM.20200417.pdf |pdf]]) * 10. Functions ([[Media:ARM.2ASM.10A.Function.20210115.pdf |pdf]]) * 11. Parameter Passing ([[Media:ARM.2ASM.11A.Parameter.20210106.pdf |pdf]]) * 12. Stack Frames ([[Media:ARM.2ASM.12A.StackFrame.20210611.pdf |pdf]]) :: :: === ARM Assembly Programming (II) === :: * 1. Thumb instruction programming ([[Media:ARM.2ASM.Thumb.20210612.pdf |pdf]]) * 2. Exceptions ([[Media:ARM.2ASM.Exception.20220722.pdf |pdf]]) * 3. Exception Programming ([[Media:ARM.2ASM.ExceptionProg.20220311.pdf |pdf]]) * 4. Exception Handlers ([[Media:ARM.2ASM.ExceptionHandler.20220131.pdf |pdf]]) * 5. Interrupt Programming ([[Media:ARM.2ASM.InterruptProg.20211030.pdf |pdf]]) * 6. Interrupt Handlers ([[Media:ARM.2ASM.InterruptHandler.20211030.pdf |pdf]]) * 7. Vectored Interrupt Programming ([[Media:ARM.2ASM.VectorInt.20220818.pdf |pdf]]) </br> * ARM Assembly Exercises ([[Media:ESys.3.A.ARM-ASM-Exercise.20160608.pdf |A.pdf]], [[Media:ESys.3.B.Assembly.20160716.pdf |B.pdf]]) :: === ARM Assembly Programming (III) === * 1. Fixed point arithmetic (integer division) * 2. Floating point arithmetic * 3. Matrix multiply === ARM Linking === * arm link ([[Media:arm_link.20211208.pdf |pdf]]) </br> === ARM Microcontroller Programming === * 1. Input / Output * 2. Serial / Parallel Port Interfacing * 3. Analog I/O Interfacing * 4. Communication </br> == '''Memory Architecture''' == </br> === '''Memory Hierarchy''' === </br> === '''System and Peripheral Buses''' === </br> === '''Architectural Support''' === * High Level Languages * System Development * Operating Systems </br> == '''Peripheral Architecture''' == </br> === '''Vectored Interrupt Controller ''' === </br> === '''Timers ''' === * Timer / Counter ([[Media:ARM.4ASM.Timer.20220801.pdf |pdf]]) * Real Time Clock * Watchdog Timer </br> === '''Serial Bus''' === * '''UART''' : Universal Asynchronous Receiver/Transmitter ([[Media:ARM.4ASM.UART.20220817.pdf |pdf]]) * '''I2C''' : Inter-Integrated Circuit * '''SPI''' : Serial Peripheral Interface * '''USB''' : Universal Serial Bus Device Controller </br> === '''I/Os ''' === * General Purpose Input/Output ports (GPIO) * Pulse Width Modulator * Analog-to-Digital Converter (ADC) * Digital-to-Analog Converter (DAC) </br> <!-- == '''Interrupts and Exceptions ''' == --> </br> == '''Synchrnoization'''== </br> === H/W and S/W Synchronization === * busy wait synchronization * handshake interface </br> === Interrupt Synchronization === * interrupt synchronization * reentrant programming * buffered IO * periodic interrupt * periodic polling </br> ==''' Interfacing '''== </br> === Time Interfacing === * input capture * output compare </br> === Serial Interfacing === * Programming UART * Programming SPI * Programming I2C * Programming USB </br> === Analog Interfacing === * OP Amp * Filters * ADC * DAC </br> == '''Old materials''' == === '''Instruction Set Architecture''' === * ARM Instruction Set :: - Overview ([[Media:ARM.2ISA.1A.Overview.20180528.pdf |pdf]]) :: - Binary Encoding ([[Media:ARM.2ISA.2A.Encoding.20180528.pdf |pdf]]) :: - Assembler Format ([[Media:ARM.2ISA.3A.Format.20180528.pdf |pdf]]) * Thumb Instruction Set * ARM Assembly Language ([[Media:ESys3.1A.Assembly.20160608.pdf |pdf]]) * ARM Machine Language ([[Media:ESys3.2A.Machine.20160615.pdf |pdf]]) </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] ewl72k0hifjmu6eundbinvxdaad3fp8 2416338 2416335 2022-08-18T22:56:59Z Young1lim 21186 /* Serial Bus */ wikitext text/x-wiki == '''Background''' == '''Combinational and Sequential Circuits''' * [[Media:DD2.B.4..Adder.20131007.pdf |Adder]] * [[Media:DD3.A.1.LatchFF.20160308.pdf |Latches and Flipflops]] '''FSM''' * [[Media:DD3.A.3.FSM.20131030.pdf |FSM]] * [[Media:CArch.2.A.Bubble.20131021.pdf |FSM Example]] '''Tiny CPU Example''' * [[Media:CDsgn6.TinyCPU.2.A.ISA.20160511.pdf |Instruction Set]] * [[Media:CDsgn6.TinyCPU.2.B.DPath.20160502.pdf |Data Path]] * [[Media:CDsgn6.TinyCPU.2.C.CPath.20160427.pdf |Control Path]] * [[Media:CDsgn6.TinyCPU.2.D.Implement.20160513.pdf |FPGA Implementation]] </br> == '''Microprocessor Architecture''' == * ARM Architecture : - Programmer's Model ([[Media:ARM.1Arch.1A.Model.20180321.pdf |pdf]]) : - Pipelined Architecture ([[Media:ARM.1Arch.2A.Pipeline.20180419.pdf |pdf]]) * ARM Organization * ARM Cortex-M Processor Architecture * ARM Processor Cores </br> == '''Instruction Set Architecture''' == * ARM Instruction Set : - Overview ([[Media:ARM.2ISA.1A.Overview.20190611.pdf |pdf]]) : - Addressing Modes ([[Media:ARM.2ISA.2A.AddrMode.20191108.pdf |pdf]]) : - Multiple Transfer ([[Media:ARM.2ISA.3A.MTransfer.20190903.pdf |pdf]]) : - Assembler Format :: - Data Processing ([[Media:ARM.2ISA.4A.Proc.Format.20200204.pdf |pdf]]) :: - Data Transfer ([[Media:ARM.2ISA.4B.Trans.Format.20200205.pdf |pdf]]) :: - Coprocessor ([[Media:ARM.2ISA.4C.CoProc.Format.20191214.pdf |pdf]]) :: - Summary ([[Media:ARM.2ISA.4D.Summary.Format.20200205.pdf |pdf]]) : - Binary Encoding ([[Media:ARM.2ISA.5A.Encoding.201901105.pdf |pdf]]) * Thumb Instruction Set </br> == '''Assembly Programming''' == === ARM Assembly Programming (I) === * 1. Overview ([[Media:ARM.2ASM.1A.Overview.20200101.pdf |pdf]]) * 2. Example Programs ([[Media:ARM.2ASM.2A.Program.20200108.pdf |pdf]]) * 3. Addressing Modes ([[Media:ARM.2ASM.3A.Address.20200127.pdf |pdf]]) * 4. Data Transfer ([[Media:ARM.2ASM.4A.DTransfer.20200206.pdf |pdf]]) * 5. Data Processing ([[Media:ARM.2ASM.5A.DProcess.20200208.pdf |pdf]]) * 6. Control ([[Media:ARM.2ASM.6A.Control.20200215.pdf |pdf]]) * 7. Arrays ([[Media:ARM.2ASM.7A.Array.20200311.pdf |pdf]]) * 8. Data Structures ([[Media:ARM.2ASM.8A.DataStruct.20200718.pdf |pdf]]) * 9. Finite State Machines ([[Media:ARM.2ASM.9A.FSM.20200417.pdf |pdf]]) * 10. Functions ([[Media:ARM.2ASM.10A.Function.20210115.pdf |pdf]]) * 11. Parameter Passing ([[Media:ARM.2ASM.11A.Parameter.20210106.pdf |pdf]]) * 12. Stack Frames ([[Media:ARM.2ASM.12A.StackFrame.20210611.pdf |pdf]]) :: :: === ARM Assembly Programming (II) === :: * 1. Thumb instruction programming ([[Media:ARM.2ASM.Thumb.20210612.pdf |pdf]]) * 2. Exceptions ([[Media:ARM.2ASM.Exception.20220722.pdf |pdf]]) * 3. Exception Programming ([[Media:ARM.2ASM.ExceptionProg.20220311.pdf |pdf]]) * 4. Exception Handlers ([[Media:ARM.2ASM.ExceptionHandler.20220131.pdf |pdf]]) * 5. Interrupt Programming ([[Media:ARM.2ASM.InterruptProg.20211030.pdf |pdf]]) * 6. Interrupt Handlers ([[Media:ARM.2ASM.InterruptHandler.20211030.pdf |pdf]]) * 7. Vectored Interrupt Programming ([[Media:ARM.2ASM.VectorInt.20220818.pdf |pdf]]) </br> * ARM Assembly Exercises ([[Media:ESys.3.A.ARM-ASM-Exercise.20160608.pdf |A.pdf]], [[Media:ESys.3.B.Assembly.20160716.pdf |B.pdf]]) :: === ARM Assembly Programming (III) === * 1. Fixed point arithmetic (integer division) * 2. Floating point arithmetic * 3. Matrix multiply === ARM Linking === * arm link ([[Media:arm_link.20211208.pdf |pdf]]) </br> === ARM Microcontroller Programming === * 1. Input / Output * 2. Serial / Parallel Port Interfacing * 3. Analog I/O Interfacing * 4. Communication </br> == '''Memory Architecture''' == </br> === '''Memory Hierarchy''' === </br> === '''System and Peripheral Buses''' === </br> === '''Architectural Support''' === * High Level Languages * System Development * Operating Systems </br> == '''Peripheral Architecture''' == </br> === '''Vectored Interrupt Controller ''' === </br> === '''Timers ''' === * Timer / Counter ([[Media:ARM.4ASM.Timer.20220801.pdf |pdf]]) * Real Time Clock * Watchdog Timer </br> === '''Serial Bus''' === * '''UART''' : Universal Asynchronous Receiver/Transmitter ([[Media:ARM.4ASM.UART.20220818.pdf |pdf]]) * '''I2C''' : Inter-Integrated Circuit * '''SPI''' : Serial Peripheral Interface * '''USB''' : Universal Serial Bus Device Controller </br> === '''I/Os ''' === * General Purpose Input/Output ports (GPIO) * Pulse Width Modulator * Analog-to-Digital Converter (ADC) * Digital-to-Analog Converter (DAC) </br> <!-- == '''Interrupts and Exceptions ''' == --> </br> == '''Synchrnoization'''== </br> === H/W and S/W Synchronization === * busy wait synchronization * handshake interface </br> === Interrupt Synchronization === * interrupt synchronization * reentrant programming * buffered IO * periodic interrupt * periodic polling </br> ==''' Interfacing '''== </br> === Time Interfacing === * input capture * output compare </br> === Serial Interfacing === * Programming UART * Programming SPI * Programming I2C * Programming USB </br> === Analog Interfacing === * OP Amp * Filters * ADC * DAC </br> == '''Old materials''' == === '''Instruction Set Architecture''' === * ARM Instruction Set :: - Overview ([[Media:ARM.2ISA.1A.Overview.20180528.pdf |pdf]]) :: - Binary Encoding ([[Media:ARM.2ISA.2A.Encoding.20180528.pdf |pdf]]) :: - Assembler Format ([[Media:ARM.2ISA.3A.Format.20180528.pdf |pdf]]) * Thumb Instruction Set * ARM Assembly Language ([[Media:ESys3.1A.Assembly.20160608.pdf |pdf]]) * ARM Machine Language ([[Media:ESys3.2A.Machine.20160615.pdf |pdf]]) </br> </br> go to [ [[Electrical_%26_Computer_Engineering_Studies]] ] s95mfy3psequmzgnmhoak7ogxo0flfv 0 234487 2416260 2398992 2022-08-18T19:14:21Z Marshallsumter 311529 wikitext text/x-wiki [[Image:Earth-grazing meteoroid, 13 October 1990 (2) TFA.jpg|right|thumb|300px|This is a scanned photograph of the bolide EN131090, originally captured on a glass photographic plate. Credit: European Fireball Network.{{tlx|free media}}]] This is a scanned photograph on the right of the bolide EN131090, originally captured on a glass photographic plate. The Earth-grazing meteoroid flew above Czechoslovakia and Poland on 13 October 1990 and left to space again. It was taken by an all-sky camera equipped with a fish-eye objective Zeiss Distagon 3.5/30mm located at the hydrometeorological station at Červená hora, Czechoslovakia (now in the Czech Republic). The bolide travels from the south to the north and its track is interrupted by a shutter rotating 12.5 times per second, which allows determining its speed. "As of 2011 the International Astronomical Union officially defines a meteoroid as a solid object moving in interplanetary space, of a size considerably smaller than an asteroid and considerably larger than an atom".<ref name=Millman>{{ cite journal |author=Peter M. Millman |year=1961 |title=A report on meteor terminology |journal=JRASC |volume=55 |pages=265–267 |bibcode=1961JRASC..55..265M }}</ref><ref name=Glossary>{{ cite book |url=http://www.imo.net/glossary |title=Glossary International Meteor Organization |publisher=Imo.net |date=2008-11-18 |accessdate=2011-09-16}}</ref> {{clear}} ==Meteors== {{main|Radiation/Meteors}} [[Image:Meteor burst.jpg|thumb|right|250px|This picture is of the Alpha-Monocerotid meteor outburst in 1995. It is a timed exposure where the meteors have actually occurred several seconds to several minutes apart. Credit: NASA Ames Research Center/S. Molau and P. Jenniskens.{{tlx|free media}}]] Some wanderers are '''meteors'''. A '''meteor''' is the visible path of a meteoroid that has entered the Earth's atmosphere. Meteors typically occur in the [[w:mesosphere|mesosphere]], and most range in altitude from 75&nbsp;km to 100&nbsp;km.<ref name=Erickson>{{cite book |url=http://www.haystack.mit.edu/~pje/meteors/ | title = Millstone Hill UHF Meteor Observations: Preliminary Results | author = Philip J. Erickson}}</ref> Millions of meteors occur in the Earth's atmosphere every day. Most meteoroids that cause meteors are about the size of a [[w:pebble|pebble]]. The [[w:Perseid|Perseid]] meteor shower, usually the richest meteor shower of the year, peaks in August. Over the course of an hour, a person watching a clear sky from a dark location might see as many as 50-100 meteors. Most meteors are actually pieces of rock that have broken off a comet and continue to orbit the Sun. The Earth travels through the comet debris in its orbit. As the small pieces enter the Earth's atmosphere, friction causes them to burn up. '''Def.''' "[a]ll other objects [not a planet or dwarf planet], except satellites, orbiting the Sun" are called collectively '''Small Solar-System Bodies'''.<ref name=Christensen>{{ cite book |author=Lars Lindberg Christensen |title=IAU 2006 General Assembly: Result of the IAU Resolution votes |publisher=International Astronomical Union |date=August 24, 2006 |url=http://www.iau.org/static/archives/releases/pdf/iau0603.pdf |accessdate=2011-10-30 }}</ref> "Coronal mass ejections (CMEs) are large‐scale expulsions of plasma and magnetic field from the solar corona to the interplanetary space. During a large CME event, ∼10¹⁶ g of coronal material with energies of ∼10³² ergs are ejected from the Sun [Hundhausen, 1997; Vourlidas et al., 2002]. While accelerating away from the Sun, CMEs present speeds between few tens up to ∼2500 km/s. CMEs with speeds exceeding the magnetosonic speed can drive fast shocks ahead of them. CME‐driven fast shocks are able to accelerate charged particles up to very high energies (∼GeV/nucleon) [Wang and Wang, 2006]."<ref name=Loesch>{{ cite journal |author=C. Loesch, M. Opher, M. V. Alves, R. M. Evans, and W. B. Manchester |title=Signatures of two distinct driving mechanisms in the evolution of coronal mass ejections in the lower corona |journal=Journal of Geophysical Research |month=April 13, |year=2011 |volume=116 |issue= |pages=A04106 |url=http://plutao.dpi.inpe.br/col/dpi.inpe.br/plutao/2011/06.11.03.21/doc/2010JA015582.pdf |arxiv= |bibcode= |doi=10.1029/2010JA015582 |pmid= |accessdate=2014-02-16 }}</ref> Current "knowledge of the orbital structure of the outer solar system, [is] mostly slanted towards that information which has been learned from the Canada-France-Ecliptic Plane Survey (CFEPS: www.cfeps.net). Based on our current datasets (inside and outside CFEPS) outer solar system modeling is now entering the erra of precission cosmogony."<ref name=Kavelaars>{{ cite journal |author=J. J. Kavelaars |title=The Outer Solar System, from Centaurs to the Detached Disk: Entering the Era of Precision Cosmogony, In: ''Asteroids, Comets, Meteors'' |publisher=Lunar and Planetary Institute |location=3600 Bay Area Boulevard, Houston, TX USA 77058 |month=May |year=2012 |editor= |volume= |issue=LPI Contribution No. 1667 |pages=6460 |url=http://adsabs.harvard.edu//abs/2012LPICo1667.6460K |arxiv= |bibcode=2012LPICo1667.6460K |doi= |pmid= |isbn= |accessdate=2013-12-20 }}</ref> "Since the discovery of the first members of the Kuiper belt (Jewitt and Luu, 1993) the growth in knowledge of the outer solar system has been marked (perhaps driven) by the discovery of individual objects whose dynamics pointed at previously unknown reserviours; for example: 1993 RO and the plutinos, 1996 TL66 and the ‘scattering disk’, 2003 CR103 and the detectatch disk, 90377 Sedna and the Inner Oort Cloud."<ref name=Kavelaars/> The "‘main Kuiper belt’ is populated by dynamically ‘hot’ and ‘cold’ subcomponents (Brown 2001), the dyncamically ‘cold’ component is further sub-divided into a ‘stirred’ and ‘kernel’ component (Petit et al., 2011). The plane of the Ecliptic does not match the ecliptic or invariable planes of the solar sytem (Elliot et al., 2005). Collisional families exists, Haumea (Brown et al., 2007)."<ref name=Kavelaars/> {{clear}} ==Theoretical meteoroids== [[Image:Ssc2005-04a medium.jpg|thumb|right|250px|Comet Encke's meteoroid trail is the long diagonal red glow. Twin jets of material can also be seen shooting away from the comet in the short, fan-shaped emission, spreading horizontally from the comet. Credit: NASA.{{tlx|free media}}]] [[Image:Sig06-011 medium.jpg|thumb|left|250px|Meteoroid trail shows up between fragments of Comet 73P (aka) Comet Schwassman-Wachmann 3. Credit: NASA/JPL-Caltech/W. Reach (SSC/Caltech).{{tlx|fairuse}}]] '''Def.''' a "fast-moving streak of light in the night sky caused by the entry of extraterrestrial matter into the earth's atmosphere"<ref name=MeteorWikt>{{ cite book |author=[[wikt:User:Xed~enwiktionary|Xed~enwiktionary]] |title=meteor |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=3 September 2004 |url=https://en.wiktionary.org/wiki/meteor |accessdate=26 June 2019 }}</ref> is called a '''meteor'''. '''Def.''' "a relatively small (sand- to boulder-sized) fragment of debris in a solar system"<ref name=MeteoroidWikt>{{ cite book |author=[[wikt:User:SnoopY|SnoopY]] |title=meteoroid |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=21 December 2005 |url=https://en.wiktionary.org/wiki/meteoroid |accessdate=2016-02-06 }}</ref> is called a '''meteoroid'''. Beech and Steel, writing in ''Quarterly Journal of the Royal Astronomical Society'', proposed a new definition where a meteoroid is between 100 µm and 10 m across.<ref name=Beach>{{ cite journal |author=Martin Beech, Duncan Steel |year=1995 |month=September |title=On the Definition of the Term Meteoroid |journal=Quarterly Journal of the Royal Astronomical Society |volume=36 |issue=3 |pages=281–284 |bibcode=1995QJRAS..36..281B }})</ref> Following the discovery and naming of asteroids below 10 m in size (e.g., 2008 TC3), Rubin and Grossman refined the Beech and Steel definition of meteoroid to objects between 10&nbsp;µm and 1 m in diameter.<ref name=Rubin>{{Cite journal |author=Rubin, A.E. |coauthors=Grossman, J.N. |year=2010 |month=January |title=Meteorite and meteoroid: New comprehensive definitions |journal=Meteoritics & Planetary Science |volume=45 |issue=1 |pages=114–122 |bibcode=2010M&PS...45..114R |doi = 10.1111/j.1945-5100.2009.01009.x }})</ref> The near-Earth object (NEO) definition includes larger objects, up to 50 m in diameter, in this category. Most of our short-period meteor showers are not from the normal water vapor drag of active comets, but the product of infrequent disintegrations, when large chunks break off a mostly dormant comet.<ref name="Jenniskens P. 2006">Jenniskens P. (2006). ''Meteor Showers and their Parent Comets''. Cambridge University Press, Cambridge, U.K., 790 pp.</ref> {{clear}} ==Alpha Capricornids== Alpha Capricornids is a meteor shower that takes place as early as 15 July and continues until around 10 August.<ref>{{cite book |url= http://encarta.msn.com/encyclopedia_761587655/alpha_capricornids.html |title= Alpha Capricornids: Encyclopedia Article |work=Encarta.msn.com |accessdate=2014-07-08|archiveurl=https://web.archive.org/web/20090419084015/http://encarta.msn.com/encyclopedia_761587655/Alpha_Capricornids.html |archivedate=2009-04-19|publisher= Encarta}}</ref> The parent body is asteroid 2002 EX12 [169P/NEAT], which in the return of 2005 was found weakly active near perihelion.<ref name="Jenniskens">{{cite journal|url=http://iopscience.iop.org/1538-3881/139/5/1822/ |title= Minor planet 2002 EX12 ( = 169P/NEAT) and the Alpha Capricornid shower|journal=Astronomical Journal |accessdate=2014-07-08 }}</ref> "Minor planet 2002 EX12 ... is identified as the parent body of the alpha Capricornid shower, based on a good agreement in the calculated and observed direction and speed of the approaching meteoroids for ejecta 4500-5000 years ago....The bulk of this matter still passes inside Earth's orbit, but will cross Earth's orbit 300 years from now. As a result, the alpha Capricornids are expected to become a major annual shower in 2220-2420 A.D., stronger than any current annual shower"<ref name="Jenniskens"/> The meteor shower was created about 3,500 to 5,000 years ago, when about half of the parent body disintegrated and fell into dust.<ref name="Jenniskens" /> The Alpha Capricornids are expected to become a major annual storm in 2220&ndash;2420 A.D., one that will be "stronger than any current annual shower."<ref name="Jenniskens" /> ==Alpha Monocerotids== Most years, those trails would miss the Earth altogether, but in some years the Earth is showered by meteors. This effect was first demonstrated from observations of the 1995 alpha Monocerotids,<ref>Jenniskens P., 1997. Meteor steram activity IV. Meteor outbursts and the reflex motion of the Sun. Astron. Astrophys. 317, 953–961.</ref><ref name = Jenniskens1997/> The swarm is visible every year from 15 to 25 November; its peak occurs on 21 or 22 November.<ref name = Jenniskens1997/> The speed of its meteors is 65&nbsp;km/s.<ref name = Jenniskens1997/> Normally it has a low Zenithal Hourly Rate (ZHR), but occasionally it produces remarkable meteor storms that last less than an hour: such outbursts were observed in 1925, 1935, 1985, and 1995.<ref name = Jenniskens1997/> The 1995 return was predicted based on the hypothesis that these outbursts were caused by the dust trail of a long period comet occasionally wandering in Earth's path due to planetary perturbations, during observations in southern Spain, assisted by a team of observers of the Dutch Meteor Society, and confirmed the brevity of Alpha Monocerotids outbursts as less than one hour, where the parent body, probably a long-period comet, is unknown.<ref name = Jenniskens1997>{{cite journal|last1=Jenniskens|first1=P.|last2=Betlem|first2=H.|last3=De Lignie|first3=M.|last4=Langbroek|first4=M.|year=1997|bibcode=1997ApJ...479..441J |title= The Detection of a Dust Trail in the Orbit of an Earth-threatening Long-Period Comet |journal=The Astrophysical Journal |volume=479 |pages=441|doi = 10.1086/303853 }}</ref> ==Andromedids== [[Image:Andromedid_meteors,_November_1872.jpg|thumb|right|300px|The Andromedids of 27 November 1872 is a product of the breakup of Biela's Comet several decades previously. Credit: Amedee Guillemin.{{tlx|free media}}]] The Andromedids meteor shower is associated with Biela's Comet, the showers occurring as Earth passes through old streams left by the comet's tail. The comet was observed to have broken up by 1846; further drift of the pieces by 1852 suggested the moment of breakup was in either 1842 or early 1843, when the comet was near Jupiter.<ref name="Space.com">[http://www.space.com/5200-mother-meteor-storms.html The Mother of All Meteor Storms] (Space.com April 2008)</ref><ref name=Jenniskens2007>{{cite journal |last=Jenniskens |first=Peter |author2=Vaubaillon, J. R. M. |title=3D/Biela and the Andromedids: Fragmenting versus Sublimating Comets |journal=Astronomical Journal |volume=134 |issue=3 |pages=1037–1045 |date=2007 |doi=10.1086/519074 |bibcode=2007AJ....134.1037J|url=http://authors.library.caltech.edu/12800/1/JENaj07b.pdf}}</ref> The breakup led to particularly spectacular showers in subsequent cycles (particularly in 1872 and 1885).<ref name="Kronk"/><ref name="Hergenrother">{{cite book |title=The Meteor Storms of November – Part I – The Andromedids |publisher=The Transient Sky Blog |author=Carl W. Hergenrother |url=http://transientsky.wordpress.com/2008/11/15/the-meteor-storms-of-november-part-i-the-andromedids/ |accessdate=2011-04-18}}</ref> Radiant of the Andromedids in December 2013 is near γ&nbsp;Cassiopeiae (near the middle of the ''W'').<ref name="Sky&Tel131208">{{cite book |date=8 December 2013 |title=An Outburst of Andromedid Meteors |publisher=Sky & Telescope |author=Kelly Beatty |url=http://www.skyandtelescope.com/astronomy-news/observing-news/an-outburst-of-andromedid-meteors/ |accessdate=2013-12-09}}</ref> Right ascension = {{RA|01|36}}<ref name="IMO2012">{{cite book |title = Meteor Activity Outlook for November 17-23, 2012 |publisher = International Meteor Organization (IMO) |url = http://www.imo.net/node/1450 |accessdate = 2012-12-11 |archive-url = https://web.archive.org/web/20130622102325/http://imo.net/node/1450 |archive-date = 2013-06-22 |df = }}</ref> and Declination = {{DEC|+37}}<ref name="IMO2012"/> Occurs during September 25 – December 6,<ref name="Kronk"/> date of peak is November&nbsp;9<ref name="IMO2012"/>, Velocity = 19 km/s<ref name="IMO2012"/> and its Zenithal hourly rate = 3<ref name="IMO2012"/>. The first known sighting of the Andromedids was December 6, 1741, over St Petersburg, Russia.<ref name="Hergenrother"/> The 1872 shower consisted mainly of faint (5th to 6th magnitude) meteors with "''broad and smoke-like''" trains and a predominantly orange or reddish colouration.<ref name=ajs153>''American Journal of Science'', Third Series, V (Jan-Jun 1873), 153</ref> The same shower produced at least 58,600 visible meteors between 5.50 and 10.30 pm, observed in England and that the meteors were much slower than the Leonids], with noises "''like very distant gun-shots''" several times to the north-west.<ref name=ajs152>''AJS'', V, 152</ref> In Burma, the 1885 shower was perceived as a fateful omen and was indeed followed swiftly by the collapse of the Konbaung dynasty and the conquest by Britain.<ref>Lonely Planet Myanmar, 10th edition page 256</ref> The November 27, 1885, shower was the occasion of the first known photograph of a meteor, taken by Austro-Hungarian astronomer, Ladislaus Weinek, who caught a 7 mm-long trail on a plate at his Prague observing station.<ref name=SHughes2012>{{cite book |last=Hughes |first=Stefan |title= Catchers of the Light: The Astrophotographers' Family History |date=2012 |url= http://www.catchersofthelight.com |publisher= S. Hughes |isbn= 162050961X |pages=457}}</ref> Since the 19th century the Andromedids have faded so substantially that they are no longer generally visible to the naked eye, though some activity is still observable each year in mid-November given suitable detection equipment.<ref name="Hergenrother"/> In recent years, peak activity had been less than three meteors per hour, around November 9<ref name="IMO2012"/> to 14.<ref name="Kronk"/> Andromedid activity of November comes from the newest streams, while that of early December comes from the oldest.<ref name="Kronk"/> On December 4, 2011, six Canadian radar stations detected 50 meteors in an hour. The activity was likely from the 1649 stream.<ref name=wieg120926>{{cite journal |title=The return of the Andromedids meteor shower |journal=Astronomical Journal |volume=145 |issue=3 |pages=70 |doi=10.1088/0004-6256/145/3/70 |date=26 Sep 2012 |author=Paul A. Wiegert |author2=Peter G. Brown |author3=Robert J. Weryk | author-link3=Robert Weryk|author4=Daniel K. Wong |arxiv = 1209.5980 |bibcode = 2013AJ....145...70W }}</ref> On December 8, 2013, Meteor specialist Peter Brown reported that the Canadian Meteor Orbit Radar had recorded an outburst from the Andromedid meteors in the past 24 hours.<ref name="Sky&Tel131208"/> Scientists postulate a somewhat weaker return in 2018, but a yield of up to 200 meteors an hour in 2023.<ref name=wieg120926/><ref name=kcros121019>{{cite book |last=Croswell |first=Ken |title=The Return of a Great 19th-Century Meteor Shower |url=http://www.scientificamerican.com/article.cfm?id=return-of-a-great-19th-century-meteor-shower |publisher=Scientific American |accessdate=19 October 2012}}</ref> Canadian Meteor Orbit Radar (CMOR) data also detected a spike of 30 meteors per hour on November 27, 2008.<ref name=wieg120926/> During the 2012 shower an inconspicuous maximum occurred on November 9.<ref name="IMO2012"/> {{clear}} ==Arietids== The '''Arietids''' are a strong meteor shower that lasts from May 22 to July 2 each year, and peaks on June 7. The Arietids, along with the Zeta Perseids, are the most intense daylight meteor showers of the year.<ref name="NASA2000">{{cite book |title=June's Invisible Meteors |publisher=[[NASA]] |author=Tony Phillips |date=2000 |url=https://science.nasa.gov/science-news/science-at-nasa/2000/ast06jun_1m |archiveurl=https://web.archive.org/web/20071102152836/https://science.nasa.gov/headlines/y2000/ast06jun_1m.htm |archivedate=November 2, 2007 |accessdate=September 7, 2007 }}</ref> The source of the shower is unknown, but scientists suspect that they come from the asteroid 1566 Icarus,<ref name="NASA2000" /><ref name="spaceweather2007">{{cite book |title=Daylight Meteors: The Arietids |publisher=spaceweather.com |url=http://www.spaceweather.com/meteors/arietids.html |archiveurl= https://web.archive.org/web/20070927225531/http://www.spaceweather.com/meteors/arietids.html |archivedate= 27 September 2007 |accessdate=September 7, 2007}}</ref> although the orbit also corresponds similarly to 96P/Machholz.<ref name=Ohtsuka2003>{{cite journal |title=On the Association among Periodic Comet 96P/Machholz, Arietids, the Marsden Comet Group, and the Kracht Comet Group. |last1=Ohtsuka |first1=Katsuhito |last2=Nakano |first2=Syuichi |last3=Yoshikawa |first3=Makoto |journal=Publications of the Astronomical Society of Japan |volume=55 |issue=1 |pages=321–324 |year=2003 |doi=10.1093/pasj/55.1.321}}</ref> First discovered at Jodrell Bank Observatory in England during the summer of 1947, the showers are caused when the Earth passes through a dense portion of two interplanetary meteoroid streams, producing an average of 60 shooting stars each hour, that originate in the sky from the constellation Aries and the constellation Perseus.<ref name="Kronk"/> However, because both constellations are so close to the Sun when these showers reach their peak, the showers are difficult to view with the naked eye.<ref name="NASA2000" /> Some of the early meteors are visible in the very early hours of the morning, usually an hour before dawn.<ref name="newsgroup">{{cite book |title=Listen...to the Arietids!! |accessdate=September 7, 2007 |publisher=The Astronomy Connection |date=1999 |author=James Turley |url=http://observers.org/tac.mailing.list/2001/june/0109.html |archiveurl=https://web.archive.org/web/20070929191104/http://observers.org/tac.mailing.list/2001/june/0109.html |archivedate=September 29, 2007 |df= }}</ref> The meteors strike Earth's atmosphere at speeds around 39&nbsp;km/s.<ref name="NASA2000" /> By June 22 the radiant has migrated to the constellation Taurus (3h 51m +27) which is the same constellation that the Beta Taurids peak on June 28.<ref name=ams20190622>[https://www.amsmeteors.org/2019/06/meteor-activity-outlook-for-june-22-28-2019 Meteor Activity Outlook for June 22-28, 2019]</ref> ==Aurigids== [[Image:Aurigids - Jeremie Vaubaillon.jpg|thumb|right|250px|The Aurigid meteor shower is observed by a group of astronomers on a NASA mission at 47,000 feet. Credit: Jeremie Vaubaillon, Caltech, NASA.{{tlx|free media}}]] Aurigids is a meteor shower occurring primarily within September.<ref>[http://www.imo.net/calendar/2009#aur © 1997-2011 International Meteor Organization] retrieved 16:55 11.10.11</ref> The comet Kiess (C/1911 N1) is the source of the material that causes the meteors, with an orbital period as approximately 1800 to 2000 years, and showers observed in the years 1935, '86, '94 and 2007 .<ref>[http://adsabs.harvard.edu/abs/2007EOSTr..88..317J Jenniskens, P. and J. Vaubaillon (2007), An unusual meteor shower on 1 September 2007, Eos Trans. AGU, 88(32), 317, ] {{doi|10.1029/2007EO320001}} 16:14 11.10.11</ref><ref> [https://web.archive.org/web/20160304023351/http://www.astro.amu.edu.pl/~jopek/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=206&colecimy=0] 4 March 2016 retrieved 16:25 11.10.11</ref> The Alpha were discovered by C. Hoffmeister and A. Teichgraeber, during the night of 31 August 1935.<ref name=Kronk>Gary W. Kronk website [https://web.archive.org/web/20170915123440/http://meteorshowersonline.com/showers/alpha_aurigids.html] ] 15 September 2017 17:35 11.10.11</ref><ref>[https://archive.is/20120913235941/http://www.skyandtelescope.com/observing/home/9332696.html article written by Joe Rao in Sky and Telescope magazine 23 August 2007] approx. 17:45 retrieved 11.10.11</ref> {{clear}} ==Beta Taurids== The Beta Taurids are normally active from June 5 to July 18.<ref name="Kronk"/> They emanate from an average radiant of right ascension 5h18m, declination +21.2 and exhibit maximum activity around June 28–29 (Solar Longitude=98.3 deg). The sun has a solar longitude (λ⊙) of 90 degress on June 21 (Summer solstice) and as there are 365 days/year moves roughly 1 degree/day. The meteor shower radiant of RA=79.4 degrees converts to 5h 18m as each hour is 15 degrees. The Zenithal Hourly Rate typically reaches about 25 km/s as seen on radar.<ref name="Kronk"/> Non-radio observers are faced with a very difficult prospect, because the Beta Taurid radiant is just 10–15 degrees west of the Sun on June 28.<ref name="meteorobs1999">{{cite book |date = 1999-06-18 |title = IMO-NEWS: 1999 Beta Taurids Alert - Possible Swarm Appearance |publisher = Meteor Observing Mailing List (meteorobs) |url = http://www.meteorobs.org/maillist/msg13477.html |accessdate = 2012-11-26 |archiveurl = https://web.archive.org/web/20080612132836/http://www.meteorobs.org/maillist/msg13477.html |archivedate = 2008-06-12 |df = }}</ref><ref>The International Meteor Organization's radiant of [https://www.imo.net/resources/calendar/meteor-shower-calendar-archive/english2015/#table7 86° +19°] will be 10° from the Sun on June 28. Gary W. Kronk's radiant of [https://web.archive.org/web/20071023040540/http://meteorshowersonline.com/showers/beta_taurids.html 79.4° +21.2°] will be 15° from the Sun on June 28.</ref> Asteroids associated with the β–Taurids include 2201 Oljato, 5143 Heracles, 6063 Jason, (8201) 1994 AH₂ and 1991 BA.<ref name=Babadzhanov2001>{{cite journal |last1=Babadzhanov |first1=P. B. |title=Search for meteor showers associated with Near-Earth Asteroids |journal=Astronomy and Astrophysics |volume=373 |issue=1 |pages=329–335 |date=2001 |doi=10.1051/0004-6361:20010583|bibcode = 2001A&A...373..329B }}</ref> 2019 will be the closest post-perihelion encounter with Earth since 1975. The Taurid swarm is expected to pass {{convert|0.06|AU|abbr=on}} below the ecliptic between June 23 – July 17.<ref name=TSC2019>{{cite journal |last1=Clark |first1=David L. |last2=Weigert |first2=Paul |last3=Brown |first3=Peter G. |title=The 2019 Taurid resonant swarm: prospects for ground detection of small NEOs |journal=Monthly Notices of the Royal Astronomical Society |volume=487 |issue=1 |pages=L35–L39 |year=2019 |doi=10.1093/mnrasl/slz076 |arxiv=1905.01260 }}</ref> During 2019 astronomers hope to search for hypothesized asteroids ~100 meters in diameter from the Taurid swarm between July 5–11, and July 21 – August 10.<ref name="Plait">{{cite book |title=Could larger space rocks be hiding in the Beta Taurid Meteor stream? We may find out this summer |publisher=Bad Astronomy |author=Phil Plait |url=https://www.syfy.com/syfywire/could-larger-space-rocks-be-hiding-in-the-beta-taurid-meteor-stream-we-may-find-out-this |accessdate=2019-05-14}}</ref> There is circumstantial evidence that the daytime June 30 Tunguska event came from the same direction in the sky as the Beta Taurids.<ref name="Plait"/> The next June close approach to the Taurid swarm is expected in 2036.<ref name="Asher">{{cite book |title=Taurid swarm years |publisher=University of Cambridge |author=David J. Asher |url=http://www.cantab.net/users/davidasher/taurid/swarmyears.html |accessdate=2019-06-16}}</ref> ==Draconids== [[Image:Draconids outburst 2012-10-08.png|thumb|right|250px|Composite image of Draconid meteors is taken from Petnica Meteor Group video camera stationed at Institute of Physics, Belgrade. Credit: Petnica Meteor Group.{{tlx|free media}}]] The October Draconids, in the past also unofficially known as the Giacobinids, are a meteor shower whose parent body is the periodic comet 21P/Giacobini-Zinner.<ref name="Kronk" /> They are named after the constellation Draco, where they seemingly come from. Almost all meteors which fall towards Earth ablate long before reaching its surface. The Draconids are best viewed after sunset in an area with a clear dark sky. RA {{RA|17.467}}<ref name="Kronk"/> and Declination = {{DEC|+54}}<ref name="Kronk"/>. Velocity = 20 km/s.<ref name=jimo37_5_161>{{citation | display-authors=1 | last1=Żołądek | first1=P. | last2=Wiśniewski | first2=M. | last3=Olech | first3=A. | last4=Krasnowski | first4=M. | last5=Kwinta | first5=M. | last6=Fietkiewicz | first6=K. | last7=Złoczewski | first7=K. | last8=Mularczyk | first8=K. | last9=Jonderko | first9=W. | last10=Spaleniak | first10=I. | last11=Gruszka | first11=Ł. | title=The 2004 Perseid meteor shower – Polish Fireball Network double station preliminary results | journal=Journal of the International Meteor Organization | volume=37 | issue=5 | pages=161–163 |date=October 2009 | bibcode=2009JIMO...37..161Z}}</ref> The 1933<ref name="KronkD">{{cite book|url=http://meteorshowersonline.com/showers/draconids.html|title=Draconids ("Giacobinids")|last=Kronk|first=Gary W |date=|website=Meteor Showers Online|archive-url=https://web.archive.org/web/20180627202229/http://meteorshowersonline.com/showers/draconids.html|archive-date=2018-06-27|accessdate=11 October 2017}}</ref><ref name="Wyle_1933">"The meteors from Giacobini's comet", Wylie, C. C., Popular Astronomy, Vol. 42, p.44, {{cite book|title=The meteors from Giacobini's comet|url=http://adsbit.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1934PA.....42...44W&db_key=AST&page_ind=0&data_type=GIF&type=SCREEN_VIEW&classic=YES|language=en|accessdate=2018-09-25}}</ref><ref name="storm1933">{{cite book |date=October 7, 2011 |title=Account of the 1933 Draconids meteor storm |publisher=International Meteor Organization (IMO) |author=John McFarland and Mark Bailey |url=http://www.imo.net/docs/ellison_1933_draconids_storm_account.txt |accessdate=2011-10-08}}</ref> and 1946<ref name="KronkD" /> Draconids had Zenithal Hourly Rates of thousands of meteors visible per hour, among the most impressive meteor storms of the 20th century. Rare outbursts in activity can occur when the Earth travels through a denser part of the cometary debris stream; for example, in 1998, rates suddenly spiked<ref>{{cite book|title=Giacobinids dazzle observers|url=https://science.nasa.gov/newhome/headlines/ast14oct98_2.htm|date=October 14, 1998}}</ref><ref>Arlt, R. [http://adsabs.harvard.edu/abs/1998JIMO...26..256A "Summary of 1998 Draconid Outburst Observations"], ''WGN, Journal of the International Meteor Organization, Vol. 26, p. 256-259'', 1998.</ref> and spiked again (less spectacularly) in 2005.<ref>Campbell-Brown, M.; Vaubaillon, J.; Brown, P.; Weryk, R. J.; Arlt, R. [http://adsabs.harvard.edu/abs/2006A&A...451..339C "The 2005 Draconid outburst"], ''Astronomy and Astrophysics, Volume 451, pp.&nbsp;339–344'', 2006.</ref> A Draconid meteor outburst occurred<ref name="2011results">{{cite book|title=Draconids show expected outburst|publisher=International Meteor Organization (IMO)|url=http://www.imo.net/draconids2011results|accessdate=2011-12-06|archiveurl=https://web.archive.org/web/20111213230937/http://www.imo.net/draconids2011results|archivedate=2011-12-13|df=}}</ref> as expected<ref name=West>{{cite news |author= Adrian West |title= The Draconid Meteor Shower – A Storm is Coming! |work= Universe Today |date= October 3, 2011 |url= http://www.universetoday.com/89430/the-draconid-meteor-shower-a-storm-is-coming/ |accessdate= 2011-10-03}}</ref><ref>{{cite book|title=Draconids Meteor Shower on 8 October 2011|url=http://www.imo.net/draconids2011|work=International Meteor Organization|archiveurl=https://web.archive.org/web/20110925171621/http://www.imo.net/draconids2011|archivedate=25 September 2011|df=}}</ref><ref>{{cite book|last=Beatty|first=Kelly|title=A Deluge of Draconids?|url=http://www.skyandtelescope.com/observing/highlights/104450349.html|work=Sky and Telescope|publisher=Highlights|accessdate=31 December 2010}}</ref> on 2011 October 8, though a waxing gibbous Moon reduced the number of meteors observed visually. "Observers in the UK and Northern Europe are ideally placed to see the peak of the Draconids. Unfortunately the peak occurs in the day time for North America. There will also be a bright Moon which may drown out many but the brightest meteors, but if predictions are correct, you will still see many. You may see Draconid meteors on the 7th an the 9th also, so it is worth going out and checking the skies."<ref name=West/> During the 2012 shower radar observations detected up to 1000 meteors per hour. The 2012 outburst may have been caused by the narrow trail of dust and debris left behind by the parent comet in 1959.<ref name="Barentsen">{{cite book |date=2012-10-08 |title=Draconids show outburst (again!) |author=Geert Barentsen |url=http://barentsen.wordpress.com/2012/10/08/draconids-2012/ |accessdate=2012-10-08}}</ref> {{clear}} ==Eta Aquariids== [[Image:Animation of 1P／Halley orbit - 1986 apparition.gif|260px|thumb|right|Animation is of 1P/Halley orbit - 1986 apparition.{{legend2|magenta| 1P/Halley }}{{legend2|Royalblue|Earth }}{{legend2|Yellow|Sun}}. Credit: [[c:user:Phoenix7777|Phoenix7777]].{{tlx|free media}}]] The current orbit of Halley's Comet does not pass close enough to the Earth to be a source of meteoric activity.<ref name="AMS2013">{{cite book |title=Viewing the 2013 Eta Aquariid Meteor Shower |publisher=American Meteor Society |author=Robert Lunsford |url=http://www.amsmeteors.org/2013/05/viewing-the-2013-eta-aquariid-meteor-shower/ |accessdate=2013-05-04}}</ref> The shower is best viewed from the equator to 30 degrees south latitude.<ref name="AMS2013"/> The meteoroids are from very old ejection from the parent 1P/Halley and are trapped probably in resonances to Jupiter's orbit (similar to the Orionids observed between 2007 and 2010).<ref name="Calendar">{{cite book |title=IMO Meteor Shower Calendar |publisher=International Meteor Organization |author=Jürgen Rendtel |url=https://www.imo.net/resources/calendar/ |accessdate=2018-05-06}}</ref> The peak ZHR reached 135 ± 16.<ref name="Rendtel">{{citation |title=Meteor Shower Workbook, p. 23-24 |publisher=International Meteor Organization |author=Jürgen Rendtel}}</ref> Updated information on the expected time and rates of the shower is provided through the annual IMO Meteor Shower Calendar.<ref name="Calendar">{{cite book |title=IMO Meteor Shower Calendar |publisher=International Meteor Organization |author=Jürgen Rendtel |url=https://www.imo.net/resources/calendar/ |accessdate=2018-05-06}}</ref> {{clear}} ==Geminids== [[Image:Geminid 121407 1.jpg|thumb|right|250px|A Geminid meteor in 2007, seen from San Francisco. Credit: [https://sites.google.com/site/thebrockeninglory/ Brocken Inaglory].{{tlx|free media}}]] The '''Geminids''' are a prolific meteor shower caused by the object 3200 Phaethon,<ref name=IAUC3881>{{cite book |author=Brian G. Marsden |title=IAUC 3881: 1983 TB and the Geminid Meteors; 1983 SA; KR Aur (Circular No. 3881) |url=http://www.cbat.eps.harvard.edu/iauc/03800/03881.html |publisher=Central Bureau for Astronomical Telegrams |date=1983-10-25 |accessdate=2009-05-18 |archiveurl=https://web.archive.org/web/20120501023951/http://www.cbat.eps.harvard.edu/iauc/03800/03881.html |archivedate=2012-05-01 |df= }}</ref> which is thought to be a Palladian asteroid<ref name="NG2010">{{cite book | author = Victoria Jaggard | title = Exploding Clays Drive Geminids Sky Show? | url = http://newswatch.nationalgeographic.com/2010/10/12/geminids_meteor_shower_driven_by_exploding_clays/ | publisher = National Geographic Society | date = 2010-10-12 | accessdate = 2010-10-18}}</ref> with a "rock&nbsp;comet" orbit.<ref name=Jewitt2010>{{cite journal | last = Jewitt | first = David | authorlink = | author2 = Li, Jing | title = Activity in Geminid Parent (3200) Phaethon | journal = The Astronomical Journal | volume = 140 | issue = 5 | pages = 1519–1527 | arxiv = 1009.2710 | date = 2010 | version = | accessdate = | doi = 10.1088/0004-6256/140/5/1519 | bibcode = 2010AJ....140.1519J }}</ref> {{clear}} ==June Bootids== "The June Bootid meteor shower is active each year from June 26th until July 2nd. It peaks on June 27th. Normally the shower is very weak, but occasional outbursts produce a hundred or more meteors per hour."<ref name=Space>{{ cite book |author=SpaceWeather |title=The Unpredictable June Bootids |publisher=SpaceWeather |location= |date=24 June 2005 |url=http://spaceweather.com/meteors/junebootids.html |accessdate=27 June 2019 }}</ref> "The shower's radiant lies in the constellation Bootes (right ascension 14h 56m, declination 48°)."<ref name=Space/> "The source of the June Bootids is periodic comet 7P/Pons-Winnecke."<ref name=Space/> "June Bootid meteoroids hit Earth's atmosphere with a velocity of 18 km/s (40,000 mph).They are considered slow-moving meteors."<ref name=Space/> "On June 27th, 1998, northern sky watchers were surprised when meteors suddenly began to stream out of the constellation Bootes. Observers saw as many as 100 meteors per hour during the 7-hour-long outburst. It wasn't the first time: similar outbursts from Bootes had been recorded in 1916, 1921 and 1927. Astronomers call these unpredictable meteors the June Bootids."<ref name=Space/> ==Kappa Cygnids== Kappa Cygnids, abbreviated KCG, is a minor meteor shower that takes place in August along with the larger Perseids meteor shower.<ref>{{cite book|url=http://www.imo.net/calendar/2009#kcg |title=IMO Meteor Shower Calendar 2009: Contents: July to September: Kappa Cygnids |publisher=IMO.net}}</ref> The Kappa Cygnids in 2009 were Active between August 3-August 25 August, with Peak of shower at August 17, and ZHR = 3 km/s.<ref>{{cite book|url=http://www.imo.net/calendar/2009#kcg|title=IMO Meteor Shower Calendar 2009: Contents: July to September: Alpha Capricornids |work=IMO.net |accessdate=2009-02-10}}</ref> ==Leonids== [[Image:Leonid Meteor.jpg|thumb|right|250px|The photograph shows the meteor, afterglow, and wake as distinct components of a meteor during the peak of the 2009 Leonid Meteor Shower. Credit: [[c:User:Navicore|Navicore]].{{tlx|free media}}]] [[Image:Leonid meteor shower as seen from space (1997).jpg|thumb|left|250px|This photograph shows the Leonids as many begin contacting the Earth's atmosphere. Credit: NASA.{{tlx|free media}}]] "The Leonid meteor shower peaked early Saturday (Nov. 17 [2012]), and some night sky watchers caught a great view. The Leonids are a yearly meteor display of shooting stars that appear to radiate out of the constellation Leo. They are created when Earth crosses the path of debris from the comet Tempel-Tuttle, which swings through the inner solar system every 33 years."<ref name=Moskowitz>{{ cite book |author=Clara Moskowitz |title=Amazing Leonid Meteor Shower Photos Captured By Stargazers |publisher=SPACE.com |location= |date=November 17, 2012 |url=http://news.yahoo.com/amazing-leonid-meteor-shower-photos-captured-stargazers-163450853.html |accessdate=2012-11-18 }}</ref> {{clear}} ==Lyrids== [[Image:02-Lyrid Meteor Shower 2022-nX-4.jpg|thumb|right|250px|Lyrid Meteor Shower 2022 is shown. Credit: [[c:user:PsamatheM|PsamatheM]].{{tlx|free media}}]] The April Lyrids (LYR, IAU shower number 6)<ref name="IAU Meteor Shower Working List">{{cite book|url=http://www.astro.amu.edu.pl/~jopek/MDC2007/index.php |title=IAU Meteor Data Center|page=1|accessdate=2013-08-06}}</ref> is a meteor shower lasting from April 16 to April 26<ref name="meteonline">{{cite book|url=http://meteorshowersonline.com/lyrids.html |title=Lyrids|publisher=Meteor Showers Online|page=1|accessdate=2008-08-05}}</ref> The source of the meteor shower is particles of dust shed by the long-period Comet C/1861 G1 Thatcher.<ref name=Jenniskens2006>{{cite journal | last=Jenniskens | first = P. | title=Meteor Showers and their Parent Comets | publisher=Cambridge University Press | date=2006 | pages=790}}</ref> The Lyrids have been observed and reported since 687 BC; no other modern shower has been recorded as far back in time.<ref>{{cite book|last1=King|first1=Bob|title=The Lyrid Shower Kicks Off Year of Great Meteor Watching|url=http://www.skyandtelescope.com/observing/lyrid-shower-kicks-off-year-of-great-meteor-watching/|website=www.skyandtelescope.com|publisher=F+W Media, Inc.|accessdate=19 April 2018|date=18 April 2018}}</ref> The shower usually peaks on around April 22 and the morning of April 23. Counts typically range from 5 to 20 meteors per hour, averaging around 10.<ref name="meteonline"/> April Lyrid meteors are usually around magnitude +2. However, some meteors can be brighter, known as "Lyrid fireballs", cast shadows for a split second and leave behind smokey debris trails that last minutes.<ref name="spaceweather">{{cite book|url=http://www.spaceweather.com/meteors/lyrids/lyrids.html|title=the Lyrid meteor shower|date=2008|publisher=spaceweather.com|page=1|accessdate=2008-08-05}}</ref> Occasionally, the shower intensifies when the planets steer the one-revolution dust trail of the comet into Earth's path, an event that happens about once every 60 years.<ref name=Jenniskens2006/> The one-revolution dust trail is dust that has completed one orbit: the stream of dust released in the return of the comet prior to the current 1862 return. This mechanism replaces earlier ideas that the outbursts were due to a cloud of dust moving in a 60-year orbit.<ref name=Arter1997>{{cite journal | last=Arter | first=T. R. |author2=Williams, I. P. | title=The mean orbit of the April Lyrids | journal=Monthly Notices of the Royal Astronomical Society | date=1997 | volume=289 | issue=3 | pages=721–728 | bibcode=1997MNRAS.289..721A | doi=10.1093/mnras/289.3.721}}</ref> In 1982, amateur astronomers counted 90 April Lyrids per hour at the peak and similar rates were seen in 1922. A stronger storm of up to 700 per hour occurred in 1803,<ref name=martinez1994>{{citation | first1=Patrick | last1=Martinez | title=The Observer's Guide to Astronomy | volume=2 | series=Practical Astronomy Handbooks | others=Translated by Storm Dunlop | publisher=Cambridge University Press | date=1994 | isbn=0521458986 | page=645 | url=https://books.google.com/books?id=TXc54LfKsSQC&pg=PA645 | postscript=. }}</ref> observed by a journalist in Richmond, Virginia: "Shooting stars. This electrical [sic] phenomenon was observed on Wednesday morning last at Richmond and its vicinity, in a manner that alarmed many, and astonished every person that beheld it. From one until three in the morning, those starry meteors seemed to fall from every point in the heavens, in such numbers as to resemble a shower of sky rockets&nbsp;...<ref name="spaceweather"/>"<ref name=martinez1994/> The oldest known outburst, the shower on March 23.7,<ref>M. Ed. Biot, 1841, Gatalogue General des Etoiles Filantes et des Autres Meteores Observes en Chine pendent 24 Siecles, Paris, Imprimerie Royale; P. Jenniskens, 2006, Meteor Showers and their Parent Comets, Cambridge University Press, 790 pp.</ref> 687 BC (proleptic Julian calendar) was recorded in ''Zuo Zhuan'', which describes the shower as "On the 4th month in the summer in the year of Sexagenary cycle (xīn-mǎo) (of year 7 of King Zhuang of the State of Lu), at night, (the sky is so bright that some) fixed stars become invisible (because of the meteor shower); at midnight, stars fell like rain."<ref>{{cite book|url=http://www.skyandtelescope.com/observing/objects/meteors/3305866.html|title=Meteors&nbsp;– April's Lyrid Meteor Shower|last=Sinnott|first=Roger W.|date=2008|publisher=Sky and Telescope|page=1|accessdate=2008-08-05}}</ref> In the Australian Aboriginal astronomy of the Boorong tribe, the Lyrids represent the scratchings of the Mallee fowl (represented by Vega), coinciding with its nest-building season.<ref>{{cite book|last=Hill|first=Tanya|title=The Lyrids meteor shower should put on a show overnight|url=http://theconversation.com/the-lyrids-meteor-shower-should-put-on-a-show-overnight-25746|work=The Conversation|accessdate=22 April 2014|author2=Brown, Michael J. I. |date=22 April 2014}}</ref> {{clear}} ==Northern Taurids== [[Image:Tauride-20201204.jpg|thumb|right|250px|Northern Taurid was photographed from Skibotn, Norway, 4 December 2020 14:30 CET. Credit: [[c:user:Medisilvanus|Medisilvanus]].{{tlx|free media}}]] Parent body = 2004 TG₁₀<ref name=JenniskensTG10>Meteor showers and their parent comets [https://books.google.com/books?id=QpajMuyXG8AC&pg=PA470&dq=TG10 pg 470] by Peter Jenniskens</ref><ref name=moore_rees2011>{{citation | first1=Patrick | last1=Moore | first2=Robin | last2=Rees | title=Patrick Moore's Data Book of Astronomy | edition=2nd | publisher=Cambridge University Press | date=2011 | isbn=0-521-89935-4 | page=275 | url=https://books.google.com/books?id=2FNfjWKBZx8C&pg=PA275 }}</ref> Radiant point = RA {{RA|03|52}} Dec = {{DEC|+22}}.<ref name="IMO2015">{{cite book |title=IMO Meteor Shower Calendar 2015 (Working list of visual meteor showers) |publisher=International Meteor Organization |url=https://www.imo.net/resources/calendar/meteor-shower-calendar-archive/english2015/#table5 |accessdate=2019-06-20}}</ref> Occurs during October 20 – December 10, with a peak at 12 November.<ref name=IMO2015/> Velocity = 29 km/s.<ref name=IMO2015/> Zenithal hourly rate is 5.<ref name=IMO2015/> The Northern Taurids originated from the asteroid 2004 TG₁₀.<ref name="Taurids 2017">{{cite book|url=https://patch.com/us/across-america/taurids-meteor-shower-fireballs-peak-dates-what-expect|title=Taurids Meteor Shower Fireballs: Peak Dates, What To Expect|author=Beth Dalbey|date=24 October 2017|accessdate=11 November 2017}}</ref> The Taurids are also made up of weightier material, pebbles instead of dust grains.<ref name="NASA2005">{{cite book |date=2005-11-03 |title=Earth is orbiting through a swarm of space debris that may be producing an unusual number of nighttime fireballs |publisher=NASA Science News |author=Dr. Tony Phillips |url=https://science.nasa.gov/science-news/science-at-nasa/2005/03nov_taurids/}}</ref> Typically, Taurids appear at a rate of about 5 per hour, moving slowly across the sky at about 28 km/s (17 mi/s), or 100,800 km/h (65,000 mph).<ref name="NASA2005"/> If larger than a pebble, these meteors may become bolides as bright as the moon and leave behind smoke trails.<ref name="NASA2005"/> The Beta Taurids could be the cause of the Tunguska event of June 30, 1908.<ref name=NG>[http://news.nationalgeographic.com/news/2003/11/1107_031107_taurids.html Meteor Shower Promises Seven Shooting Stars an Hour] – National Geographic News (November 7, 2003)</ref> In 1962 and 1963, the Mars 1 probe recorded one micrometeorite strike every two minutes at altitudes ranging from {{convert|6000|to|40000|km|mi|abbr=on}} from Earth's surface due to the Taurids meteor shower, and also recorded similar densities at distances from {{convert|20|to|40|e6km|mi}} from Earth.<ref name='Sputnik 23'>{{cite book | url = http://jtgnew.sjrdesign.net/exploration_space_planetary_mars.html | title = "Journey Through the Galaxy" Mars Program: Mars ~ 1960-1974 | accessdate = 2014-01-26 | last = Robbins | first = Stuart | date = 2008 | publisher = SJR Design}}</ref><ref name='Mihos 2006'>{{cite book | url = http://burro.astr.cwru.edu/stu/advanced/20th_soviet_mars.html | title = Mars (1960-1974): Mars 1 | accessdate = 2014-01-26 | last = Mihos | first = Chris | date = 11 January 2006 | work = Department of Astronomy, Case Western Reserve University. | archiveurl = https://web.archive.org/web/20131013211415/http://burro.astr.cwru.edu/stu/advanced/20th_soviet_mars.html | archivedate = 13 October 2013 | df = }}</ref> The Taurid stream has a cycle of activity that peaks roughly every 2,500 to 3,000 years,<ref name=NG/> when the core of the stream passes nearer to Earth and produces more intense showers. In fact, because of the separate "branches" (night-time in one part of the year and daytime in another; and Northern/Southern in each case) there are two (possibly overlapping) peaks separated by a few centuries, every 3000 years. The next peak is expected around 3000 AD.<ref name=NG/> Over Poland in 1995, all-sky cameras imaged an absolute magnitude –17 Taurid bolide that was estimated to be 900 kg and perhaps a meter in diameter.<ref name=Jenniskens-pg467>Meteor showers and their parent comets [https://books.google.com/books?id=QpajMuyXG8AC&pg=PA467&dq=meter pg 467] by Peter Jenniskens</ref> In 1993, it was predicted that there would be a swarm of activity in 2005.<ref name="NASA2005"/> Around Halloween in 2005, many fireballs were witnessed that affected people's night vision.<ref name="NASA2005"/> Astronomers have taken to calling these the "Halloween fireballs."<ref name="NASA2005"/> The Tunguska event may have been caused by a Beta Taurid.<ref name="TunguskaTaurid">{{cite news |author=Joel Achenbach |date=2018-12-25 |title=Incoming! A June meteor swarm could be loaded with surprises. |url=https://www.washingtonpost.com/science/2018/12/25/incoming-june-meteor-swarm-could-be-loaded-with-surprises/?utm_term=.3d66234e3b12 |website=www.washingtonpost.com |location=Washington DC, US |accessdate=2019-05-04}}</ref> A brief flash of light from a lunar impact event was recorded on November 7, 2005, while testing a new 250 mm (10 in) telescope and video camera built to monitor the Moon for meteor strikes.<ref>[http://news.bbc.co.uk/1/hi/sci/tech/4570730.stm BBC News: Nasa team sees explosion on Moon] (3 January 2006)</ref> This may be the first photographic record of such a strike.<ref>[http://neo.jpl.nasa.gov/news/news118.html An Eyewitness Impact Debunked]{{Dead link|date=June 2019}}</ref> {{clear}} ==Orionids== [[Image:Orionids Meteor shower.jpg|thumb|right|250px|Orionids Meteor shower is captured. Credit: [[c:user:Wacool|Wacool]].{{tlx|free media}}]] "The Orionid meteor shower [leftover bits of Halley's Comet] is scheduled to reach its maximum before sunrise on Sunday morning (Oct. 21 [2012]). This will be an excellent year to look for the Orionids, since the [[moon]] will set around 11 p.m. local time on Saturday night (Oct. 20) and will not be a hindrance at all ... The orbit of Halley's Comet closely approaches the Earth's orbit at two places. One point is in the early part of May producing a meteor display known as the Eta Aquarids. The other point comes in the middle to latter part of October, producing the Orionids."<ref name=Rao>{{ cite book |author=Joe Rao |title=Orionid Meteor Shower Spawned by Halley's Comet Peaks This Weekend |publisher=SPACE.com |location= |date=October 19, 2012 |url=http://news.yahoo.com/orionid-meteor-shower-spawned-halleys-comet-peaks-weekend-160214151.html |accessdate=2012-10-19 }}</ref> {{clear}} ==Perseid meteor showers== [[Image:PSM V18 D201 Shower of perseids sept 6 and 7.jpg|right|thumb|300px|Perseid meteor shower is from September 6 and 7, 1880-81. Credit: unknown.{{tlx|free media}}]] [[Image:Perseid meteor 2007.jpg|thumb|left|300px|A Perseid shower occurs in 2007. Credit: [https://sites.google.com/site/thebrockeninglory/ Brocken Inaglory].{{tlx|free media}}]] [[Image:Animation of 109P／Swift–Tuttle orbit.gif|right|thumb|300px|Animation of 109P/Swift–Tuttle orbit from 1875 to 2100.<br>{{legend2| Yellow| Sun}}{{·}}{{legend2|RoyalBlue|Earth}}{{·}}{{legend2| Lime| Jupiter }}{{·}}{{legend2|OrangeRed|Saturn}}{{·}}{{legend2|Gold|Uranus}}{{·}}{{legend2|Magenta|109P/Swift–Tuttle}}. Credit: [[c:User:Phoenix7777|Phoenix7777]].{{tlx|free media}}]] [[Image:Perseidak 2006.jpg|left|thumb|300px|Radiant point is from August 8, 2006. Credit: Olga Berrios.{{tlx|free media}}]] In 1835, Adolphe Quetelet identified the shower as emanating from the constellation Perseus.<ref name="NASA Chat">{{cite book|url=http://www.nasa.gov/pdf/677215main_Perseids2012.pdf |title=NASA Chat: Stay ‘Up All Night’ to Watch the Perseids! |publisher=NASA |author= Dr. Bill Cooke |author2= Danielle Moser |author3= Rhiannon Blaauw | page=55|date=2012-08-11|accessdate=2013-08-16}}</ref><ref name="Kronk"/> Right ascension = {{RA|03|04}}<ref name=moore_rees2011/> and Declination = {{DEC|+58}}<ref name=moore_rees2011/> The Perseid meteor shower, usually the richest meteor shower of the year, peaks in August. Over the course of an hour, a person watching a clear sky from a dark location might see as many as 50-100 meteors. Parent body is Comet Swift–Tuttle.<ref name=moore_rees2011/> The first record is from 36 CE.<ref name="NASA Chat"/><ref name="Kronk"/> The radiant point image on the right is from September 6 and 7, 1880-81.<ref>[https://archive.org/details/popularsciencemo18newy ''Popular Science Monthly'', Volume 18]</ref> Velocity = 58 km/s<ref name=jimo37_5_161>{{citation | display-authors=1 | last1=Żołądek | first1=P. | last2=Wiśniewski | first2=M. | last3=Olech | first3=A. | last4=Krasnowski | first4=M. | last5=Kwinta | first5=M. | last6=Fietkiewicz | first6=K. | last7=Złoczewski | first7=K. | last8=Mularczyk | first8=K. | last9=Jonderko | first9=W. | last10=Spaleniak | first10=I. | last11=Gruszka | first11=Ł. | title=The 2004 Perseid meteor shower – Polish Fireball Network double station preliminary results | journal=Journal of the International Meteor Organization | volume=37 | issue=5 | pages=161–163 |date=October 2009 | bibcode=2009JIMO...37..161Z}}</ref> and Zenithal hourly rate = 100<ref name=moore_rees2011/>. The stream of debris is called the Perseid cloud and stretches along the orbit of the comet Swift–Tuttle. The cloud consists of particles ejected by the comet as it travels on its 133-year orbit.<ref>{{cite book|author=Dan Vergano|url=https://www.usatoday.com/story/tech/sciencefair/2013/08/08/perseids-2013-meteor/2628521/ |title=Perseid meteor shower to light up night sky this weekend |publisher=Usatoday.com |date=2010-08-07 |accessdate=2013-08-12}}</ref> Most of the particles have been part of the cloud for around a thousand years. However, there is also a relatively young filament of dust in the stream that was pulled off the comet in 1865, which can give an early mini-peak the day before the maximum shower.<ref>{{cite book |title = The 2004 Perseid Meteor Shower |date = June 25, 2004 |author = Dr. Tony Phillips |publisher = Science@NASA |accessdate = 2010-03-12 |url = https://science.nasa.gov/headlines/y2004/25jun_perseids2004.htm |archiveurl = https://web.archive.org/web/20100320071509/http://science.nasa.gov/headlines/y2004/25jun_perseids2004.htm |archivedate = March 20, 2010 |df = }}</ref> The dimensions of the cloud in the vicinity of the Earth are estimated to be approximately 0.1 AU across and 0.8 AU along the latter's orbit, spread out by annual interactions with the Earth's gravity.<ref>{{Cite book | title = The Cosmic Dust Connection | editor = J. Mayo Greenberg | date = 1996 | publisher = Springer Science & Business Media | article = Cometary Dust Loss: Meteoroid Streams and the Inner Solar System Dust Cloud | author= D.W. Hughes | page = 375 | url = https://books.google.com/books?id=5MH0CAAAQBAJ&pg=PA375 }}</ref> The shower is visible from mid-July each year, with the peak in activity between 9 and 14 August, depending on the particular location of the stream. During the peak, the rate of meteors reaches 60 or more per hour. They can be seen all across the sky; however, because of the shower's radiant in the constellation of Perseus, the Perseids are primarily visible in the Northern Hemisphere.<ref>{{cite book|url=https://www.timeanddate.com/astronomy/meteor-shower/perseid.html|publisher=timeanddate.com|title=Perseids Meteor Shower 2018 |accessdate=2018-07-30}}</ref> As with many meteor showers the visible rate is greatest in the pre-dawn hours, since more meteoroids are scooped up by the side of the Earth moving forward into the stream, corresponding to local times between midnight and noon, as can be seen in the accompanying diagram.<ref>http://meteorshowersonline.com/what_is.html</ref> While many meteors arrive between dawn and noon, they are usually not visible due to daylight. Some can also be seen before midnight, often grazing the Earth's atmosphere to produce long bright trails and sometimes fireballs. Most Perseids burn up in the atmosphere while at heights above {{convert|80|km|mi}}.<ref name="Meteorwatch">{{cite book |date=2012-08-11 |title=NASA All Sky Fireball Network: Perseid End Height |publisher=NASA Meteor Watch on Facebook |url=https://www.facebook.com/photo.php?fbid=402054346520464 |accessdate=2012-11-19}}</ref> {{clear}} ==Phoenicids== The Phoenicids get their name from the location of their radiant, which is in the constellation Phoenix, active from 29 November to 9 December, with a peak occurring around 5/6 December each year,<ref>http://meteorshowersonline.com/showers/phoenicids.html</ref> and are best seen from the Southern Hemisphere. The Phoenicids appear to be associated with a stream of material from the disintegrating comet D/1819 W1 (Blanpain).<ref>P. Jenniskens and E. Lyytinen, METEOR SHOWERS FROM THE DEBRIS OF BROKEN COMETS: D/1819 W1 (BLANPAIN), 2003 WY25, AND THE PHOENICIDS.The Astronomical Journal, 130:1286–90, 2005 September</ref> A very minor meteor shower with a radiant in Phoenix also occurs in July; this shower is referred to as the '''July Phoenicids'''.<ref>http://meteorshowersonline.com/showers/july_phoenicids.html</ref> ==Pi Puppids== "The Pi Puppids are a meteor shower associated with the comet 26P/Grigg-Skjellerup."<ref name=PiPuppids>{{ cite book |author=[[w:User:Wikibob|Wikibob]] |title=Pi Puppids |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=26 June 2004 |url=https://en.wikipedia.org/wiki/Pi_Puppids |accessdate=28 June 2019 }}</ref> "The Pi Puppids get their name because their radiant appears to lie in the constellation Puppis, at around Right ascension 112 degrees and Declination -45 degrees."<ref name=PiPuppids/> ==Quadrantids== The Quadrantids ('''QUA''') are a January meteor shower, with the zenithal hourly rate (ZHR) of this shower as high as that of two other reliably rich meteor showers, the Perseids in August and the Geminids in December.<ref name="amsmeteors.org">{{ cite book |url=http://www.amsmeteors.org/meteor-showers/meteor-faq/#7 |title=Does the published meteor rate for a shower really represent what I should expect to see? |publisher=American Meteor Society |accessdate=2012-12-29 }}</ref> The meteor rates exceed one-half of their highest value for only about eight hours (compared to two days for the August Perseids), which means that the stream of particles that produces this shower is narrow, and apparently deriving within the last 500 years from some orbiting body.<ref name="space.com">{{ cite book |url=http://www.space.com/spacewatch/081219-ns-meteor-shower.html |title=Stellar Meteor Shower Jan. 3 |publisher=Space.com |accessdate=2009-01-03 }}</ref> The parent body of the Quadrantids was tentatively identified in 2003<ref name="sjaa">{{ cite book |date=Dec 8, 2003 |title=2003 EH1 is the Quadrantid shower parent comet |publisher=The Ephemeris (San Jose Astronomical Association newsletter) |author=Peter Jenniskens |url=http://ephemeris.sjaa.net/0401/e.html |accessdate=2004-12-17}}</ref> as the minor planet (196256) 2003 EH1, which in turn may be related to the comet C/1490 Y1<ref name=Jenniskens2004>{{ cite journal |last=Jenniskens |first=Peter |title=2003 EH1 Is the Quadrantid Shower Parent Comet |journal=The Astronomical Journal |volume=127 |issue=5 |pages=3018–3022 |date=2004 |doi=10.1086/383213 |bibcode=2004AJ....127.3018J}}</ref> that was observed by Chinese, Japanese and Korean astronomers some 500 years ago. ==Southern Delta Aquariids== Meteors radiating from near the star Delta Aquarii (declension "-i") are called the Delta Aquariids. ==Southern Taurids== During the Southern Taurid meteor shower in 2013, fireball sightings were spotted over southern California, Arizona, Nevada, and Utah.<ref name="BSkyB">{{cite news |author=Sky News US Team |date=2013-11-07 |title=Meteor 'Fireball' Lights Up California Sky |url=http://news.sky.com/story/1165152/meteor-fireball-lights-up-california-sky |website=news.sky.com |location=London, UK |publisher=BSkyB |accessdate=2013-11-07}}</ref> The Southern Taurids originated from Comet Encke, while the Northern Taurids originated from the asteroid 2004 TG₁₀.<ref name=Dalbey>{{cite book|url=https://patch.com/us/across-america/taurids-meteor-shower-fireballs-peak-dates-what-expect|title=Taurids Meteor Shower Fireballs: Peak Dates, What To Expect|author=Beth Dalbey|date=24 October 2017|accessdate=11 November 2017}}</ref> Encke and the Taurids are believed to be remnants of a much larger comet, which has disintegrated over the past 20,000 to 30,000 years.<ref name=Babadzhanov2008>{{cite journal |last1=Babadzhanov |first1=P. B. |last2=Williams |first2=I. P. |last3=Kokhirova |first3=G. I. |title=Near-Earth Objects in the Taurid complex |journal=Monthly Notices of the Royal Astronomical Society |volume=386 |issue=3 |pages=1436–1442 |year=2008 |doi=10.1111/j.1365-2966.2008.13096.x|bibcode = 2008MNRAS.386.1436B }}</ref> ==Ursids== The Ursids were probably discovered by William F. Denning who observed them for several years around the start of the 20th century.<ref name="KronkU">{{cite book |title=Observing the Ursids |publisher=Meteor Showers Online |author=Gary W. Kronk |url=http://meteorshowersonline.com/showers/ursids.html |accessdate=2012-11-17}}</ref> While there were sporadic observations after, the first coordinated studies of shower didn't begin until Dr. A. Bečvář observed an outburst of 169 per hour in 1945.<ref name="Kronk"/> Further observations in the 1970s and ongoing to current have established a relationship with comet 8P/Tuttle.<ref name="KronkU"/> Parent body = 8P/Tuttle.<ref name="KronkU"/> Right ascension = {{RA|14|28}}<ref name=moore_rees2011/>, Declination = {{DEC|+78}}<ref name=moore_rees2011/> Constellation = Ursa Minor (near Kochab) Occurs during December 17 – December 26.<ref name="KronkU"/> Date of peak = December&nbsp;22.<ref name="KronkU"/> Velocity = 33 km/s.<ref name=IMO2012P>[http://www.imo.net/calendar/2012#urs IMO Meteor Shower Calendar 2012: Ursids (URS)]</ref> Zenithal hourly rate = 10.<ref name="KronkU"/> ==Asteroids== {{main|Radiation astronomy/Asteroids}} [[Image:298Bap-LB1-mag15.jpg|thumb|right|250px|298 Baptistina (center) is one of the largest presumed remnants of the Baptistina family. Credit: [[w:user:Kevin Heider|Kevin Heider]].{{tlx|free media}}]] '''Def.''' a "naturally occurring solid object, [which is] smaller than a planet<ref name=AsteroidWikt>{{ cite book |author=[[wikt:User:Paul G|Paul G]] |title=asteroid |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=18 August 2005 |url=https://en.wiktionary.org/wiki/asteroid |accessdate=31 August 2015 }}</ref> and is not a comet,<ref name=AsteroidWikt2>{{ cite book |author=[[wikt:User:Kwamikagami|Kwamikagami]] |title=asteroid |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=19 January 2012 |url=https://en.wiktionary.org/wiki/asteroid |accessdate=31 August 2015 }}</ref> that orbits a star"<ref name=AsteroidWikt1>{{ cite book |author=[[wikt:User:41.240.160.214|41.240.160.214]] |title=asteroid |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=19 March 2008 |url=https://en.wiktionary.org/wiki/asteroid |accessdate=31 August 2015 }}</ref> is called an '''asteroid'''. '''Usage notes''' "The term "asteroid" has never been precisely defined. It was coined for objects which looked like stars in a telescope but moved like planets. These were known from the asteroid belt between Mars and Jupiter, and were later found co-orbiting with Jupiter (Trojan asteroids) and within the orbit of Mars. They were naturally distinguished from comets, which did not look at all starlike. Starting in the 1970s, small non-cometary bodies were found outside the orbit of Jupiter, and usage became divided as to whether to call these "asteroids" as well. Some astronomers restrict the term "asteroid" to rocky or rocky-icy bodies with orbits up to Jupiter. They may retain the term planetoid for all small bodies, and thus tend to use it for icy or rocky-icy bodies beyond Jupiter, or may use dedicated words such as centaurs, Kuiper belt objects, transneptunian objects, etc. for the latter. Other astronomers use "asteroid" for all non-cometary bodies smaller than a planet, even large ones such as Sedna and (occasionally) Pluto. However, the distinction between asteroid and comet is an artificial one; many outer "asteroids" would become comets if they ventured nearer the Sun. The official terminology since 2006 has been small Solar System body for any body that orbits the Sun directly and whose shape is not dominated by gravity."<ref name=AsteroidWikt/> From these trajectory measurements, meteoroids have been found to have many different orbits, some clustering in streams (see Meteor showers) often associated with a parent comet, others apparently sporadic. Debris from meteoroid streams may eventually be scattered into other orbits. ... Meteoroids travel around the Sun in a variety of orbits and at various velocities. The fastest ones move at about 26 miles per second (42 kilometers per second) through space in the vicinity of Earth's orbit. The Earth travels at about 18 miles per second (29 kilometers per second). Thus, when meteoroids meet the Earth's atmosphere head-on (which would only occur if the meteors were in a retrograde orbit), the combined speed may reach about 44 miles per second (71 kilometers per second). Meteoroids moving through the earth's orbital space average about 20&nbsp;km/s.<ref name=OrbitalDebris>{{ cite book |title=Report on Orbital Debris |url=http://hdl.handle.net/2060/19900003319 |work=NASA |publisher=NASA Technical Reports Server |accessdate=1 September 2012 }}</ref> {{clear}} ==Rocky objects== {{main|Rocks/Rocky objects}} [[Image:Meteoroid track through aerogel from EURECA mission.jpg|thumb|right|250px|A 1.5 mm long impact track of a meteoroid captured in aerogel exposed to space by the European Retrievable Carrier (EURECA) spacecraft. Credit: NASA/ESA.{{tlx|free media}}]] The composition of meteoroids can be determined as they pass through Earth's atmosphere from their trajectories and the light spectra of the resulting meteor. Their effects on radio signals also give information, especially useful for daytime meteors which are otherwise very difficult to observe. The light spectra, combined with trajectory and light curve measurements, have yielded various compositions and densities, ranging from fragile snowball-like objects with density about a quarter that of ice,<ref name=Povenmire>Povenmire, H. [http://www.lpi.usra.edu/meetings/lpsc2000/pdf/1183.pdf PHYSICAL DYNAMICS OF THE UPSILON PEGASID FIREBLL – EUROPEAN NETWORK 190882A]. Florida Institute of Technology</ref> to nickel-iron rich dense rocks. In meteoroid ablation spheres from deep-sea sediments, "[t]he silicate spheres are the most dominant group."<ref name=Blanchard>{{ cite journal |author=M.B. Blanchard, D.E. Brownlee, T.E. Bunch, P.W. Hodge, F.T. Kyte |title=Meteoroid ablation spheres from deep-sea sediments |journal=Earth and Planetary Science Letters |month=January |year=1980 |volume=46 |issue=2 |pages=178-90 |url=http://www.sciencedirect.com/science/article/pii/0012821X80900047 |arxiv= |bibcode= |doi=10.1016/0012-821X(80)90004-7 |pmid= |accessdate=2012-01-02 }}</ref> {{clear}} ==Micrometeoroids== [[Image:SMM panel hole.jpg|thumb|right|250px|Electron micrograph image shows an orbital debris hole made in the panel of the Solar Maximum Mission satellite. Credit: NASA image.{{tlx|free media}}]] '''Def.''' "an extraterrestrial particle less than a millimeter in size"<ref name=MicrometeoroidWikt>{{ cite book |author=[[wikt:User:SemperBlotto|SemperBlotto]] |title=micrometeoroid |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=14 April 2006 |url=https://en.wiktionary.org/wiki/micrometeoroid |accessdate=26 June 2019 }}</ref> is called a '''micrometeoroid'''. Very small meteoroids are known as '''micrometeoroids''' (see also interplanetary dust). {{clear}} ==Y asteroids== [[Image:YarkovskyEffect.svg|thumb|right|300px|Yarkovsky effect:<br />1. Radiation from asteroid's surface <br />2. Prograde rotating asteroid <br />2.1 Location with "Afternoon" <br />3. Asteroid's orbit <br />4. Radiation from Sun. Credit: [[w:user:Graevemoore|Graevemoore]].{{tlx|free media}}]] The possible importance of the Yarkovsky effect is the movement of meteoroids about the [[Solar System]].<ref>{{cite journal |first=E. J. |last=Öpik |title=Collision probabilities with the planets and the distribution of interplanetary matter |journal=Proceedings of the Royal Irish Academy |volume=54A |issue= |pages=165–199 |year=1951 |jstor=20488532 }}</ref> The diurnal effect is the dominant component for bodies with diameter greater than about 100 m.<ref name=Bottke06review>{{cite journal |last=Bottke, Jr. |first=William F., ''et al.'' |title=The Yarkovsky and YORP Effects: Implications for Asteroid Dynamics |journal=Annual Review of Earth and Planetary Sciences |volume=34 |pages=157–191 |year=2006 |doi=10.1146/annurev.earth.34.031405.125154 |bibcode = 2006AREPS..34..157B }}</ref> On very long timescales over which the spin axis of the body may be repeatedly changed due to collisions (and hence also the direction of the diurnal effect changes), the seasonal effect will also tend to dominate.<ref name=Bottke06review /> The effect was first measured in 1991–2003 on the asteroid 6489 Golevka which drifted 15&nbsp;km from its predicted position over twelve years (the orbit was established with great precision by a series of radar observations in 1991, 1995 and 1999 from the Arecibo Observatory radio telescope).<ref>{{cite journal |last=Chesley |first=Steven R. |title=Direct Detection of the Yarkovsky Effect via Radar Ranging to Asteroid 6489 Golevka |journal=Science |volume=302 |issue=5651 |pages=1739–1742 |year=2003 |doi=10.1126/science.1091452 |bibcode = 2003Sci...302.1739C |display-authors=etal}}</ref> The "population of asteroids in comet-like orbits using available asteroid size and albedo catalogs of data taken with the ''Infrared Astronomical Satellite'' [I], ''AKARI'' [A], and the ''Wide-field Infrared Survey Explorer'' [W] on the basis of their orbital properties (i.e., the Tisserand parameter with respect to Jupiter, T_J, and the aphelion distance, Q, [is] 123 asteroids in comet-like orbits [with] Q < 4.5 AU and T_J < 3, [including] a considerable number (i.e., 25 by our criteria) of asteroids in comet-like orbits have high albedo, pv > 0.1. [As] such high-albedo objects mostly consist of small (D < 3 km) bodies distributed in near-Earth space (with perihelion distance of q < 1.3 AU) [may be] susceptible to the Yarkovsky effect and drifted into comet-like orbits via chaotic resonances with planets."<ref name=Kim>{{ cite journal |author=Yoonyoung Kim, Masateru Ishiguro and Fumihiko Usui |title=Physical properties of asteroids in comet-like orbits in infrared asteroid survey catalogs |journal=The Astrophysical Journal |date=24 June 2014 |volume=789 |issue=2 |pages=151 |url=https://arxiv.org/pdf/1405.2989.pdf |arxiv= |bibcode= |doi=10.1088/0004-637X/789/2/151 |pmid= |accessdate=22 June 2019 }}</ref> "There are 138,285 asteroids whose albedos and sizes are given in the I–A–W catalog. [...] nearly all high-albedo [asteroids in comet-like orbits] ACOs consist of small asteroids at q < 1.3 AU. This trend cannot be explained by the observational bias. Because the result is obtained based on the mid-infrared data, which, unlike optical observations, are less sensitive to albedo values, it provides reliable sets of asteroid albedo information. If there are big ACOs with high albedo beyond q = 1.3 AU, they would be detected easily. Although further dynamical study is essential to evaluate the population quantitatively, we propose that such ACOs with high albedos were injected from the domain of T_J > 3 via the Yarkovsky effect, because small objects with higher surface temperature are susceptible to the thermal drag force and gradually change their orbital elements to be observed as ACOs in our list."<ref name=Kim/> "Although there are uncertainties in the dynamical simulation such as the value of the Yarkovsky force and the rocket force (for active comets), we conservatively consider that these three objects (2000 SU236, 2008 UM7, and 2009 SC298) are ACOs and PDCs. ["potential dormant comet" (PDC) is one having a low albedo (pv < 0.1) among ACOs. The second term is a paronomasia associating the spectra of potential dormant comets with spectra similar to P-type, D-type, or C-type asteroids (Licandro et al. 2008; DeMeo & Binzel 2008).]"<ref name=Kim/> "Let us consider how the Yarkovsky effect moves an asteroid into a comet-like orbit. As shown [...], high-albedo ACOs concentrate in a range of 2 < a < 3.5 AU, similar to main-belt asteroids and [Jupiter-family comets] JFCs. The Tisserand parameter is a function of a, e, and i, [the semimajor axis, eccentricity, and inclination, respectively] while the Yarkovsky effect changes a. Due to the similarity in a between high-albedo ACOs and main-belt asteroids, we conjecture that subsequent dynamical effects may change e and i. Widely known as a standard model for orbital evolution of near-Earth asteroids, the Yarkovsky effect could move small main-belt asteroids' orbits until they are close to resonances with planets, and subsequently, these resonances can push them into terrestrial planet crossing orbits (see, e.g., Morbidelli et al. 2002). Numerical simulations demonstrated that chaotic resonances cause a significant increase in the e and i of test particles in the resonance regions (Gladman et al. 1997). Bottke et al. (2002) suggested that some objects on T_J < 3 (or even T_J < 2) can result from chaotic resonances. [...] Although there are a couple of ACOs close to resonances, their semimajor axes are not related to these major resonances. Therefore, it may be reasonable to think that encounters with terrestrial planets as well as chaotic resonances with massive planets can drift main-belt asteroids into comet-like orbits."<ref name=Kim/> "In particular, we stress again the significance of high-albedo ACOs. As we discussed through our ground-based observation with the Subaru Telescope, high-albedo ACOs, which may have composition similar to silicaceous asteroids, definitively exist in the I–A–W database. Considering the very low T_J as well as the small size and perihelion distance, we suggest that such high-albedo ACOs have been injected via nongravitational forces, most likely the Yarkovsky effect."<ref name=Kim/> {{clear}} ==Apollo asteroids== [[Image:Minor Planets - Apollo.svg|thumb|right|250px|This a diagram showing the Apollo asteroids, compared to the orbits of the terrestrial planets of the [[Solar System]]. {| style="width: 100%; margin-bottom: 8px; margin-left: 12px;" |- | valign=top width=45% | {{legend2|#e61d30|border=1px solid #770D18|Mars}}&nbsp;(M)<br> {{legend2|#000000|border=1px solid black|Venus}}&nbsp;(V) {{legend2|#aaa|border=1px solid #444|Mercury}}&nbsp;(H) | valign=top | {{legend2|#FFFF00|border=1px solid #B3B300|[[Stars/Sun|Sun]]}}<br> {{legend2|#00CC00|border=1px solid #003300|'''Apollo asteroids'''}}<br> {{legend2|#1708a6|border=1px solid black|Earth}}&nbsp;(E) |} Credit: [[c:User:AndrewBuck|AndrewBuck]].{{tlx|free media}}]] [[Image:162173 Ryugu.jpg|left|thumb|300px|Photograph of the full disc of the asteroid 162173 Ryugu, as it appeared to the Hayabusa2 spacecraft's Optical Navigation Camera – Telescopic (ONC-T) at a distance of 20 kilometres (12 miles) at 03:50 UTC on 26 June 2018. Credit: JAXA, University of Tokyo, Kochi University, Rikkyo University, Nagoya University, Chiba Institute of Technology, Meiji University, University of Aizu, AIST.{{tlx|fairuse}}]] [[Image:Asteroid-Bennu-OSIRIS-RExArrival-GifAnimation-20181203.gif|right|thumb|300px|Asteroid Bennu imaged by the OSIRIS-REx probe on arrival 3 December 2018. Credit: NASA/Goddard/University of Arizona.{{tlx|free media}}]] [[Image:Bennuarrival.png|thumb|left|250px|Photo of 101955 Bennu was taken by the OSIRIS-REx probe on 3 December 2018. Credit: NASA/Goddard/University of Arizona.{{tlx|free media}}]] Note that sizes and distances of bodies and orbits are ''not'' to scale in the image on the right. As of 2015, the Apollo asteroid group includes a total of 6,923 known objects of which 991 are numbered ([http://ssd.jpl.nasa.gov/sbdb_query.cgi JPL SBDB]). Ryugu shown on the left was discovered on 10 May 1999 by astronomers with the Lincoln Near-Earth Asteroid Research at the Lincoln Laboratory's Experimental Test Site near Socorro, New Mexico, in the United States.<ref name="MPC-object">{{cite book | title = 162173 Ryugu (1999 JU3) | url = https://www.minorplanetcenter.net/db_search/show_object?object_id=162173 | accessdate= 22 June 2018}}</ref> The asteroid was officially named "Ryugu" by the Minor Planet Center on 28 September 2015.<ref name="MPC-Circulars-Archive">{{cite book | title = MPC/MPO/MPS Archive | url = https://www.minorplanetcenter.net/iau/ECS/MPCArchive/MPCArchive_TBL.html | accessdate= 22 June 2018}}</ref> Initial images taken by the ''Hayabusa-2'' spacecraft on approach at a distance of 700&nbsp;km were released on 14 June 2018 and revealed a diamond shaped body and confirmed its retrograde rotation.<ref>{{cite book|title=From a distance of about 700km, Ryugu's rotation was observed|url=http://www.hayabusa2.jaxa.jp/topics/20180616je/index_e.html|publisher=JAXA|accessdate= 18 June 2018}}</ref> Between 17 and 18 June 2018, Hayabusa 2 went from 330&nbsp;km to 240&nbsp;km from Ryugu and captured a series of additional images from the closer approach.<ref>Plait, Phil, [http://www.syfy.com/syfywire/asteroid-ryugu-starts-to-come-into-focus "Asteroid Ryugu Starts to Come Into Focus"], SyFy Wire, 20 June 2018. Accessed 20 June 2018.</ref> On 21 September 2018, the first two of these rovers, which will hop around the surface of the asteroid, were released from ''Hayabusa2''.<ref>{{cite book |last1=Wall |first1=Mike |title=Japanese Probe Deploys Tiny Hopping Robots Toward Big Asteroid Ryugu |url=https://www.space.com/41898-hayabusa2-deploys-hopping-robots-asteroid-ryugu.html |accessdate=21 September 2018}}</ref> On September 22, 2018, JAXA confirmed the two rovers had successfully touched down on Ryugu's surface which marks the first time a mission has completed a successful landing on a fast-moving asteroid body.<ref name="MINERVA-II-1 Landings">{{cite book |last1=Yoshimitsu |first1=Tetsuo |last2=Kubota |first2=Takashi |last3=Tsuda |first3=Yuichi |last4=Yoshikawa |first4=Makoto |title=MINERVA-II1: Successful image capture, landing on Ryugu and hop! |url=http://www.hayabusa2.jaxa.jp/en/topics/20180922e/ |publisher=JAXA |accessdate=24 September 2018}}</ref> "This series of images [second down on the right] taken by the OSIRIS-REx spacecraft shows Bennu in one full rotation from a distance of around 50 miles (80 km). The spacecraft’s PolyCam camera obtained the 36 2.2-millisecond frames over a period of four hours and 18 minutes."<ref name=Drake>{{ cite book |author=Michael J. Drake |title=Bennu Full Rotation at a Distance of 50 Miles |publisher=NASA |location=Washington, DC USA |date=November 25, 2018 |url=https://www.asteroidmission.org/?attachment_id=12182 |accessdate=5 December 2018 }}</ref> 101955 Bennu (provisional designation 1999 RQ₃₆<ref name="contest">{{ cite book |author=Diane Murphy |title=Nine-Year-Old Names Asteroid Target of NASA Mission in Competition Run By The Planetary Society |url=http://www.planetary.org/press-room/releases/2013/nine-year-old-names-asteroid.html |accessdate=20 August 2016 |publisher=The Planetary Society |date=1 May 2013 }}</ref>, a C-type carbonaceous asteroid in the Apollo group discovered by the Lincoln Near-Earth Asteroid Research (LINEAR) Project on September 11, 1999, is a potentially hazardous object that is listed on the Sentry monitoring system, Sentry Risk Table, with the second-highest cumulative rating on the Palermo Technical Impact Hazard Scale.<ref name="risk-table">{{ cite book | title=Sentry Risk Table | publisher=NASA/JPL Near-Earth Object Program Office | accessdate=2018-03-20 |url=https://web.archive.org/web/20160911172609/http://neo.jpl.nasa.gov/risk/ }}</ref> It has a cumulative 1-in-2,700 chance of impacting Earth between 2175 and 2199.<ref name="Sentry">{{cite book | url=https://cneos.jpl.nasa.gov/sentry/details.html#?des=101955 | title=101955 1999 RQ36: Earth Impact Risk Summary | publisher=Jet Propulsion Laboratory | date=25 March 2016 | accessdate=20 March 2018 }}</ref><ref name="Meteoritics">{{ cite journal |last1=Lauretta|first1=D. S.|last2=Bartels|first2=A. E.|last3=Barucci|first3=M. A.|last4=Bierhaus|first4=E. B.|last5=Binzel|first5=R. P.|last6=Bottke|first6=W. F.|last7=Campins|first7=H.|last8=Chesley|first8=S. R.|last9=Clark|first9=B. C.|last10=Clark|first10=B. E.|last11=Cloutis|first11=E. A.|last12=Connolly|first12=H. C.|last13=Crombie|first13=M. K.|last14=Delbó|first14=M.|last15=Dworkin|first15=J. P.|last16=Emery|first16=J. P.|last17=Glavin|first17=D. P.|last18=Hamilton|first18=V. E.|last19=Hergenrother|first19=C. W.|last20=Johnson|first20=C. L.|last21=Keller|first21=L. P.|last22=Michel|first22=P.|last23=Nolan|first23=M. C.|last24=Sandford|first24=S. A.|last25=Scheeres|first25=D. J.|last26=Simon|first26=A. A.|last27=Sutter|first27=B. M.|last28=Vokrouhlický|first28=D.|last29=Walsh|first29=K. J. |title=The OSIRIS-REx target asteroid (101955) Bennu: Constraints on its physical, geological, and dynamical nature from astronomical observations |journal=Meteoritics & Planetary Science |date=April 2015 |volume=50 |issue=4 |pages=834–849 |doi=10.1111/maps.12353 |bibcode=2015M&PS...50..834L |citeseerx=10.1.1.723.9955 }}</ref> 101955 Bennu has a mean diameter of approximately {{convert|492|m|ft mi|abbr=on}} and has been observed extensively with the Arecibo Observatory planetary radar and the Goldstone Deep Space Communications Complex NASA Deep Space Network.<ref name="Nolan2013">{{cite journal |last1=Nolan|first1=M. C.|last2=Magri|first2=C.|last3=Howell|first3=E. S.|last4=Benner|first4=L. A. M.|last5=Giorgini|first5=J. D.|last6=Hergenrother|first6=C. W.|last7=Hudson|first7=R. S.|last8=Lauretta|first8=D. S.|last9=Margot|first9=J. L.|last10=Ostro|first10=S. J.|last11=Scheeres|first11=D. J. |title=Shape model and surface properties of the OSIRIS-REx target Asteroid (101955) Bennu from radar and lightcurve observations |journal=Icarus |volume=226 |issue=1 |year=2013 |pages=629–640 |issn=0019-1035 |doi=10.1016/j.icarus.2013.05.028 |bibcode=2013Icar..226..629N | url=http://www.escholarship.org/uc/item/6mh8b2t5|format=Submitted manuscript }}</ref><ref name="GDSCC-img">{{cite book | url= http://echo.jpl.nasa.gov/~lance/1999RQ36/1999RQ36.html | title=Goldstone Delay-Doppler Images of 1999 RQ36 | publisher=Jet Propulsion Laboratory }}</ref><ref name="Hudson et al. 2000">{{cite journal |author=Hudson, R. S. |author2=Ostro, S. J. |author3=Benner, L. A. M. |bibcode=2000DPS....32.0710H |title=Recent Delay-Doppler Radar Asteroid Modeling Results: 1999 RQ36 and Craters on Toutatis |journal=Bulletin of the American Astronomical Society |volume=32 |page=1001 | year=2000}}</ref> Asteroid Bennu has a roughly spheroidal shape, resembling a spinning top, with the direction of rotation about its axis retrograde with respect to its orbit and a fairly smooth shape with one prominent 10–20 m boulder on its surface, in the southern hemisphere.<ref name="Meteoritics" /> There is a well-defined ridge along the equator of asteroid Bennu that suggests that fine-grained regolith particles have accumulated in this area, possibly because of its low gravity and fast rotation.<ref name="Meteoritics" /> Observations of this minor planet by the Spitzer Space Telescope in 2007 gave an effective diameter of {{Val|484|10|u=m}}, which is in line with other studies. It has a low visible geometric albedo of {{Val|0.046|0.005}}. The thermal inertia was measured and found to vary by ±19% during each rotational period suggesting that the regolith grain size is moderate, ranging from several millimeters up to a centimeter, and evenly distributed. No emission from a potential dust coma has been detected around asteroid Bennu, which puts a limit of 10⁶&nbsp;g of dust within a radius of 4750&nbsp;km.<ref name=Emery2014>{{ cite book | title=Thermal infrared observations and thermophysical characterization of the OSIRIS-REx target asteroid (101955) Bennu, In: Conference Proceedings Asteroids, Comets, Meteors 2014 |last1=Emery | first1=J. | last2=Fernandez | first2=Y. | last3=Kelley | first3=M. | last4=Warden | first4=K. | last5=Hergenrother | first5=C. | last6=Lauretta | first6=D. | last7=Drake | first7=M. | last8=Campins | first8=H. | last9=Ziffer | first9=J. | pages=148 | editor=K. Muinonen | date=July 2014 | bibcode=2014acm..conf..148E }}</ref> Astrometric observations between 1999 and 2013 have demonstrated that 101955 Bennu is influenced by the Yarkovsky effect, causing the semimajor axis to drift on average by {{Val|284|1.5}}&nbsp;meters/year; analysis of the gravitational and thermal effects give a bulk density of ρ = {{Val|1260|70}}&nbsp;kg/m³, which is only slightly denser than water, the predicted macroporosity is {{Val|40|10}}%, suggesting that the interior has a rubble pile structure, with an estimated mass is {{Val|7.8|0.9|e=10|u=kg}}.<ref name="ChesleyFarnocchia2014">{{cite journal|last1=Chesley|first1=Steven R.|last2=Farnocchia|first2=Davide|last3=Nolan|first3=Michael C.|last4=Vokrouhlický|first4=David|last5=Chodas|first5=Paul W.|last6=Milani|first6=Andrea|last7=Spoto|first7=Federica|last8=Rozitis|first8=Benjamin|last9=Benner|first9=Lance A.M.|last10=Bottke|first10=William F.|last11=Busch|first11=Michael W.|last12=Emery|first12=Joshua P.|last13=Howell|first13=Ellen S.|last14=Lauretta|first14=Dante S.|last15=Margot|first15=Jean-Luc|last16=Taylor|first16=Patrick A.|title=Orbit and bulk density of the OSIRIS-REx target Asteroid (101955) Bennu|journal=Icarus|volume=235|date=2014|pages=5–22|issn=0019-1035|doi=10.1016/j.icarus.2014.02.020 | bibcode=2014Icar..235....5C|arxiv=1402.5573 }}</ref> Photometric observations of Bennu in 2005 yielded a synodic rotation period of {{Val|4.2905|0.0065|u=h}}, a B-type asteroid classification, which is a sub-category of C-type asteroid or carbonaceous asteroids. Polarimetric observations show that Bennu belongs to the rare F-type asteroid or F subclass of carbonaceous asteroids, which is usually associated with cometary features.<ref name="Cellino2018" /> Measurements over a range of phase angles show a phase function slope of 0.040 magnitudes per degree, which is similar to other near-Earth asteroids with low albedo.<ref name=Hergenrother2013>{{ cite journal | title=Lightcurve, Color and Phase Function Photometry of the OSIRIS-REx Target Asteroid (101955) Bennu | last1=Hergenrother | first1=Carl W. | last2=Nolan | first2=Michael C. | last3=Binzel | first3=Richard P. | last4=Cloutis | first4=Edward A. | last5=Barucci | first5=Maria Antonietta | last6=Michel | first6=Patrick | last7=Scheeres | first7=Daniel J. | last8=d'Aubigny | first8=Christian Drouet | last9=Lazzaro | first9=Daniela | last10=Pinilla-Alonso | first10=Noemi | last11=Campins | first11=Humberto | last12=Licandro | first12=Javier | last13=Clark | first13=Beth E. | last14=Rizk | first14=Bashar | last15=Beshore | first15=Edward C. | last16=Lauretta | first16=Dante S. | journal=Icarus | volume=226 | issue=1 | pages=663–670 | date=September 2013 | doi=10.1016/j.icarus.2013.05.044 | bibcode=2013Icar..226..663H }}</ref> Asteroid Bennu's basic [[mineralogy]] and chemical nature would have been established during the first 10 million years of the Solar System's formation, where the carbonaceous material underwent some geologic heating and chemical transformation into more complex minerals.<ref name="Meteoritics" /> Bennu probably began in the inner asteroid belt as a fragment from a larger body with a diameter of 100&nbsp;km, where simulations suggest a 70% chance it came from the Polana family and a 30% chance it derived from the 495 Eulalia (Eulalia family).<ref name=Bottke2015>{{cite journal | title=In search of the source of asteroid (101955) Bennu: Applications of the stochastic YORP model | last1=Bottke | first1=William F. | last2=Vokrouhlický | first2=David | last3=Walsh | first3=Kevin J. | last4=Delbo | first4=Marco | last5=Michel | first5=Patrick | last6=Lauretta | first6=Dante S. | last7=Campins | first7=Humberto | last8=Connolly | first8=Harold C. | last9=Scheeres | first9=Daniel J. | last10=Chelsey | first10=Steven R. | journal=Icarus | volume=247 | pages=191–217 | date=February 2015 | doi=10.1016/j.icarus.2014.09.046 | bibcode=2015Icar..247..191B }}</ref> Subsequently, the orbit drifted as a result of the Yarkovsky effect and mean motion resonances with the giant planets, such as Jupiter and Saturn modified the asteroid, possibly changing its spin, shape, and surface features.<ref name=Lauretta2015>{{ cite journal | title=The OSIRIS-REx target asteroid (101955) Bennu: Constraints on its physical, geological, and dynamical nature from astronomical observations | last1=Lauretta | first1=D. S. | last2=Bartels | first2=A. E. | last3=Barucci | first3=M. A. | last4=Bierhaus | first4=E. B. | last5=Binzel | first5=R. P. | last6=Bottke | first6=W. F. | last7=Campins | first7=H. | last8=Chesley | first8=S. R. | last9=Clark | first9=B. C. | last10=Clark | first10=B. E. | last11=Cloutis | first11=E. A. | last12=Connolly | first12=H. C. | last13=Crombie | first13=M. K. | last14=Delbó | first14=M. | last15=Dworkin | first15=J. P. | last16=Emery | first16=J. P. | last17=Glavin | first17=D. P. | last18=Hamilton | first18=V. E. | last19=Hergenrother | first19=C. W. | last20=Johnson | first20=C. L. | last21=Keller | first21=L. P. | last22=Michel | first22=P. | last23=Nolan | first23=M. C. | last24=Sandford | first24=S. A. | last25=Scheeres | first25=D. J. | last26=Simon | first26=A. A. | last27=Sutter | first27=B. M. | last28=Vokrouhlický | first28=D. | last29=Walsh | first29=K. J. | journal=Meteoritics & Planetary Science | volume=50 | issue=4 | pages=834–849 | date=April 2015 | doi=10.1111/maps.12353 | bibcode=2015M&PS...50..834L | citeseerx=10.1.1.723.9955}}</ref> A possible cometary origin for Bennu, based on similarities of its spectroscopic properties with known comets, with the estimated fraction of comets in the population of Near Earth asteroids is {{val|8|5|u=%}}.<ref name="Cellino2018">{{Cite arXiv | title=Unusual polarimetric properties of (101955) Bennu: similarities with F-class asteroids and cometary bodies | eprint=1808.07812 | last1=Hergenrother | first1=Carl W | author2=Maria Antonietta Barucci | last3=Barnouin | first3=Olivier | last4=Bierhaus | first4=Beau | last5=Binzel | first5=Richard P | last6=Bottke | first6=William F | last7=Chesley | first7=Steve | last8=Clark | first8=Ben C | last9=Clark | first9=Beth E | last10=Cloutis | first10=Ed | author11=Christian Drouet d'Aubigny | last12=Delbo | first12=Marco | last13=Emery | first13=Josh | last14=Gaskell | first14=Bob | last15=Howell | first15=Ellen | last16=Keller | first16=Lindsay | last17=Kelley | first17=Michael | last18=Marshall | first18=John | last19=Michel | first19=Patrick | last20=Nolan | first20=Michael | last21=Rizk | first21=Bashar | last22=Scheeres | first22=Dan | last23=Takir | first23=Driss | last24=Vokrouhlický | first24=David D | last25=Beshore | first25=Ed | last26=Lauretta | first26=Dante S | class=astro-ph.EP | year=2018 }}</ref> {{clear}} ==Earth crossers== [[Image:2008VK184-year2014.png|thumb|right|250px|The close approach of apollo asteroid 2007 VK184 was in May 2014. Credit: Osamu Ajiki (AstroArts) and Ron Baalke (JPL).{{tlx|free media}}]] EC denotes Earth-crossing.<ref name=Michel>{{ cite journal |author=Patrick Michel, Fabbio Migliorini, Alessandro Morbidelli, Vincenzo Zappalà |title=The Population of Mars-Crossers: Classification and Dynamical Evolution |journal=Icarus |month=June |year=2000 |volume=145 |issue=2 |pages=332-47 |url=http://www.obs-nice.fr/morby/papers/6358a.pdf |bibcode= |doi=10.1006/icar.2000.6358 |pmid= |accessdate=2011-10-06 }}</ref> "50 % of the MB Mars-crossers [MCs] become ECs within 59.9 Myr and [this] contribution ... dominates the production of ECs".<ref name= Michel /> [[Image:SmallAsteroidImpacts-Frequency-Bolide-20141114.jpg|thumb|upright=2|center|500px|This diagram maps the data gathered from 1994-2013 on small asteroids impacting Earth's atmosphere. Credit: NASA/Planetary Science.{{tlx|free media}}]] "This diagram [center] maps the data gathered from 1994-2013 on small asteroids impacting Earth's atmosphere to create very bright meteors, technically called "bolides" and commonly referred to as "fireballs". Sizes of [yellow] dots (daytime impacts) and blue dots (nighttime impacts) are proportional to the optical radiated energy of impacts measured in billions of Joules (GJ) of energy, and show the location of impacts from objects about 1 meter (3 feet) to almost 20 meters (60 feet) in size."<ref name=Agle>{{ cite book |author=DC Agle |title=New Map Shows Frequency of Small Asteroid Impacts, Provides Clues on Larger Asteroid Population |publisher=NASA's Jet Propulsion Laboratory |location=Pasadena, California |date=14 November 2014 |editor= |pages= |url=https://www.jpl.nasa.gov/news/news.php?release=2014-397 |arxiv= |bibcode= |doi= |pmid= |isbn= |accessdate=2018-04-01 }}</ref> "A map released [...] by NASA's Near Earth Object (NEO) Program reveals that small asteroids frequently enter and disintegrate in the Earth's atmosphere with random distribution around the globe. Released to the scientific community, the map visualizes data gathered by U.S. government sensors from 1994 to 2013. The data indicate that Earth's atmosphere was impacted by small asteroids, resulting in a bolide (or fireball), on 556 separate occasions in a 20-year period. Almost all asteroids of this size disintegrate in the atmosphere and are usually harmless. The notable exception was the Chelyabinsk event which was the largest asteroid to hit Earth in this period."<ref name=Agle/> {{clear}} ==2008 TC₃== [[Image:2008TC3-groundpath-rev.png|right|thumb|300px|Estimated path and altitude of the meteor in red, with the possible location for the METEOSAT IR fireball (bolide) as orange crosshairs and the infrasound detection of the explosion in green. Credit: George William Herbert (graphic overlay) / US Government (original map).{{tlx|free media}}]] [[Image:2008 TC3 Tumbling (reduced).gif|thumb|left|250px|An animation of 2008 TC3's excited rotation prior to entering the atmosphere is shown. Credit: Astronomical Institute of the Charles University: Josef Ďurech, Vojtěch Sidorin.{{tlx|free media}}]] [[Image:Eumetsat-composite-2008TC3-impact.jpg|thumb|left|250px|Meteosat 8/EUMETSAT infrared image is of the explosion. Credit: EUMETSAT.{{tlx|fairuse}}]] [[Image:ELG webcam record of 2008 TC3 frame 0005.png|thumb|right|250px|This webcam frame was shot. Credit: Webcam at [http://www.kitepower-elgouna.com/en/ kitepower], Mangroovy Beach, El Gouna, Red Sea governate, Egypt.{{tlx|free media}}]] [[Image:323213main Petersmeteorites 946-710.jpg|thumb|250px|2008 TC₃ fragment was found on February 28, 2009 by Peter Jenniskens, with help from students and staff of the University of Khartoum. Nubian Desert, Sudan. Credit: NASA / SETI / P. Jenniskens.{{tlx|free media}}]] [[Image:M8-HRV-200810070245.jpg|thumb|right|250px|Meteosat 8 / EUMETSAT visual image is first light flare from 2008 TC₃ with lat/long reference. Credit: EUMETSAT.{{tlx|fairuse}}]] [[Image:M8-NCOL-200810070245.jpg|thumb|right|250px|Meteosat 8 / EUMETSAT IR image of main fireball from 2008 TC₃. Credit: EUMETSAT.{{tlx|fairuse}}]] [[Image:Img asteroid hrv ir108.jpg|thumb|right|250px|Meteosat images combined, showing offset from first light flare to main IR flare. Credit: EUMETSAT.{{tlx|fairuse}}]] 2008 TC₃ (Catalina Sky Survey temporary designation 8TA9D69) was an {{convert|80|t|ton|sigfig=1|adj=on}}, {{convert|4.1|m|ft|sigfig=2|sp=us|adj=on}} diameter asteroid<ref name=" Jenniskens2009">{{Cite journal |last=Jenniskens |first=P. |date=2009 |volume=458 |issue=7237 |pages=485–488 |title=The impact and recovery of asteroid 2008 TC₃ |journal=Nature |bibcode=2009Natur.458..485J |doi=10.1038/nature07920 |pmid=19325630 |display-authors=etal}}</ref> that entered Earth's atmosphere on October 7, 2008.<ref name="discovermagazine">{{cite book |last=Plait |first=P. |date=October 6, 2008 |title=Incoming!!! |url=http://blogs.discovermagazine.com/badastronomy/2008/10/06/incoming-2/ |publisher=Bad Astronomy |accessdate=October 8, 2008 |archiveurl=https://web.archive.org/web/20081007190747/http://blogs.discovermagazine.com/badastronomy/2008/10/06/incoming-2/ |archivedate=October 7, 2008 }}</ref> It exploded at an estimated {{convert|37|km|sigfig=2|sp=us}} above the Nubian Desert in Sudan. Some 600 meteorites, weighing a total of {{convert|10.5|kg|sigfig=3}}, were recovered; many of these belonged to a rare type known as ureilites, which contain, among other minerals, nanodiamonds.<ref name="Jenniskens2009"/><ref name="nytimes">{{cite news |last=Chang |first=K. |date=March 25, 2009 |title=Recovered Pieces of Asteroid Hold Clues to Early History |url=https://www.nytimes.com/2009/03/26/science/space/26asteroid.html?ref=us |work=The New York Times |accessdate=March 26, 2009 |archiveurl=https://web.archive.org/web/20131113234704/http://www.nytimes.com/2009/03/26/science/space/26asteroid.html?ref=us |archivedate=November 13, 2013 }}</ref><ref name=AP>{{cite news|last=Borenstein |first=S. |date=March 26, 2009 |title=Astronomers catch a shooting star for 1st time |url=http://abcnews.go.com/Technology/Space/WireStory?id=7175837&page=2 |publisher=ABC news |agency=The Associated Press |accessdate=September 15, 2009 |archiveurl=https://web.archive.org/web/20090329084214/http://abcnews.go.com/Technology/Space/WireStory?id=7175837&page=2 |archivedate=March 29, 2009 }}</ref> It was the first time that an asteroid impact had been predicted prior to its entry into the atmosphere as a meteor.<ref name="Roylance2008">{{cite news |last=Roylance |first=F. |date=October 7, 2008 |title=Predicted meteor may have been sighted |url=http://weblogs.marylandweather.com/2008/10/predicted_meteor_may_have_been.html |publisher=Maryland Weather |accessdate=October 8, 2008 |archiveurl=https://web.archive.org/web/20081010045620/http://weblogs.marylandweather.com/2008/10/predicted_meteor_may_have_been.html |archivedate=October 10, 2008 }}</ref> The asteroid was discovered by Richard A. Kowalski at the Catalina Sky Survey (CSS) 1.5-meter telescope at Mount Lemmon, north of Tucson, Arizona, US, on October 6, 06:39 UTC, 19 hours before the impact.<ref>{{cite book |last=Williams |first=G. V. |date=October 6, 2008 |title=MPEC 2008-T50 |url=http://www.cfa.harvard.edu/mpec/K08/K08T50.html |publisher=Minor Planet Center |accessdate=October 8, 2008 |archiveurl=https://web.archive.org/web/20081009221534/http://www.cfa.harvard.edu/mpec/K08/K08T50.html |archivedate=October 9, 2008 }}</ref><ref>{{cite book |last=Huntington |first=J. |date=October 7, 2008 |url=http://www.efluxmedia.com/news_Small_Asteroid_Enters_Earths_Atmosphere_26024.html |title=Small Asteroid Enters Earth's Atmosphere |publisher=eFluxMedia |accessdate=October 8, 2008 |archiveurl=https://web.archive.org/web/20081010095304/http://www.efluxmedia.com/news_Small_Asteroid_Enters_Earths_Atmosphere_26024.html |archivedate=October 10, 2008 |df= }}</ref><ref name="jplneo">{{cite book |last1=Chesley |first1=S. |last2=Chodas |first2=P. |last3=Yeomans |first3=D. |date=November 4, 2008 |title=Asteroid 2008 TC3 Strikes Earth: Predictions and Observations Agree |url=http://neo.jpl.nasa.gov/news/2008tc3.html |work=Near Earth Object Program |publisher=NASA |accessdate=June 18, 2009 |archiveurl=https://web.archive.org/web/20090618045538/http://neo.jpl.nasa.gov/news/2008tc3.html |archivedate=June 18, 2009 }}</ref> It was notable as the first such body to be observed and tracked prior to reaching Earth.<ref name="Roylance2008" /> The process of detecting and tracking a near-Earth object, an effort sometimes referred to as Spaceguard, was put to the test. In total, 586 astrometric and almost as many photometric observations were performed by 27 amateur and professional observers in less than 19 hours and reported to the Minor Planet Center, which in eleven hours issued 25 Minor Planet Electronic Circulars with new orbit solutions as observations poured in. On October 7, 01:49 UTC,<ref name="jplneo"/> the asteroid entered the shadow of the Earth, which made further observations impossible. Impact predictions were performed by University of Pisa's CLOMON 2 semi-automatic monitoring system<ref>{{cite book |title=NEODys Main Risk Page |url=http://newton.dm.unipi.it/neodys/ |accessdate=October 8, 2008 }}</ref><ref>{{cite book |title=NEODys 2008 TC₃ page |url=http://newton.dm.unipi.it/cgi-bin/neodys/neoibo?objects:2008TC3;main |accessdate=October 8, 2008 }}</ref> as well as Jet Propulsion Laboratory's Sentry system. Spectral observations that were performed by astronomers at the 4.2-meter William Herschel Telescope at La Palma, Canary Islands are consistent with either a C-type or M-type asteroid. The meteor entered Earth's atmosphere above northern Sudan at 02:46 UTC (05:46 local time) on October 7, 2008 with a velocity of {{convert|12.8|km/s|mph|sp=us}} at an azimuth of 281 degrees and an altitude angle of 19 degrees to the local horizon. It exploded tens of kilometers above the ground with the energy of 0.9 to 2.1 kilotons of TNT over a remote area of the Nubian Desert, causing a large fireball or bolide.<ref>{{cite book |date=October 6, 2008 |title=Astronomers predict shooting star over Sudan from space boulder |url=http://afp.google.com/article/ALeqM5jOHHF4wcyteiUj5AlJP69dIMb0yA |publisher=Agence France-Presse |accessdate=October 8, 2008 |archiveurl=https://web.archive.org/web/20081011054910/http://afp.google.com/article/ALeqM5jOHHF4wcyteiUj5AlJP69dIMb0yA |archivedate=October 11, 2008 |df= }}</ref> The meteor's "light was so intense that it lit up the sky like a full moon and an airliner {{convert|1,400|km|mi|abbr=on}} away reported seeing the bright flash."<ref>{{cite news |last=Simon |first=P. |date=October 8, 2008 |title=Weather Eye: NASA spots asteroid before annihilation |url=http://www.timesonline.co.uk/tol/news/weather/article4902427.ece |work=The Times |accessdate=October 8, 2008 }}</ref> A webcam captured the flash lighting up El-Gouna beach 725 kilometres north of the explosion (see [[:File:ELG webcam record of 2008 TC3 frame 0005.png|this webcam frame]]).<ref name=Javaux>{{cite book |last=Javaux |first=G. |title=2008 TC3... Première détection d'un astéroïde avant son impact sur Terre... quelques heures plus tard |url=http://pagesperso-orange.fr/pgj/2008_TC3.htm |language=French |accessdate=September 15, 2009 }}</ref> "Une webcam de surveillance, située sur la plage de la Mer Rouge à El Gouna en Egypte, a enregistré indirectement le flash de l'explosion qui s'est produit à environ 725 km plus au sud."<ref name=Javaux/> A low-resolution image of the explosion was captured by the weather satellite Meteosat 8.<ref name="SpaceWeather2008">{{cite book |date=October 8, 2008 |title=Asteroid Impact |url=http://www.spaceweather.com/archive.php?view=1&day=08&month=10&year=2008 |publisher=SpaceWeather.com |accessdate=October 8, 2008 |archiveurl=https://web.archive.org/web/20081010193852/http://www.spaceweather.com/archive.php?view=1&day=08&month=10&year=2008 |archivedate=October 10, 2008 }}</ref> The Meteosat images place the fireball at {{Coord|21.00|32.15|name=2008 TC₃ fireball}}.<ref name="Meteosat 8 HRV">{{cite book |date=October 8, 2008 |title=m8 HRV 200810070245 |url=http://www.eumetsat.int/groups/ops/documents/image/img_asteroid_impact_hrv.jpg |accessdate=October 8, 2008 |archiveurl=https://web.archive.org/web/20081017142511/http://www.eumetsat.int/groups/ops/documents/image/img_asteroid_impact_hrv.jpg |archivedate=October 17, 2008 }}</ref> Infrasound detector arrays in Kenya also detected a sound wave from the direction of the expected impact corresponding to energy of 1.1 to 2.1 kilotons of TNT.<ref name="NASA-160">{{cite book |date=October 7, 2008 |title=Impact of Asteroid 2008 TC3 Confirmed |url=http://neo.jpl.nasa.gov/news/news160.html |publisher=Near Earth Object Program] |accessdate=October 8, 2008 |archiveurl=https://web.archive.org/web/20081010010227/http://neo.jpl.nasa.gov/news/news160.html |archivedate=October 10, 2008 |df= }}</ref> Asteroids of this size hit Earth about two or three times a year.<ref>{{cite news |last=Borenstein |first=S. |date=October 6, 2008 |title=Small Asteroid Headed for Light Show Over Africa |url=http://abcnews.go.com/Technology/wireStory?id=5968166 |publisher=ABC News |agency=The Associated Press |accessdate=September 15, 2009 }} {{Dead link|date=September 2010|bot=H3llBot}}</ref> The trajectory showed intersection with Earth's surface at roughly {{Coord|20.3|N|33.5|E|name=2008 TC₃ projected impact}}<ref>{{cite book |author=Mike |date=October 6, 2008 |title=Very near NEO Meteoroid impact! |url=http://mkvh.de/ast/serendipity/index.php?/archives/9-Very-near-NEO-Meteoroid-impact-!.html |archive-url=https://web.archive.org/web/20081010123315/http://mkvh.de/ast/serendipity/index.php?%2Farchives%2F9-Very-near-NEO-Meteoroid-impact-%21.html |archive-date=October 10, 2008 |publisher=MKVH.de |df= }}</ref> though the object was expected to break up perhaps {{convert|100|-|200|km|mi|-1|sp=us}} west as it descended, somewhat east of the Nile River, and about {{convert|100|km|mi|-1|sp=us}} south of the Egypt–Sudan border. According to U.S. government sources<ref>{{cite book |date=October 15, 2008 |title=Asteroid Update |url=http://www.spaceweather.com/archive.php?month=10&day=15&year=2008&view=view |publisher=SpaceWeather.com |accessdate=October 18, 2008}}</ref><ref>{{cite book |date=October 15, 2008 |title=Fireball detection |url=http://meteor.uwo.ca/~pbrown/usaf/usg282.txt |publisher=University of Western Ontario |accessdate=October 18, 2008}}</ref> U.S. satellites detected the impact at 02:45:40 UT, with the initial detection at {{Coord|20.9|N|31.4|E|name=2008 TC₃ initial detection}} at {{convert|65.4|km|mi}} altitude and final explosion at {{Coord|20.8|N|32.2|E|name=2008 TC₃ final explosion}} at {{convert|37|km|mi}} altitude. These images have not been publicly released. A search of the impact zone that began on December 6, 2008, turned up {{convert|10.5|kg}} of rock in some 600 fragments. These meteorites are collectively named '''Almahata Sitta''',<ref>{{cite book |date=August 12, 2013 |title=Almahata Sitta |url=http://www.lpi.usra.edu/meteor/metbull.php?code=48915 |work=Meteoritical Bulletin Database |accessdate=August 13, 2013}}</ref> which means "Station Six"<ref>{{cite book |last=Gebauer |first=S. |date=April 16, 2008 |title=Station Nr. 6. – Nubian Desert |url=https://www.panoramio.com/photo/9417877 |publisher=Panoramio |accessdate=June 15, 2009 |archiveurl=https://www.webcitation.org/5hbspSSJd?url=http://www.panoramio.com/photo/9417877 |archivedate=June 17, 2009 |df= }}</ref> in Arabic and is a train station between Wadi Halfa and Khartoum, Sudan. This search was led by Peter Jenniskens from the SETI Institute, California and Muawia Shaddad of the University of Khartoum in Sudan and carried out with the collaboration of students and staff of the University of Khartoum. The initial 15 meteorites were found in the first three days of the search. Numerous witnesses were interviewed, and the hunt was guided with a search grid and specific target area produced by NASA's Jet Propulsion Laboratory in Pasadena, California.<ref name="NASA2009">{{cite book |date=March 27, 2009 |title=NASA Team Finds Riches in Meteorite Treasure Hunt |url=http://www.nasa.gov/topics/solarsystem/features/asteroid_treasure_hunt.html |publisher=NASA |accessdate=April 5, 2009 |archiveurl=https://web.archive.org/web/20090331094810/http://www.nasa.gov/topics/solarsystem/features/asteroid_treasure_hunt.html |archivedate=March 31, 2009 }}</ref><ref>{{cite journal |date=February 25, 2009 |title=Found: Pieces of meteorite spotted before impact |url=https://www.newscientist.com/article/mg20126974.800-found-pieces-of-meteorite-spotted-before-impact.html |journal=New Scientist |issue=2697 |page=15 }}</ref><ref>{{cite journal |last=Shiga |first=D. |date=February 19, 2009 |title=First tracked space rock recovered after impact |url=https://www.newscientist.com/article/dn16635-first-tracked-space-rock-recovered-after-impact.html |journal=New Scientist }}</ref><ref>{{cite journal |last=Courtland |first=R. |date=March 25, 2009 |title=Meteorite hunters 'strike gold' in Sudan |url=https://www.newscientist.com/article/dn16843-meteorite-hunters-strike-gold-in-sudan.html |journal=New Scientist |accessdate=April 4, 2009 |archiveurl=https://www.webcitation.org/5hDZNqGfI?url=http://www.newscientist.com/article/dn16843-meteorite-hunters-strike-gold-in-sudan.html |archivedate=June 1, 2009 |df= }}</ref><ref>{{cite journal |last=Kwok |first=R. |date=March 26, 2009 |title=Astronomy: The Rock That Fell to Earth |url=http://www.nature.com/news/2009/090325/full/458401a.html |journal=Nature |volume=458 |issue= 7237|pages=401–403 |doi=10.1038/458401a |pmid=19325604 }}</ref> Samples of the Almahata Sitta meteorite were sent for analysis to a consortium of researchers led by Jenniskens, the Almahata Sitta consortium, including NASA Ames Research Center in California, the Johnson Space Center in Houston, the Carnegie Institution of Washington, and Fordham University in New York City. The first sample measured was an anomalous ultra-fine-grained porous polymict ureilite achondrite, with large carbonaceous grains. Reflectance spectra of the meteorite, combined with the astronomal observations, identified asteroid 2008 TC₃ as an F-type asteroid class. These fragile anomalous dark carbon-rich ureilites are now firmly linked to the group of F-class asteroids.<ref name="Jenniskens2009" /> Amino acids have been found on the meteorite.<ref>{{cite book |date=December 16, 2010 |title=Life's Building Blocks Found on Surprising Meteorite |url=http://www.space.com/scienceastronomy/asteroid-collision-meteoroid-amino-acids-life-101215.html |publisher=Space.com |accessdate=August 13, 2013}}</ref> The nanodiamonds found in the meteorite were shown to have grown slowly, implying that the source is another planet in the solar system.<ref>{{cite journal | last=Nabiei | first=Farhang | last2=Badro | first2=James | last3=Dennenwaldt | first3=Teresa | last4=Oveisi | first4=Emad | last5=Cantoni | first5=Marco | last6=Hébert | first6=Cécile | last7=El Goresy | first7=Ahmed | last8=Barrat | first8=Jean-Alix | last9=Gillet | first9=Philippe | title=A large planetary body inferred from diamond inclusions in a ureilite meteorite | journal=Nature Communications | publisher=Springer Nature | volume=9 | issue=1 | pages=1327 | date=April 17, 2018 | issn=2041-1723 | doi=10.1038/s41467-018-03808-6 | pmid=29666368 | pmc=5904174 |bibcode=2018NatCo...9.1327N|url=https://www.nature.com/articles/s41467-018-03808-6 |display-authors=1}}</ref> Richard Kowalski, who discovered the object, received a tiny fragment of Almahatta Sitta, a gift from friends and well-wishers on the Minor Planet Mailing List, which Kowalski founded in order to help connect professional and amateur astronomers.<ref>{{cite book |last=Lakdawalla |first=E. |date=September 20, 2009 |title=A piece of an asteroid returns to the telescope that discovered it |url=http://www.planetary.org/blog/article/00002104/ |publisher=The Planetary Society |accessdate=October 7, 2009}}</ref> {{clear}} ==Meteorites== {{main|Meteorites}} [[Image:Mackinac Island.jpg|thumb|right|250px|This is a natural color image of the weathered iron meteorite "Mackinac Island". Credit: NASA.{{tlx|free media}}]] The visible path of a meteoroid that enters the Earth's atmosphere (or another body's) atmosphere is called a '''''meteor''''', or colloquially a '''''shooting star''''' or '''''falling star'''''. If a meteoroid reaches the ground and survives impact, then it is called a meteorite. {{clear}} ==See also== {{div col|colwidth=20em}} * [[Radiation astronomy/Asteroids|Asteroid radiation astronomy]] * [[Radiation astronomy/Comets|Cometary radiation astronomy]] * [[Radiation astronomy/Fieries|Fiery meteor radiation astronomy]] * [[Radiation astronomy/Kuiper belts|Kuiper belt radiation astronomy]] * [[Radiation/Meteors|Meteor radiation astronomy]] * [[Radiation astronomy/Showers|Meteor shower radiation astronomy]] * [[Radiation astronomy/Oort clouds|Oort cloud radiation astronomy]] * [[Radiation astronomy/Rocks|Rock radiation astronomy]] {{Div col end}} ==References== {{reflist|2}} ==External links== <!-- footer templates --> {{Principles of radiation astronomy}}{{tlx|Radiation astronomy resources}}{{Sisterlinks|Meteoroid radiation astronomy}} <!-- footer categories --> [[Category:Radiation astronomy/Lectures]] ivbeyxhfrt9eso5x274dwqi8u2m5jdk 0 234509 2416458 2398996 2022-08-19T08:50:35Z Marshallsumter 311529 /* Eta Virginids */ wikitext text/x-wiki [[Image:Aurigids - Jeremie Vaubaillon.jpg|right|thumb|300px|The Aurigid meteor shower is observed by a group of astronomers on a NASA mission at 47,000 feet. Credit: Jeremie Vaubaillon, Caltech, NASA.{{tlx|free media}}]] Meteors may occur in showers, which arise when the Earth passes through a trail of debris left by a comet, or as "random" or "sporadic" meteors, not associated with a specific single cause. A number of specific meteors have been observed, largely by members of the public and largely by accident, but with enough detail that orbits of the meteoroids producing the meteors have been calculated. All of the orbits passed through the asteroid belt.<ref name=Uregina>{{ cite book |url=http://uregina.ca/~astro/mb_5.html |title=Diagram 2: the orbit of the Peekskill meteorite along with the orbits derived for several other meteorite falls |publisher=Uregina.ca |date= |accessdate=2011-09-16 }}</ref> "Named meteor showers recur at approximately the same dates each year. They appear to "radiate" from a certain point in the sky (the radiant) and vary in the speed, frequency and brightness of the meteors."<ref name=ListofMeteorShowers>{{ cite web |author=[[w:User:Noyster|Noyster]] |title=List of meteor showers |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=7 September 2017 |url=https://en.wikipedia.org/wiki/List_of_meteor_showers |accessdate=29 June 2019 }}</ref> As of November 2018, there are 112 established meteor showers.<ref>[https://www.ta3.sk/IAUC22DB/MDC2007/Roje/sortowanie_roje.php?corobic_roje=1 Meteor Data Centre, IAU] </ref> {{clear}} ==Radiants== [[Image:Radiantrp.jpg|right|thumb|upright=1.2|250px|Image shows a meteor shower, with the radiant marked by o. Credit: [[c:user:Anton~commonswiki|Anton~commonswiki]].]] The radiant, apparent radiant, or radiant point of a meteor shower is the celestial sphere point in the sky from the point of view of a terrestrial observer that the paths of meteors appear to originate.<ref>[http://earthobservatory.nasa.gov/Glossary/index.php?mode=alpha&seg=q&segend=s Earth Observatory Glossary: Radiant] on NASA.gov</ref> {{clear}} ==Nomenclature Rules for Meteor Showers== "The following nomenclature rules are adopted for meteor showers, keeping in mind that it is not always known precisely during discovery when is the peak of a meteor shower and what is the position of the radiant at that time. For known showers, the Working Group may choose a traditionally accepted name (e.g., alpha-Monocerotids) over the more correct name after a radiant has been established (which would have suggested the name of delta-Canis Minorids)."<ref name=Rudawska>{{ cite web |author=R. Rudawska and T.J. Jopek |title=Nomenclature Rules for Meteor Showers |publisher=Meteor Data Center |location= |date=25 June 2019 |url=https://www.ta3.sk/IAUC22DB/MDC2007/Dokumenty/shower_nomenclature.php |accessdate=30 June 2019 }}</ref> "The general rule is that a meteor shower (and a meteoroid stream) should be named after the constellation that contains the nearest star to the radiant point, using the possessive Latin form. The possessive Latin name for the constellations end in one of seven declensions:"<ref name=Rudawska/> * ae (e.g., Lyrae), * is (e.g., Leonis), * i (e.g., Ophiuchi), * us (e.g.,Doradus), * ei (e.g., Equulei), * ium (e.g., Piscium), or * orum (e.g., Geminorum). "Custom is to replace the final suffix for '-id', or plural '-ids'. Meteors from Aquarius (Aquarii) are Aquariids, not Aquarids. An exception is made for meteors from the constellation of Hydrus,which will be called 'Hydrusids', in order not to confuse with meteors from the constellation of Hydra."<ref name=Rudawska/> "When the constellation name has two parts, only the second declension is to be replaced by 'id'. Hence, meteors from Canes Venatici (Canum Venaticorum) would be 'Canum Venaticids'. When two constellations are grouped together, a bracket is used and both constellation names will have 'id'. Hence, Puppids-Velids. As a guideline, the order of the constellations should be in the same sequence as the radiant daily motion."<ref name=Rudawska/> "If a higher precision is needed, then the shower is named after the nearest (if in doubt: brightest) star with a Greek letter assigned, as first introduced in the Uranometria atlas by Johann Bayer (1603), or one with a later introduced Roman letter. If in doubt, the radiant position at the time of the peak of the shower (in the year of discovery) should be taken. Hence, the meteors of comet IRAS-Araki-Alcock would be named 'eta-Lyrids'."<ref name=Rudawska/> "In that case, if a meteor shower radiant is near the border of a constellation and the nearest such star is in the neighboring constellation, then the shower is named after that star."<ref name=Rudawska/> "Following existing custom, one may add the name of the month to distinguish among showers from the same constellation. In this case, one could call the shower from comet IRAS-Araki-Alcock the 'May Lyrids', in order to differentiate from the more familiar 'April Lyrids'."<ref name=Rudawska/> "For daytime showers, it is custom to add 'Daytime', hence the name for the 'Daytime Arietids' in June as opposed to the Arietids in October. As a guideline, the stream radiant should be less than 32 degs from the Sun to be called a daytime shower. This ensures that no where is the radiant more than 20 degs above the horizon at the start of local Nautical twilight."<ref name=Rudawska/> "South and North refer to 'branches' of a shower south and north of the ecliptic plane (stricktly the orbital plane of Jupiter), resulting from meteoroids of the same (original) parent body. Because they have nearly the same longitude of perihelion at a given solar longitude (the argument of perihelion and longitude of ascending node differing by 180 degrees between South and North), the two branches are active over about the same time period."<ref name=Rudawska/> "If the meteoroid stream is encountered at the other node, it is customary to speak of 'twin showers'. The Orionids and eta-Aquariids are twin showers, even though each represent dust deposited at different times and are now in quite different orbits. As a matter of custom, twin showers and the north and south branches of a stream carry different names. Meteor showers are not to be named after their parent bodies (e.g., Giacobinids, IRAS-Araki-Alcockids). The names of comets tend not to be Latin, making the naming not unique. Also, comet names can change when they get lost and are recovered."<ref name=Rudawska/> "The Working Group for Meteor Shower Nomenclature will choose among possible alternative proposed names for newly identified meteor showers, in order to establish a unique name for each meteor shower (e.g., eta-Lyrids, not May Lyrids)."<ref name=Rudawska/> ===Potential meteor shower nomenclatures=== {{div col|colwidth=12em}} # Andromeda Andromedids # Anser Anserids # Antinous Antinousids # Antlia Antliids # Apus Apodids # Aquarius Aquariids # Aquila Aquilids # Ara Arids # Argo Navis Argus Navids # Aries Arietids # Officina Typographica Officinae Typographicids # Auriga Aurigids # Globus Aerostaticus Globus Aerostaticids # Boötes Boötids # Caelum Caelids # Camelopardalis Camelopardalids # Cancer Cancrids # Canes Venatici Canum Venaticids # Canis Major Canis Majorids # Canis Minor Canis Minorids # Capricornus Capricornids # Caput Medusae Caput Medusids # Carina Carinids # Cassiopeia Cassiopeiids # Centaurus Centaurids # Cepheus Cepheids # Cerberus Cerberids # Cetus Cetids # Chamaeleon Chamaeleontids # Circinus Circinids # Columba Columbids # Coma Berenices Comae Berenicids # Corona Australis Coronae Australids # Corona Borealis Coronae Borealids # Corvus Corvids # Crater Craterids # Crux Crucids # Custos Messium Custus Messiids # Cygnus Cygnids # Delphinus Delphinids # Dong’ou Dong’ouids # Dorado Doradids # Draco Draconids # Equuleus Equuleids # Eridanus Eridanids # Felis Felids # Fornax Fornacids # Gemini Geminids # Gloria Frederici Gloriae Fredericids # Grus Gruids # Hercules Herculids # Horologium Horologiids # Hut Hutids # Hydra Hydrids # Hydrus Hydrusids # Indus Indids # Lacerta Lacertids # Legs Legsids # Leo Leonids # Leo Minor Leonis Minorids # Lepus Leporids # Libra Librids # Lupus Lupids # Lynx Lyncids # Lyra Lyrids # Machina Electrica Machinae Electricids # Mensa Mensids # Microscopium Microscopiids # Monoceros Monocerotids # Mons Mænalus Mons Mænalids # Musca Muscids # Noctua Noctuids # Norma Normids # Octans Octantids # Ophiuchus Ophiuchids # Orion Orionids # Pavo Pavonids # Pegasus Pegasids # Perseus Perseids # Phoenix Phoenicids # Pictor Pictorids # Pisces Piscids # Piscis Australis Piscis Austrinids # Psalterium Georgii Psalterii Georgiids # Puppis Puppids # Pyxis Pyxidids # Quadrans Muralis Quadrantids # Rangifer Rangiferids # Reticulum Reticulids # Sagitta Sagittids # Sagittarius Sagittariids # Sarvvis Sarvvids # Scorpius Scorpiids # Sculptor Sculptorids # Scutum Scutids # Serpens Caput Serpentids # Serpens Cauda Serpentids # Serpentarius Serpentariids # Sextans Sextantids # Tarandus Tarandids # Taurus Taurids # Telescopium Telescopiids # Telescopium Hershelii Telescopii Hersheliids # Tianmiao Tianmiaids # Triangulum Triangulids # Triangulum Australe Trianguli Australids # Tucana Tucanids # Taiwei Taiweids # Ursa Major Ursae Majorids # Ursa Minor Ursae Minorids # Ursus Ursids # Vela Velorids # Virgo Virginids # Volans Volantids # Vulpecula Vulpeculids {{Div col end}} ==Andromedids== [[Image:Andromedid_meteors,_November_1872.jpg|thumb|right|300px|The Andromedids of 27 November 1872 is a product of the breakup of Biela's Comet several decades previously. Credit: Amedee Guillemin.{{tlx|free media}}]] The Andromedids meteor shower is associated with Biela's Comet, the showers occurring as Earth passes through old streams left by the comet's tail. The comet was observed to have broken up by 1846; further drift of the pieces by 1852 suggested the moment of breakup was in either 1842 or early 1843, when the comet was near Jupiter.<ref name="Space.com">[http://www.space.com/5200-mother-meteor-storms.html The Mother of All Meteor Storms] (Space.com April 2008)</ref><ref name=Jenniskens2007>{{cite journal |last=Jenniskens |first=Peter |author2=Vaubaillon, J. R. M. |title=3D/Biela and the Andromedids: Fragmenting versus Sublimating Comets |journal=Astronomical Journal |volume=134 |issue=3 |pages=1037–1045 |date=2007 |doi=10.1086/519074 |bibcode=2007AJ....134.1037J|url=http://authors.library.caltech.edu/12800/1/JENaj07b.pdf}}</ref> The breakup led to particularly spectacular showers in subsequent cycles (particularly in 1872 and 1885).<ref name="Kronk"/><ref name="Hergenrother">{{cite web |title=The Meteor Storms of November – Part I – The Andromedids |publisher=The Transient Sky Blog |author=Carl W. Hergenrother |url=http://transientsky.wordpress.com/2008/11/15/the-meteor-storms-of-november-part-i-the-andromedids/ |accessdate=2011-04-18}}</ref> Radiant of the Andromedids in December 2013 is near γ&nbsp;Cassiopeiae (near the middle of the ''W'').<ref name="Sky&Tel131208">{{cite web |date=8 December 2013 |title=An Outburst of Andromedid Meteors |publisher=Sky & Telescope |author=Kelly Beatty |url=http://www.skyandtelescope.com/astronomy-news/observing-news/an-outburst-of-andromedid-meteors/ |accessdate=2013-12-09}}</ref> Right ascension = {{RA|01|36}}<ref name="IMO2012">{{cite web |title = Meteor Activity Outlook for November 17-23, 2012 |publisher = International Meteor Organization (IMO) |url = http://www.imo.net/node/1450 |accessdate = 2012-12-11 |archive-url = https://web.archive.org/web/20130622102325/http://imo.net/node/1450 |archive-date = 2013-06-22 |df = }}</ref> and Declination = {{DEC|+37}}<ref name="IMO2012"/> Occurs during September 25 – December 6,<ref name="Kronk"/> date of peak is November&nbsp;9<ref name="IMO2012"/>, Velocity = 19 km/s<ref name="IMO2012"/> and its Zenithal hourly rate = 3<ref name="IMO2012"/>. The first known sighting of the Andromedids was December 6, 1741, over St Petersburg, Russia.<ref name="Hergenrother"/> The 1872 shower consisted mainly of faint (5th to 6th magnitude) meteors with "''broad and smoke-like''" trains and a predominantly orange or reddish colouration.<ref name=ajs153>''American Journal of Science'', Third Series, V (Jan-Jun 1873), 153</ref> The same shower produced at least 58,600 visible meteors between 5.50 and 10.30 pm, observed in England and that the meteors were much slower than the Leonids], with noises "''like very distant gun-shots''" several times to the north-west.<ref name=ajs152>''AJS'', V, 152</ref> In Burma, the 1885 shower was perceived as a fateful omen and was indeed followed swiftly by the collapse of the Konbaung dynasty and the conquest by Britain.<ref>Lonely Planet Myanmar, 10th edition page 256</ref> The November 27, 1885, shower was the occasion of the first known photograph of a meteor, taken by Austro-Hungarian astronomer, Ladislaus Weinek, who caught a 7 mm-long trail on a plate at his Prague observing station.<ref name=SHughes2012>{{cite book |last=Hughes |first=Stefan |title= Catchers of the Light: The Astrophotographers' Family History |date=2012 |url= http://www.catchersofthelight.com |publisher= S. Hughes |isbn= 162050961X |pages=457}}</ref> Since the 19th century the Andromedids have faded so substantially that they are no longer generally visible to the naked eye, though some activity is still observable each year in mid-November given suitable detection equipment.<ref name="Hergenrother"/> In recent years, peak activity had been less than three meteors per hour, around November 9<ref name="IMO2012"/> to 14.<ref name="Kronk"/> Andromedid activity of November comes from the newest streams, while that of early December comes from the oldest.<ref name="Kronk"/> On December 4, 2011, six Canadian radar stations detected 50 meteors in an hour. The activity was likely from the 1649 stream.<ref name=wieg120926>{{cite journal |title=The return of the Andromedids meteor shower |journal=Astronomical Journal |volume=145 |issue=3 |pages=70 |doi=10.1088/0004-6256/145/3/70 |date=26 Sep 2012 |author=Paul A. Wiegert |author2=Peter G. Brown |author3=Robert J. Weryk | author-link3=Robert Weryk|author4=Daniel K. Wong |arxiv = 1209.5980 |bibcode = 2013AJ....145...70W }}</ref> On December 8, 2013, Meteor specialist Peter Brown reported that the Canadian Meteor Orbit Radar had recorded an outburst from the Andromedid meteors in the past 24 hours.<ref name="Sky&Tel131208"/> Scientists postulate a somewhat weaker return in 2018, but a yield of up to 200 meteors an hour in 2023.<ref name=wieg120926/><ref name=kcros121019>{{cite web |last=Croswell |first=Ken |title=The Return of a Great 19th-Century Meteor Shower |url=http://www.scientificamerican.com/article.cfm?id=return-of-a-great-19th-century-meteor-shower |publisher=Scientific American |accessdate=19 October 2012}}</ref> Canadian Meteor Orbit Radar (CMOR) data also detected a spike of 30 meteors per hour on November 27, 2008.<ref name=wieg120926/> During the 2012 shower an inconspicuous maximum occurred on November 9.<ref name="IMO2012"/> {{clear}} ==Anthelion== "The center of the large Anthelion (ANT) radiant is currently located at 19:24 (291) -22. This position lies in eastern Sagittarius, 1 degree east of the bright planet Saturn. This area of the sky is best placed near 02:00 LDST when it lies highest above the southern horizon. Due to the large size of this radiant, anthelion activity may also appear from Scutum as well as Sagittarius. Rates at this time should be near 2 per hour as seen from mid-northern latitudes (45 N) and 3 per hour as seen from the southern tropics (S 25). With an entry velocity of 30 km/sec., the average anthelion meteor would be of slow velocity."<ref name=Lunsford>{{ cite journal |author=Robert Lunsford |title=Meteor Activity Outlook for June 29-July 5, 2019 |journal= |date=June 2019 |volume= |issue= |pages= |url=https://amsmeteors.org/2019/06/meteor-activity-outlook-for-june-29-july-5-2019/ |arxiv= |bibcode= |doi= |pmid= |accessdate=1 July 2019 }}</ref> RA 18:24 (291 RA in degrees) Declination -22, entry velocity 30 km/s, hourly rate = 2-3, Class 11.<ref name=Lunsford/> ==Alpha Antliids== "From the IMO working list of meteor showers, 𝛅-Leonids and meteors from the Antihelion Source were detected. For the first time, 𝛂-Antliids were detected in the optical domain."<ref name=Molau>{{ cite journal |author=Sirko Molau and Javor Kac |title=Results of the IMO Video Meteor Network - February 2009 |journal=WGN |date=April 2009 |volume=37 |issue=2 |pages=75-76 |url=http://adsabs.harvard.edu/full/2009JIMO...37...75M |arxiv= |bibcode=2009JIMO...37...75M |doi= |pmid= |accessdate=1 July 2019 }}</ref> A "possible weak shower from February 2 to 7 (solar longitude 313-318°), [is] based on just 66 shower members. On February 4, the average radiant lies at 𝛂 = 162°, 𝛅 = -14° [...]. The mean meteor shower velocity is 45 km/s."<ref name=Molau/> The "IAU Working list of meteor showers lists the 𝛂-Antliids (AAN) shower (No. 110) with quite similar characteristics (𝛂 = 140°, 𝛅 = -10°, v_ini = 43 km/s) on February 2. It appears that the shower was first detected by the Advanced Meteor Orbit Radar (AMOR) at 𝛂 = 162°, 𝛅 = -13°, v_ini = 43 km/s (Galligan & Baggaley, 2002). The shower was also found by the Canadian Meteor Orbit Radar (CMOR) (Brown ''et al.'', 2008). According to their latest analysis, 𝛂-Antliids are active between solar longitude 308 and 321 degrees. On February 4 the shower lies at 𝛂 = 162°, 𝛅 = -12° [...]. Also the meteor shower velocity (44 km/s) matches well to the video data. Thus the existence of the 𝛂-Antliids is now also proved in the optical domain. [...] The activity profile [...] shows that the ZHR will hardly exceed 1 in the full activity interval."<ref name=Molau/> ==Eta Aquariids== [[Image:Animation of 1P／Halley orbit - 1986 apparition.gif|260px|thumb|right|Animation is of 1P/Halley orbit - 1986 apparition.{{legend2|magenta| 1P/Halley }}{{legend2|Royalblue|Earth }}{{legend2|Yellow|Sun}}. Credit: [[c:user:Phoenix7777|Phoenix7777]].]] The current orbit of Halley's Comet does not pass close enough to the Earth to be a source of meteoric activity.<ref name="AMS2013">{{cite web |title=Viewing the 2013 Eta Aquariid Meteor Shower |publisher=American Meteor Society |author=Robert Lunsford |url=http://www.amsmeteors.org/2013/05/viewing-the-2013-eta-aquariid-meteor-shower/ |accessdate=2013-05-04}}</ref> The shower is best viewed from the equator to 30 degrees south latitude.<ref name="AMS2013"/> The meteoroids are from very old ejection from the parent 1P/Halley and are trapped probably in resonances to Jupiter's orbit (similar to the Orionids observed between 2007 and 2010).<ref name="Calendar">{{cite web |title=IMO Meteor Shower Calendar |publisher=International Meteor Organization |author=Jürgen Rendtel |url=https://www.imo.net/resources/calendar/ |accessdate=2018-05-06}}</ref> The peak ZHR reached 135 ± 16.<ref name="Rendtel">{{citation |title=Meteor Shower Workbook, p. 23-24 |publisher=International Meteor Organization |author=Jürgen Rendtel}}</ref> Updated information on the expected time and rates of the shower is provided through the annual IMO Meteor Shower Calendar.<ref name="Calendar">{{cite web |title=IMO Meteor Shower Calendar |publisher=International Meteor Organization |author=Jürgen Rendtel |url=https://www.imo.net/resources/calendar/ |accessdate=2018-05-06}}</ref> "At 66 kilometers (41 miles) per second, they appear as fast streaks, faster by a hair than their sisters, the Eta Aquarids of May. And like the Eta Aquarids, the brightest of family tend to leave long-lasting trains. Fireballs are possible three days after maximum."<ref name=Levy>{{ cite book |author=David H. Levy and Stephen J. Edberg |title=Observe Meteors: The Association of Lunar and Planetary Observers Meteor Observer's Guide |publisher=Astronomical League |location= |year=1986 |editor= |pages=55 |url=https://books.google.com/books?id=0cfgAAAAMAAJ |arxiv= |bibcode= |doi= |pmid= |isbn= |accessdate=2012-10-19 }}</ref> {{clear}} ==Southern Delta Aquariids== Meteors radiating from near the star Delta Aquarii (declension "-i") are called the Delta Aquariids. Parent body = 96P/Machholz<ref name="Jenniskens2006" /> Zenithal hourly rate = 16<ref name="Jenniskens2006" /> The Southern Delta Aquariids<ref name="imo2009">International Meteor Organization (IMO) [http://www.imo.net/calendar/2009#sda Meteor Shower Calendar 2009]</ref> are a meteor shower visible from mid July to mid August each year with peak activity on 28 or 29 July. The shower originated from the breakup of what are now the Marsden and Kracht Sungrazing comets.<ref name="Jenniskens2006"/> Both accurate velocity and orbit of the δ Aquariids was determined with a "more selective beamed aerial" (echo radio) to identify probable member meteors and plotted an accurate orbital plane, as a broad "system of orbits" that are probably "connected and produced by one extended stream."<ref>{{cite journal |first1=Ernst |last1=Öpik |date=September 1950 |title=Interstellar Meteors and Related Problems |journal=Irish Astronomical Journal |volume=1 |issue=3 |pages=80–96 |bibcode=1950IrAJ....1...80O}}</ref> Radiant: 22:40 -16.4°, Velocity: 26 miles/sec (medium - 41km/sec).<ref name= Hankey >{{ cite web |author=Mike Hankey |title=METEOR SHOWER CALENDAR 2019-2020 |publisher=American Meteor Society |location= |date=2019 |url=https://www.amsmeteors.org/meteor-showers/meteor-shower-calendar/#2009 |accessdate=30 June 2019 }}</ref> ==Arietids== The '''Arietids''' are a strong meteor shower that lasts from May 22 to July 2 each year, and peaks on June 7. The Arietids, along with the Zeta Perseids, are the most intense daylight meteor showers of the year.<ref name="NASA2000">{{Cite web |title=June's Invisible Meteors |publisher=[[NASA]] |author=Tony Phillips |date=2000 |url=https://science.nasa.gov/science-news/science-at-nasa/2000/ast06jun_1m |archiveurl=https://web.archive.org/web/20071102152836/https://science.nasa.gov/headlines/y2000/ast06jun_1m.htm |archivedate=November 2, 2007 |accessdate=September 7, 2007 }}</ref> The source of the shower is unknown, but scientists suspect that they come from the asteroid 1566 Icarus,<ref name="NASA2000" /><ref name="spaceweather2007">{{Cite web |title=Daylight Meteors: The Arietids |publisher=spaceweather.com |url=http://www.spaceweather.com/meteors/arietids.html |archiveurl= https://web.archive.org/web/20070927225531/http://www.spaceweather.com/meteors/arietids.html |archivedate= 27 September 2007 |accessdate=September 7, 2007}}</ref> although the orbit also corresponds similarly to 96P/Machholz.<ref name=Ohtsuka2003>{{cite journal |title=On the Association among Periodic Comet 96P/Machholz, Arietids, the Marsden Comet Group, and the Kracht Comet Group. |last1=Ohtsuka |first1=Katsuhito |last2=Nakano |first2=Syuichi |last3=Yoshikawa |first3=Makoto |journal=Publications of the Astronomical Society of Japan |volume=55 |issue=1 |pages=321–324 |year=2003 |doi=10.1093/pasj/55.1.321}}</ref> First discovered at Jodrell Bank Observatory in England during the summer of 1947, the showers are caused when the Earth passes through a dense portion of two interplanetary meteoroid streams, producing an average of 60 shooting stars each hour, that originate in the sky from the constellation Aries and the constellation Perseus.<ref name="Kronk"/> However, because both constellations are so close to the Sun when these showers reach their peak, the showers are difficult to view with the naked eye.<ref name="NASA2000" /> Some of the early meteors are visible in the very early hours of the morning, usually an hour before dawn.<ref name="newsgroup">{{Cite web |title=Listen...to the Arietids!! |accessdate=September 7, 2007 |publisher=The Astronomy Connection |date=1999 |author=James Turley |url=http://observers.org/tac.mailing.list/2001/june/0109.html |archiveurl=https://web.archive.org/web/20070929191104/http://observers.org/tac.mailing.list/2001/june/0109.html |archivedate=September 29, 2007 |df= }}</ref> The meteors strike Earth's atmosphere at speeds around 39&nbsp;km/s.<ref name="NASA2000" /> By June 22 the radiant has migrated to the constellation Taurus (3h 51m +27) which is the same constellation that the Beta Taurids peak on June 28.<ref name=ams20190622>[https://www.amsmeteors.org/2019/06/meteor-activity-outlook-for-june-22-28-2019 Meteor Activity Outlook for June 22-28, 2019]</ref> ==Aurigids== Aurigids is a meteor shower occurring primarily within September.<ref>[http://www.imo.net/calendar/2009#aur © 1997-2011 International Meteor Organization] retrieved 16:55 11.10.11</ref> The comet Kiess (C/1911 N1) is the source of the material that causes the meteors, with an orbital period as approximately 1800 to 2000 years, and showers observed in the years 1935, '86, '94 and 2007 .<ref>[http://adsabs.harvard.edu/abs/2007EOSTr..88..317J Jenniskens, P. and J. Vaubaillon (2007), An unusual meteor shower on 1 September 2007, Eos Trans. AGU, 88(32), 317, ] {{doi|10.1029/2007EO320001}} 16:14 11.10.11</ref><ref> [https://web.archive.org/web/20160304023351/http://www.astro.amu.edu.pl/~jopek/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=206&colecimy=0] 4 March 2016 retrieved 16:25 11.10.11</ref> The Alpha were discovered by C. Hoffmeister and A. Teichgraeber, during the night of 31 August 1935.<ref name=Kronk>Gary W. Kronk website [https://web.archive.org/web/20170915123440/http://meteorshowersonline.com/showers/alpha_aurigids.html] ] 15 September 2017 17:35 11.10.11</ref><ref>[https://archive.is/20120913235941/http://www.skyandtelescope.com/observing/home/9332696.html article written by Joe Rao in Sky and Telescope magazine 23 August 2007] approx. 17:45 retrieved 11.10.11</ref> ==June Bootids== [[Image:Bootiden 2016.png|right|thumb|300px|The meteor shower Bootids during the maximum in 2016. Credit: [[c:user:HemelWaarnemen|HemelWaarnemen]].{{tlx|free media}}]] "The June Bootid meteor shower is active each year from June 26th until July 2nd. It peaks on June 27th. Normally the shower is very weak, but occasional outbursts produce a hundred or more meteors per hour."<ref name=Space>{{ cite web |author=SpaceWeather |title=The Unpredictable June Bootids |publisher=SpaceWeather |location= |date=24 June 2005 |url=http://spaceweather.com/meteors/junebootids.html |accessdate=27 June 2019 }}</ref> "The shower's radiant lies in the constellation Bootes (right ascension 14h 56m, declination 48°)."<ref name=Space/> "The source of the June Bootids is periodic comet 7P/Pons-Winnecke."<ref name=Space/> "June Bootid meteoroids hit Earth's atmosphere with a velocity of 18 km/s (40,000 mph).They are considered slow-moving meteors."<ref name=Space/> "On June 27th, 1998, northern sky watchers were surprised when meteors suddenly began to stream out of the constellation Bootes. Observers saw as many as 100 meteors per hour during the 7-hour-long outburst. It wasn't the first time: similar outbursts from Bootes had been recorded in 1916, 1921 and 1927. Astronomers call these unpredictable meteors the June Bootids."<ref name=Space/> {{clear}} ==Delta Cancrids== The Delta Cancrids is a medium strength meteor shower lasting from December 14 to February 14,<ref name="MeteonlineCancrids">{{cite web|url=http://meteorshowersonline.com/showers/delta_cancrids.html|title=Delta Cancrids|publisher=Meteor Showers Online|pages=1|accessdate=2008-08-05}}</ref> the main shower from January 1 to January 24.<ref name="berlin">{{cite web |url=http://www.surveyor.in-berlin.de/himmel/sky.07.01.html |title=The Sky over Berlin 1"07 |year=2007 |publisher=The Sky over Berlin |pages=1 |accessdate=2008-08-05 |archiveurl=https://web.archive.org/web/20080925220043/http://www.surveyor.in-berlin.de/himmel/sky.07.01.html |archivedate=2008-09-25 |df= }}</ref><ref name="imo">{{cite web|url=http://www.imo.net/calendar/2004/winter?PHPSESSID=1fb18a1401ae2541b2a2c.|title=January to March|year=2004|publisher=International Meteor Organization|pages=1|accessdate=2008-08-05|archiveurl=https://web.archive.org/web/20110610153126/http://www.imo.net/calendar/2004/winter?PHPSESSID=1fb18a1401ae2541b2a2c.|archivedate=2011-06-10|df=}}</ref> The radiant is located in the constellation of Cancer, near Delta Cancri. It peaks on January 17 each year, with only four meteors per hour.<ref name="berlin"/><ref name="imo"/> It was first discovered in 1872, but the first solid evidence of this phenomenon came in 1971.<ref name="MeteonlineCancrids"/> The source of this meteor shower is unknown, it has been suggested that it is similar to the orbit of asteroid 2001 YB5.<ref>{{cite web|url=http://www.meteorobs.org/maillist/msg24836.html|title=(meteorobs) asteroid 2001 YB5 and delta Cancrids|last=Langbroek|first=Marco|year=2002|publisher=meteorbs|pages=1|accessdate=2008-08-05|archiveurl=https://web.archive.org/web/20080704141212/http://www.meteorobs.org/maillist/msg24836.html|archivedate=2008-07-04|df=}}</ref> ==Alpha Capricornids== Alpha Capricornids is a meteor shower that takes place as early as 15 July and continues until around 10 August.<ref>{{cite web |url= http://encarta.msn.com/encyclopedia_761587655/alpha_capricornids.html |title= Alpha Capricornids: Encyclopedia Article |work=Encarta.msn.com |accessdate=2014-07-08|archiveurl=https://web.archive.org/web/20090419084015/http://encarta.msn.com/encyclopedia_761587655/Alpha_Capricornids.html |archivedate=2009-04-19|publisher= Encarta}}</ref> The parent body is asteroid 2002 EX12 [169P/NEAT], which in the return of 2005 was found weakly active near perihelion.<ref name="Jenniskens">{{cite journal|url=http://iopscience.iop.org/1538-3881/139/5/1822/ |title= Minor planet 2002 EX12 ( = 169P/NEAT) and the Alpha Capricornid shower|journal=Astronomical Journal |accessdate=2014-07-08 }}</ref> "Minor planet 2002 EX12 ... is identified as the parent body of the alpha Capricornid shower, based on a good agreement in the calculated and observed direction and speed of the approaching meteoroids for ejecta 4500-5000 years ago....The bulk of this matter still passes inside Earth's orbit, but will cross Earth's orbit 300 years from now. As a result, the alpha Capricornids are expected to become a major annual shower in 2220-2420 A.D., stronger than any current annual shower"<ref name="Jenniskens"/> The meteor shower was created about 3,500 to 5,000 years ago, when about half of the parent body disintegrated and fell into dust.<ref name="Jenniskens" /> The Alpha Capricornids are expected to become a major annual storm in 2220&ndash;2420 A.D., one that will be "stronger than any current annual shower."<ref name="Jenniskens" /> ==Alpha Centaurids== "The Alpha Centaurids are a meteor shower in the constellation Centaurus, peaking in early February each year. The average magnitude is around 2.5, with a peak of about three meteors an hour."<ref name=AlphaCentaurids>{{ cite web |author=[[w:User:Andrew Gray|Andrew Gray]] |title=Alpha Centaurids |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=28 September 2006 |url=https://en.wikipedia.org/wiki/Alpha_Centaurids |accessdate=29 June 2019 }}</ref> "They have been observed since 1969".<ref name=AlphaCentaurids/> "The Alpha Centaurids emanate from α=216°, δ=-60° [...] The Alpha Centaurids have maximum hourly rates of 3 [...] The Alpha Centaurids have an average magnitude of 2.45 [...] During 1979, members of the Western Australia Meteor Section (WAMS) managed to observe the "Alpha Centaurids" during February 2-18. At maximum on February 7, the radiant is at α=216°, δ=-59°. [...] First of all, the Alpha Centaurids are apparently a consistent shower, with Buhagiar assigning an hourly rate of 3, and WAMS observers detecting high rates of 2 (ZHR calculated as 8.56+/-4.94) in 1979. [...] The Alpha Centaurids may have been detected by radar at Adelaide Observatory during 1969. G. Gartrell and W. G. Elford operated the radar system during February 10-17. Two meteors were noted from a radiant of α=223°, δ=-61°, with the date of nodal passage being determined as February 15. Assuming these meteors are members of the Alpha Centaurids, then this stream orbit has an inclination near 105°, and a semimajor axis near 2.5 AU. This identification would also indicate that the radiant's daily motion is very close to +1° in α. The movement in δ can not be determined from the available observations."<ref name=Kronk2018>{{ cite web |author=Gary W. Kronk |title=Alpha & Beta Centaurids |publisher=Meteor Showers Online |location= |date=21 October 2018 |url=https://web.archive.org/web/20071130152917/http://meteorshowersonline.com/showers/alpha-beta_centaurids.html |accessdate=29 June 2019 }}</ref> ==Beta Centaurids== The "Beta Centaurids have a radiant of α=208°, δ=-58°. [...] the Beta Centaurids can reach hourly rates as high as 14. [...] the Beta Centaurids are probably about 1.6. [...] M. Buhagiar (Western Australia),[...] obtained observations of both Centaurid radiants during 1969-1980. In his "Southern Hemisphere Meteor Stream List" of 1980, Buhagiar listed two radiants which reached maximum on February 7. "Radiant 290" was active during February 6-8, from α=206°, δ=-57°, while "radiant 299" was active during February 5-9, from α=214°, δ=-64°. Both radiants were referred to as "Beta Centaurids." [...] During 1980, the same group observed members of the "Alpha Centaurids" during February 2-24. [...] the 1980 radiant is the Beta Centaurids. [...] The Beta Centaurids are apparently variable in activity, according to Buhagiar, with his 1969-1980 observations revealing high rates of 10 meteors per hour. WAMS observers obtained maximum rates of 11-14 per hour (ZHR calculated as 28.48+/-4.88) during a one-hour interval on 1980 February 8/9. [...] during 1980, 169 Beta Centaurids revealed an average magnitude of 1.6 (the latter number is an approximation by the Author based on a table published in the October 1980 issue of Meteor News)."<ref name=Kronk2018/> ==Kappa Cygnids== Kappa Cygnids, abbreviated KCG, is a minor meteor shower that takes place in August along with the larger Perseids meteor shower.<ref>{{cite web|url=http://www.imo.net/calendar/2009#kcg |title=IMO Meteor Shower Calendar 2009: Contents: July to September: Kappa Cygnids |publisher=IMO.net}}</ref> The Kappa Cygnids in 2009 were Active between August 3-August 25 August, with Peak of shower at August 17, and ZHR = 3 km/s.<ref>{{cite web|url=http://www.imo.net/calendar/2009#kcg|title=IMO Meteor Shower Calendar 2009: Contents: July to September: Alpha Capricornids |work=IMO.net |accessdate=2009-02-10}}</ref> ==Draconids== [[Image:Draconids outburst 2012-10-08.png|right|thumb|300px|Composite image of Draconid meteors taken from Petnica Meteor Group (http://www.meteori.rs) video camera stationed at Institute of Physics, Belgrade. Credit: Petnica Meteor Group.{{tlx|free media}}]] The October Draconids, in the past also unofficially known as the Giacobinids, are a meteor shower whose parent body is the periodic comet 21P/Giacobini-Zinner.<ref name="Kronk" /> They are named after the constellation Draco, where they seemingly come from. Almost all meteors which fall towards Earth ablate long before reaching its surface. The Draconids are best viewed after sunset in an area with a clear dark sky. RA {{RA|17.467}}<ref name="Kronk"/> and Declination = {{DEC|+54}}<ref name="Kronk"/>. Velocity = 20 km/s.<ref name=jimo37_5_161>{{citation | display-authors=1 | last1=Żołądek | first1=P. | last2=Wiśniewski | first2=M. | last3=Olech | first3=A. | last4=Krasnowski | first4=M. | last5=Kwinta | first5=M. | last6=Fietkiewicz | first6=K. | last7=Złoczewski | first7=K. | last8=Mularczyk | first8=K. | last9=Jonderko | first9=W. | last10=Spaleniak | first10=I. | last11=Gruszka | first11=Ł. | title=The 2004 Perseid meteor shower – Polish Fireball Network double station preliminary results | journal=Journal of the International Meteor Organization | volume=37 | issue=5 | pages=161–163 |date=October 2009 | bibcode=2009JIMO...37..161Z}}</ref> The 1933<ref name="KronkD">{{cite web|url=http://meteorshowersonline.com/showers/draconids.html|title=Draconids ("Giacobinids")|last=Kronk|first=Gary W |date=|website=Meteor Showers Online|archive-url=https://web.archive.org/web/20180627202229/http://meteorshowersonline.com/showers/draconids.html|archive-date=2018-06-27|accessdate=11 October 2017}}</ref><ref name="Wyle_1933">"The meteors from Giacobini's comet", Wylie, C. C., Popular Astronomy, Vol. 42, p.44, {{cite web|title=The meteors from Giacobini's comet|url=http://adsbit.harvard.edu/cgi-bin/nph-iarticle_query?bibcode=1934PA.....42...44W&db_key=AST&page_ind=0&data_type=GIF&type=SCREEN_VIEW&classic=YES|language=en|accessdate=2018-09-25}}</ref><ref name="storm1933">{{cite web |date=October 7, 2011 |title=Account of the 1933 Draconids meteor storm |publisher=International Meteor Organization (IMO) |author=John McFarland and Mark Bailey |url=http://www.imo.net/docs/ellison_1933_draconids_storm_account.txt |accessdate=2011-10-08}}</ref> and 1946<ref name="KronkD" /> Draconids had Zenithal Hourly Rates of thousands of meteors visible per hour, among the most impressive meteor storms of the 20th century. Rare outbursts in activity can occur when the Earth travels through a denser part of the cometary debris stream; for example, in 1998, rates suddenly spiked<ref>{{cite web|title=Giacobinids dazzle observers|url=https://science.nasa.gov/newhome/headlines/ast14oct98_2.htm|date=October 14, 1998}}</ref><ref>Arlt, R. [http://adsabs.harvard.edu/abs/1998JIMO...26..256A "Summary of 1998 Draconid Outburst Observations"], ''WGN, Journal of the International Meteor Organization, Vol. 26, p. 256-259'', 1998.</ref> and spiked again (less spectacularly) in 2005.<ref>Campbell-Brown, M.; Vaubaillon, J.; Brown, P.; Weryk, R. J.; Arlt, R. [http://adsabs.harvard.edu/abs/2006A&A...451..339C "The 2005 Draconid outburst"], ''Astronomy and Astrophysics, Volume 451, pp.&nbsp;339–344'', 2006.</ref> A Draconid meteor outburst occurred<ref name="2011results">{{cite web|title=Draconids show expected outburst|publisher=International Meteor Organization (IMO)|url=http://www.imo.net/draconids2011results|accessdate=2011-12-06|archiveurl=https://web.archive.org/web/20111213230937/http://www.imo.net/draconids2011results|archivedate=2011-12-13|df=}}</ref> as expected<ref name=West>{{cite news |author= Adrian West |title= The Draconid Meteor Shower – A Storm is Coming! |work= Universe Today |date= October 3, 2011 |url= http://www.universetoday.com/89430/the-draconid-meteor-shower-a-storm-is-coming/ |accessdate= 2011-10-03}}</ref><ref>{{cite web|title=Draconids Meteor Shower on 8 October 2011|url=http://www.imo.net/draconids2011|work=International Meteor Organization|archiveurl=https://web.archive.org/web/20110925171621/http://www.imo.net/draconids2011|archivedate=25 September 2011|df=}}</ref><ref>{{cite web|last=Beatty|first=Kelly|title=A Deluge of Draconids?|url=http://www.skyandtelescope.com/observing/highlights/104450349.html|work=Sky and Telescope|publisher=Highlights|accessdate=31 December 2010}}</ref> on 2011 October 8, though a waxing gibbous Moon reduced the number of meteors observed visually. "Observers in the UK and Northern Europe are ideally placed to see the peak of the Draconids. Unfortunately the peak occurs in the day time for North America. There will also be a bright Moon which may drown out many but the brightest meteors, but if predictions are correct, you will still see many. You may see Draconid meteors on the 7th an the 9th also, so it is worth going out and checking the skies."<ref name=West/> During the 2012 shower radar observations detected up to 1000 meteors per hour. The 2012 outburst may have been caused by the narrow trail of dust and debris left behind by the parent comet in 1959.<ref name="Barentsen">{{cite web |date=2012-10-08 |title=Draconids show outburst (again!) |author=Geert Barentsen |url=http://barentsen.wordpress.com/2012/10/08/draconids-2012/ |accessdate=2012-10-08}}</ref> {{clear}} ==Geminids== [[Image:Geminid 121407 1.jpg|thumb|right|250px|A Geminid meteor in 2007, seen from San Francisco. Credit: [https://sites.google.com/site/thebrockeninglory/ Brocken Inaglory].]] [[Image:Geminidák meteorraj maximuma 2007-ben.jpg|thumb|left|250px|Geminid meteors clearly show the position of the radiant. Credit: [[:hu:user:Berkó Ernő|Berkó Ernő]].]] The '''Geminids''' are a prolific meteor shower caused by the object 3200 Phaethon,<ref name=IAUC3881>{{cite web |author=Brian G. Marsden |title=IAUC 3881: 1983 TB and the Geminid Meteors; 1983 SA; KR Aur (Circular No. 3881) |url=http://www.cbat.eps.harvard.edu/iauc/03800/03881.html |publisher=Central Bureau for Astronomical Telegrams |date=1983-10-25 |accessdate=2009-05-18 |archiveurl=https://web.archive.org/web/20120501023951/http://www.cbat.eps.harvard.edu/iauc/03800/03881.html |archivedate=2012-05-01 |df= }}</ref> which is thought to be a Palladian asteroid<ref name="NG2010">{{cite web | author = Victoria Jaggard | title = Exploding Clays Drive Geminids Sky Show? | url = http://newswatch.nationalgeographic.com/2010/10/12/geminids_meteor_shower_driven_by_exploding_clays/ | publisher = National Geographic Society | date = 2010-10-12 | accessdate = 2010-10-18}}</ref> with a "rock&nbsp;comet" orbit.<ref name=Jewitt2010>{{cite journal | last = Jewitt | first = David | authorlink = | author2 = Li, Jing | title = Activity in Geminid Parent (3200) Phaethon | journal = The Astronomical Journal | volume = 140 | issue = 5 | pages = 1519–1527 | arxiv = 1009.2710 | date = 2010 | version = | accessdate = | doi = 10.1088/0004-6256/140/5/1519 | bibcode = 2010AJ....140.1519J }}</ref> {{clear}} ==Leonids== [[Image:Leonid Meteor.jpg|thumb|right|250px|The photograph shows the meteor, afterglow, and wake as distinct components of a meteor during the peak of the 2009 Leonid Meteor Shower. Credit: [[commons:User:Navicore|Navicore]].]] [[Image:Leonid meteor shower as seen from space (1997).jpg|thumb|left|250px|This photograph shows the Leonids as many begin contacting the Earth's atmosphere. Credit: NASA.]] "The Leonid meteor shower peaked early Saturday (Nov. 17 [2012]), and some night sky watchers caught a great view. The Leonids are a yearly meteor display of shooting stars that appear to radiate out of the constellation Leo. They are created when Earth crosses the path of debris from the comet Tempel-Tuttle, which swings through the inner solar system every 33 years."<ref name=Moskowitz>{{ cite book |author=Clara Moskowitz |title=Amazing Leonid Meteor Shower Photos Captured By Stargazers |publisher=SPACE.com |location= |date=November 17, 2012 |url=http://news.yahoo.com/amazing-leonid-meteor-shower-photos-captured-stargazers-163450853.html |accessdate=2012-11-18 }}</ref> {{clear}} ==Leonis Minorids== Parent body: C/1739 K1 (Zanotti).<ref name=Lignie>{{ cite web |author=Marc de Lignie and Hans Betlem |title=A double-station video look on the October meteor showers |journal=Work Group News (WGN) |volume=27 |issue=3-4 |pages=195-201 |date=August 1999 |url=http://adsabs.harvard.edu/abs/1999JIMO...27..195D |bibcode=1999JIMO...27..195D |accessdate=30 June 2019 }}</ref> ==June Librids== The "June Librid shower, which had not been observed since 1937, was active in 1992."<ref name=Elford>{{ cite journal |author=W. G. Elford, M. A. Cervera and D. I. Steel |title=Single station identification of radar meteor shower activity: the June Librids in 1992 |journal=Monthly Notices of the Royal Astronomical Society |date=15 September 1994 |volume=270 |issue=2 |pages=401–408 |url=https://academic.oup.com/mnras/article-pdf/270/2/401/3984847/mnras270-0401.pdf |arxiv= |bibcode= |doi=10.1093/mnras/270.2.401 |pmid= |accessdate=2 July 2019 }}</ref> The "Librid meteor shower (RA = 227.2°, Dec = -28.3°) [...] was optically observed by Cuno Hoffmeister on 1937 June 8-9, but, to the knowledge of the authors, not since (Hoffmeister 1948; Cook 1973; M. Gyssens & P. Roggemans, personal communication, 1992)."<ref name=Elford/> An "initial set of data collected on 1992 June 6-9 [...] indicate that the June Librid shower was active in 1992, having only previously been observed (visually) in 1937."<ref name=Elford/> Only "about 10 June Librids were detected in total."<ref name=Elford/> "The observations by Link (1975) had shown that there was a certain correlation between enhanced twilight scattering above 70 km and the occurrences of Orionid, Geminid and Librid meteor showers."<ref name=Thomsen>{{ cite journal |author=Nina Mateshvili, Iuri Mateshvili, Giuli Mateshvili, Lev Gheondjian and Zurab Kapanadze |title=Dust Particles in the Atmosphere during the Leonid Meteor Showers of 1998 and 1999 |journal=Leonid Storm Research |date=2000 |volume= |issue= |pages=489-504 |url=https://www.researchgate.net/profile/Joe_Zender/publication/226257591_The_ESOC_Meteor_alert_center_MALC_and_its_application_during_the_1999_Leonid_shower/links/0c9605328078e2c9da000000.pdf#page=494 |arxiv= |bibcode= |doi=10.1007/978-94-017-2071-7_35 |pmid= |accessdate=2 July 2019 }}</ref> ==Lyrids== [[Image:Lyrid meteor shower radiant point.jpeg|thumb|right|300px|Radiant point of the April Lyrid meteor shower is shown, active each year around April 22. Credit: NASA/Don Pettit.]] [[Image:Stunning Lyrids Over Earth at Night.jpg|left|thumb|300px|On the night of April 21, the 2012 Lyrid meteor shower peaked in the skies over Earth. While NASA allsky cameras were looking up at the night skies, astronaut Don Pettit aboard the International Space Station trained his video camera on Earth below. Credit: NASA/Don Pettit.{{tlx|free media}}]] The April Lyrids (LYR, IAU shower number 6)<ref name="IAU Meteor Shower Working List">{{cite web|url=http://www.astro.amu.edu.pl/~jopek/MDC2007/index.php |title=IAU Meteor Data Center|page=1|accessdate=2013-08-06}}</ref> is a meteor shower lasting from April 16 to April 26<ref name="MeteonlineLyrids">{{cite web|url=http://meteorshowersonline.com/lyrids.html |title=Lyrids|publisher=Meteor Showers Online|page=1|accessdate=2008-08-05}}</ref> The source of the meteor shower is particles of dust shed by the long-period Comet C/1861 G1 Thatcher.<ref name=Jenniskens2006>{{cite journal | last=Jenniskens | first = P. | title=Meteor Showers and their Parent Comets | publisher=Cambridge University Press | date=2006 | pages=790}}</ref> The Lyrids have been observed and reported since 687 BC; no other modern shower has been recorded as far back in time.<ref>{{cite web|last1=King|first1=Bob|title=The Lyrid Shower Kicks Off Year of Great Meteor Watching|url=http://www.skyandtelescope.com/observing/lyrid-shower-kicks-off-year-of-great-meteor-watching/|website=www.skyandtelescope.com|publisher=F+W Media, Inc.|accessdate=19 April 2018|date=18 April 2018}}</ref> The shower usually peaks on around April 22 and the morning of April 23. Counts typically range from 5 to 20 meteors per hour, averaging around 10.<ref name="MeteonlineLyrids"/> April Lyrid meteors are usually around magnitude +2. However, some meteors can be brighter, known as "Lyrid fireballs", cast shadows for a split second and leave behind smokey debris trails that last minutes.<ref name="spaceweather">{{cite web|url=http://www.spaceweather.com/meteors/lyrids/lyrids.html|title=the Lyrid meteor shower|date=2008|publisher=spaceweather.com|page=1|accessdate=2008-08-05}}</ref> Occasionally, the shower intensifies when the planets steer the one-revolution dust trail of the comet into Earth's path, an event that happens about once every 60 years.<ref name=Jenniskens2006/> The one-revolution dust trail is dust that has completed one orbit: the stream of dust released in the return of the comet prior to the current 1862 return. This mechanism replaces earlier ideas that the outbursts were due to a cloud of dust moving in a 60-year orbit.<ref name=Arter1997>{{cite journal | last=Arter | first=T. R. |author2=Williams, I. P. | title=The mean orbit of the April Lyrids | journal=Monthly Notices of the Royal Astronomical Society | date=1997 | volume=289 | issue=3 | pages=721–728 | bibcode=1997MNRAS.289..721A | doi=10.1093/mnras/289.3.721}}</ref> In 1982, amateur astronomers counted 90 April Lyrids per hour at the peak and similar rates were seen in 1922. A stronger storm of up to 700 per hour occurred in 1803,<ref name=martinez1994>{{citation | first1=Patrick | last1=Martinez | title=The Observer's Guide to Astronomy | volume=2 | series=Practical Astronomy Handbooks | others=Translated by Storm Dunlop | publisher=Cambridge University Press | date=1994 | isbn=0521458986 | page=645 | url=https://books.google.com/books?id=TXc54LfKsSQC&pg=PA645 | postscript=. }}</ref> observed by a journalist in Richmond, Virginia: "Shooting stars. This electrical [sic] phenomenon was observed on Wednesday morning last at Richmond and its vicinity, in a manner that alarmed many, and astonished every person that beheld it. From one until three in the morning, those starry meteors seemed to fall from every point in the heavens, in such numbers as to resemble a shower of sky rockets&nbsp;...<ref name="spaceweather"/>"<ref name=martinez1994/> The oldest known outburst, the shower on March 23.7,<ref>M. Ed. Biot, 1841, Gatalogue General des Etoiles Filantes et des Autres Meteores Observes en Chine pendent 24 Siecles, Paris, Imprimerie Royale; P. Jenniskens, 2006, Meteor Showers and their Parent Comets, Cambridge University Press, 790 pp.</ref> 687 BC (proleptic Julian calendar) was recorded in ''Zuo Zhuan'', which describes the shower as "On the 4th month in the summer in the year of Sexagenary cycle (xīn-mǎo) (of year 7 of King Zhuang of the State of Lu), at night, (the sky is so bright that some) fixed stars become invisible (because of the meteor shower); at midnight, stars fell like rain."<ref>{{cite web|url=http://www.skyandtelescope.com/observing/objects/meteors/3305866.html|title=Meteors&nbsp;– April's Lyrid Meteor Shower|last=Sinnott|first=Roger W.|date=2008|publisher=Sky and Telescope|page=1|accessdate=2008-08-05}}</ref> In the Australian Aboriginal astronomy of the Boorong tribe, the Lyrids represent the scratchings of the Mallee fowl (represented by Vega), coinciding with its nest-building season.<ref>{{cite web|last=Hill|first=Tanya|title=The Lyrids meteor shower should put on a show overnight|url=http://theconversation.com/the-lyrids-meteor-shower-should-put-on-a-show-overnight-25746|work=The Conversation|accessdate=22 April 2014|author2=Brown, Michael J. I. |date=22 April 2014}}</ref> {{clear}} ==Monocerotids== "Monocerotids is a reliable minor meteor shower that takes place from December 20th to December 7th and peaks on December 9th."<ref name=Monocerotids>{{ cite web |author=[[w:User:Italiantechie|Italiantechie]] |title=Monocerotids |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=29 September 2009 |url=https://en.wikipedia.org/wiki/Monocerotids |accessdate=30 June 2019 }}</ref> Parent body: C/1917 F1 (Mellish).<ref name=IAUMonocerotids>{{ cite web |author=IAU |title=Meteor Data Center |publisher=IAU |location= |date=2019 |url=https://www.ta3.sk/IAUC22DB/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=00019&colecimy=1&kodmin=00001&kodmax=00569&sortowanie= |accessdate=30 June 2019 }}</ref> ==Alpha Monocerotids== [[Image:Meteor burst.jpg|thumb|right|200px|This picture is of the Alpha-Monocerotid meteor outburst in 1995. It is a timed exposure where the meteors have actually occurred several seconds to several minutes apart. Credit: NASA Ames Research Center/S. Molau and P. Jenniskens.]] Most years, those trails would miss the Earth altogether, but in some years the Earth is showered by meteors. This effect was first demonstrated from observations of the 1995 alpha Monocerotids,<ref>Jenniskens P., 1997. Meteor steram activity IV. Meteor outbursts and the reflex motion of the Sun. Astron. Astrophys. 317, 953–961.</ref><ref name = Jenniskens1997/> The swarm is visible every year from 15 to 25 November; its peak occurs on 21 or 22 November.<ref name = Jenniskens1997/> The speed of its meteors is 65&nbsp;km/s.<ref name = Jenniskens1997/> Normally it has a low Zenithal Hourly Rate (ZHR), but occasionally it produces remarkable meteor storms that last less than an hour: such outbursts were observed in 1925, 1935, 1985, and 1995.<ref name = Jenniskens1997/> The 1995 return was predicted based on the hypothesis that these outbursts were caused by the dust trail of a long period comet occasionally wandering in Earth's path due to planetary perturbations, during observations in southern Spain, assisted by a team of observers of the Dutch Meteor Society, and confirmed the brevity of Alpha Monocerotids outbursts as less than one hour, where the parent body, probably a long-period comet, is unknown.<ref name = Jenniskens1997>{{cite journal|last1=Jenniskens|first1=P.|last2=Betlem|first2=H.|last3=De Lignie|first3=M.|last4=Langbroek|first4=M.|year=1997|bibcode=1997ApJ...479..441J |title= The Detection of a Dust Trail in the Orbit of an Earth-threatening Long-Period Comet |journal=The Astrophysical Journal |volume=479 |pages=441|doi = 10.1086/303853 }}</ref> {{clear}} ==Gamma Normids== Discovery date = 1929<ref name=Kronk2014>{{cite book|last1=Kronk|first1=Gary W|title=Meteor Showers: An Annotated Catalog|date=2014|publisher=Springer|isbn=978-1-4614-7896-6|pages=61–63|edition=2nd|url=https://books.google.com.au/books?id=t026BAAAQBAJ&pg=PA61&lpg=PA61&source=bl&ots=07PVtZdxUu&sig=0ys3YDjyyqrtBXdpKT5cl_c9-yY&hl=en&sa=X&redir_esc=y#v=onepage&f=false|accessdate=28 August 2015}}</ref> Right ascension = {{RA|16|24}}, Declination = {{DEC|-51}}<ref name=AMS>{{cite web |last1=Lunsford |first1=Robert |title=Meteor Activity Outlook for March 21-27, 2015 |url=http://www.amsmeteors.org/2015/03/meteor-activity-outlook-for-march-21-27-2015/ |website=American Meteor Society |accessdate=28 August 2015 |archiveurl=https://www.webcitation.org/6b7OVoq6R?url=http://www.amsmeteors.org/2015/03/meteor-activity-outlook-for-march-21-27-2015/ |archivedate=28 August 2015 |df= }}</ref> Date of peak = March&nbsp;15<ref name="Kronk2014"/> Velocity = 68 km/s<ref name=AMS /> Zenithal hourly rate = <1-2<ref name=AMS /> The first observations were made by R A McIntosh from Auckland, New Zealand in 1929, with confirmation coming from observations made by M. Geddes in 1932.<ref name=Kronk2014/> The shower was virtually ignored until radar equipment used by A A Weiss in Adelaide, South Australia detected activity 15–16 March 1953.<ref name=Kronk2014/> An attempt to observe the shower with radar in 1956 was unsuccessful, however the shower was observed again with radar in 1969.<ref name=Kronk2014/> Members of the Western Australia Meteor Section made extensive observations in the 1970s and 1980s. In 1983 the average magnitude of the 63 meteors was 2.68 and 9.5% had trains with the highest Zenithal Hourly Rate (ZHR) of 9.6±2.3 recorded on the night of March 13/14. In 1986 273 meteors were observed, and the highest ZHR (3.49) was recorded on March 14/15. Nearly 20% of the meteors left trains.<ref name=Kronk2014/> In 2005 the Liga IberoAmericana De Astronomía noted meteors from this stream every night during the observation period of March 8–17. The highest number of meteors seen was 5, on the night of March 10/11 with a ZHR of 14 ± 6.<ref name=Kronk2014/> ==Orionids== [[Image:Two orionids and milky way.jpg|right|thumb|300px|Two Orionids meteors and the Milky Way are shown. Credit: [https://sites.google.com/site/thebrockeninglory/ Brocken Inaglory].{{tlx|free media}}]] "The Orionid meteor shower [leftover bits of Halley's Comet] is scheduled to reach its maximum before sunrise on Sunday morning (Oct. 21 [2012]). This will be an excellent year to look for the Orionids, since the [[moon]] will set around 11 p.m. local time on Saturday night (Oct. 20) and will not be a hindrance at all ... The orbit of Halley's Comet closely approaches the Earth's orbit at two places. One point is in the early part of May producing a meteor display known as the Eta Aquarids. The other point comes in the middle to latter part of October, producing the Orionids."<ref name=Rao>{{ cite book |author=Joe Rao |title=Orionid Meteor Shower Spawned by Halley's Comet Peaks This Weekend |publisher=SPACE.com |location= |date=October 19, 2012 |url=http://news.yahoo.com/orionid-meteor-shower-spawned-halleys-comet-peaks-weekend-160214151.html |accessdate=2012-10-19 }}</ref> {{clear}} ==Perseids== [[Image:PSM V18 D201 Shower of perseids sept 6 and 7.jpg|right|thumb|300px|Perseid meteor shower is from September 6 and 7, 1880-81. Credit: unknown.{{tlx|free media}}]] [[Image:Perseid meteor 2007.jpg|thumb|left|300px|A Perseid shower occurs in 2007. Credit: [https://sites.google.com/site/thebrockeninglory/ Brocken Inaglory].]] [[Image:Animation of 109P／Swift–Tuttle orbit.gif|right|thumb|300px|Animation of 109P/Swift–Tuttle orbit from 1875 to 2100.<br>{{legend2| Yellow| Sun}}{{·}}{{legend2|RoyalBlue|Earth}}{{·}}{{legend2| Lime| Jupiter }}{{·}}{{legend2|OrangeRed|Saturn}}{{·}}{{legend2|Gold|Uranus}}{{·}}{{legend2|Magenta|109P/Swift–Tuttle}}. Credit: [[c:User:Phoenix7777|Phoenix7777]].{{tlx|free media}}]] [[Image:Perseidak 2006.jpg|left|thumb|300px|Radiant point is from August 8, 2006. Credit: Olga Berrios.{{tlx|free media}}]] In 1835, Adolphe Quetelet identified the shower as emanating from the constellation Perseus.<ref name="NASA Chat">{{cite web|url=http://www.nasa.gov/pdf/677215main_Perseids2012.pdf |title=NASA Chat: Stay ‘Up All Night’ to Watch the Perseids! |publisher=NASA |author= Dr. Bill Cooke |author2= Danielle Moser |author3= Rhiannon Blaauw | page=55|date=2012-08-11|accessdate=2013-08-16}}</ref><ref name="Kronk"/> Right ascension = {{RA|03|04}}<ref name=moore_rees2011/> and Declination = {{DEC|+58}}<ref name=moore_rees2011/> The Perseid meteor shower, usually the richest meteor shower of the year, peaks in August. Over the course of an hour, a person watching a clear sky from a dark location might see as many as 50-100 meteors. Parent body is Comet Swift–Tuttle.<ref name=moore_rees2011/> The first record is from 36 CE.<ref name="NASA Chat"/><ref name="Kronk"/> The radiant point image on the right is from September 6 and 7, 1880-81.<ref>[https://archive.org/details/popularsciencemo18newy ''Popular Science Monthly'', Volume 18]</ref> Velocity = 58 km/s<ref name=jimo37_5_161>{{citation | display-authors=1 | last1=Żołądek | first1=P. | last2=Wiśniewski | first2=M. | last3=Olech | first3=A. | last4=Krasnowski | first4=M. | last5=Kwinta | first5=M. | last6=Fietkiewicz | first6=K. | last7=Złoczewski | first7=K. | last8=Mularczyk | first8=K. | last9=Jonderko | first9=W. | last10=Spaleniak | first10=I. | last11=Gruszka | first11=Ł. | title=The 2004 Perseid meteor shower – Polish Fireball Network double station preliminary results | journal=Journal of the International Meteor Organization | volume=37 | issue=5 | pages=161–163 |date=October 2009 | bibcode=2009JIMO...37..161Z}}</ref> and Zenithal hourly rate = 100<ref name=moore_rees2011/>. The stream of debris is called the Perseid cloud and stretches along the orbit of the comet Swift–Tuttle. The cloud consists of particles ejected by the comet as it travels on its 133-year orbit.<ref>{{cite web|author=Dan Vergano|url=https://www.usatoday.com/story/tech/sciencefair/2013/08/08/perseids-2013-meteor/2628521/ |title=Perseid meteor shower to light up night sky this weekend |publisher=Usatoday.com |date=2010-08-07 |accessdate=2013-08-12}}</ref> Most of the particles have been part of the cloud for around a thousand years. However, there is also a relatively young filament of dust in the stream that was pulled off the comet in 1865, which can give an early mini-peak the day before the maximum shower.<ref>{{Cite web |title = The 2004 Perseid Meteor Shower |date = June 25, 2004 |author = Dr. Tony Phillips |publisher = Science@NASA |accessdate = 2010-03-12 |url = https://science.nasa.gov/headlines/y2004/25jun_perseids2004.htm |archiveurl = https://web.archive.org/web/20100320071509/http://science.nasa.gov/headlines/y2004/25jun_perseids2004.htm |archivedate = March 20, 2010 |df = }}</ref> The dimensions of the cloud in the vicinity of the Earth are estimated to be approximately 0.1 AU across and 0.8 AU along the latter's orbit, spread out by annual interactions with the Earth's gravity.<ref>{{Cite book | title = The Cosmic Dust Connection | editor = J. Mayo Greenberg | date = 1996 | publisher = Springer Science & Business Media | article = Cometary Dust Loss: Meteoroid Streams and the Inner Solar System Dust Cloud | author= D.W. Hughes | page = 375 | url = https://books.google.com/books?id=5MH0CAAAQBAJ&pg=PA375}}</ref> The shower is visible from mid-July each year, with the peak in activity between 9 and 14 August, depending on the particular location of the stream. During the peak, the rate of meteors reaches 60 or more per hour. They can be seen all across the sky; however, because of the shower's radiant in the constellation of Perseus, the Perseids are primarily visible in the Northern Hemisphere.<ref>{{cite web|url=https://www.timeanddate.com/astronomy/meteor-shower/perseid.html|publisher=timeanddate.com|title=Perseids Meteor Shower 2018 |accessdate=2018-07-30}}</ref> As with many meteor showers the visible rate is greatest in the pre-dawn hours, since more meteoroids are scooped up by the side of the Earth moving forward into the stream, corresponding to local times between midnight and noon, as can be seen in the accompanying diagram.<ref>http://meteorshowersonline.com/what_is.html</ref> While many meteors arrive between dawn and noon, they are usually not visible due to daylight. Some can also be seen before midnight, often grazing the Earth's atmosphere to produce long bright trails and sometimes fireballs. Most Perseids burn up in the atmosphere while at heights above {{convert|80|km|mi}}.<ref name="Meteorwatch">{{cite web |date=2012-08-11 |title=NASA All Sky Fireball Network: Perseid End Height |publisher=NASA Meteor Watch on Facebook |url=https://www.facebook.com/photo.php?fbid=402054346520464 |accessdate=2012-11-19}}</ref> {{clear}} ==Phoenicids== The Phoenicids get their name from the location of their radiant, which is in the constellation Phoenix, active from 29 November to 9 December, with a peak occurring around 5/6 December each year,<ref>http://meteorshowersonline.com/showers/phoenicids.html</ref> and are best seen from the Southern Hemisphere. The Phoenicids appear to be associated with a stream of material from the disintegrating comet D/1819 W1 (Blanpain).<ref>P. Jenniskens and E. Lyytinen, METEOR SHOWERS FROM THE DEBRIS OF BROKEN COMETS: D/1819 W1 (BLANPAIN), 2003 WY25, AND THE PHOENICIDS.The Astronomical Journal, 130:1286–90, 2005 September</ref> A very minor meteor shower with a radiant in Phoenix also occurs in July; this shower is referred to as the '''July Phoenicids'''.<ref>http://meteorshowersonline.com/showers/july_phoenicids.html</ref> ==Pi Puppids== "The Pi Puppids are a meteor shower associated with the comet 26P/Grigg-Skjellerup."<ref name=PiPuppids>{{ cite web |author=[[w:User:Wikibob|Wikibob]] |title=Pi Puppids |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=26 June 2004 |url=https://en.wikipedia.org/wiki/Pi_Puppids |accessdate=28 June 2019 }}</ref> "The Pi Puppids get their name because their radiant appears to lie in the constellation Puppis, at around Right ascension 112 degrees and Declination -45 degrees."<ref name=PiPuppids/> ==Quadrantids== The Quadrantids ('''QUA''') are a January meteor shower, with the zenithal hourly rate (ZHR) of this shower as high as that of two other reliably rich meteor showers, the Perseids in August and the Geminids in December.<ref name="amsmeteors.org">{{ cite web |url=http://www.amsmeteors.org/meteor-showers/meteor-faq/#7 |title=Does the published meteor rate for a shower really represent what I should expect to see? |publisher=American Meteor Society |accessdate=2012-12-29 }}</ref> The meteor rates exceed one-half of their highest value for only about eight hours (compared to two days for the August Perseids), which means that the stream of particles that produces this shower is narrow, and apparently deriving within the last 500 years from some orbiting body.<ref name="space.com">{{ cite web |url=http://www.space.com/spacewatch/081219-ns-meteor-shower.html |title=Stellar Meteor Shower Jan. 3 |publisher=Space.com |accessdate=2009-01-03 }}</ref> The parent body of the Quadrantids was tentatively identified in 2003<ref name="sjaa">{{ cite web |date=Dec 8, 2003 |title=2003 EH1 is the Quadrantid shower parent comet |publisher=The Ephemeris (San Jose Astronomical Association newsletter) |author=Peter Jenniskens |url=http://ephemeris.sjaa.net/0401/e.html |accessdate=2004-12-17}}</ref> as the minor planet (196256) 2003 EH1, which in turn may be related to the comet C/1490 Y1<ref name=Jenniskens2004>{{ cite journal |last=Jenniskens |first=Peter |title=2003 EH1 Is the Quadrantid Shower Parent Comet |journal=The Astronomical Journal |volume=127 |issue=5 |pages=3018–3022 |date=2004 |doi=10.1086/383213 |bibcode=2004AJ....127.3018J}}</ref> that was observed by Chinese, Japanese and Korean astronomers some 500 years ago. ==Beta Taurids== The Beta Taurids are normally active from June 5 to July 18.<ref name="Kronk"/> They emanate from an average radiant of right ascension 5h18m, declination +21.2 and exhibit maximum activity around June 28–29 (Solar Longitude=98.3 deg). The sun has a solar longitude (λ⊙) of 90 degress on June 21 (Summer solstice) and as there are 365 days/year moves roughly 1 degree/day. The meteor shower radiant of RA=79.4 degrees converts to 5h 18m as each hour is 15 degrees. The Zenithal Hourly Rate typically reaches about 25 km/s as seen on radar.<ref name="Kronk"/> Non-radio observers are faced with a very difficult prospect, because the Beta Taurid radiant is just 10–15 degrees west of the Sun on June 28.<ref name="meteorobs1999">{{cite web |date = 1999-06-18 |title = IMO-NEWS: 1999 Beta Taurids Alert - Possible Swarm Appearance |publisher = Meteor Observing Mailing List (meteorobs) |url = http://www.meteorobs.org/maillist/msg13477.html |accessdate = 2012-11-26 |archiveurl = https://web.archive.org/web/20080612132836/http://www.meteorobs.org/maillist/msg13477.html |archivedate = 2008-06-12 |df = }}</ref><ref>The International Meteor Organization's radiant of [https://www.imo.net/resources/calendar/meteor-shower-calendar-archive/english2015/#table7 86° +19°] will be 10° from the Sun on June 28. Gary W. Kronk's radiant of [https://web.archive.org/web/20071023040540/http://meteorshowersonline.com/showers/beta_taurids.html 79.4° +21.2°] will be 15° from the Sun on June 28.</ref> Asteroids associated with the β–Taurids include 2201 Oljato, 5143 Heracles, 6063 Jason, (8201) 1994 AH₂ and 1991 BA.<ref name=Babadzhanov2001>{{cite journal |last1=Babadzhanov |first1=P. B. |title=Search for meteor showers associated with Near-Earth Asteroids |journal=Astronomy and Astrophysics |volume=373 |issue=1 |pages=329–335 |date=2001 |doi=10.1051/0004-6361:20010583|bibcode = 2001A&A...373..329B }}</ref> 2019 will be the closest post-perihelion encounter with Earth since 1975. The Taurid swarm is expected to pass {{convert|0.06|AU|abbr=on}} below the ecliptic between June 23 – July 17.<ref name=TSC2019>{{cite journal |last1=Clark |first1=David L. |last2=Weigert |first2=Paul |last3=Brown |first3=Peter G. |title=The 2019 Taurid resonant swarm: prospects for ground detection of small NEOs |journal=Monthly Notices of the Royal Astronomical Society |volume=487 |issue=1 |pages=L35–L39 |year=2019 |doi=10.1093/mnrasl/slz076 |arxiv=1905.01260 }}</ref> During 2019 astronomers hope to search for hypothesized asteroids ~100 meters in diameter from the Taurid swarm between July 5–11, and July 21 – August 10.<ref name="Plait">{{cite web |title=Could larger space rocks be hiding in the Beta Taurid Meteor stream? We may find out this summer |publisher=Bad Astronomy |author=Phil Plait |url=https://www.syfy.com/syfywire/could-larger-space-rocks-be-hiding-in-the-beta-taurid-meteor-stream-we-may-find-out-this |accessdate=2019-05-14}}</ref> There is circumstantial evidence that the daytime June 30 Tunguska event came from the same direction in the sky as the Beta Taurids.<ref name="Plait"/> The next June close approach to the Taurid swarm is expected in 2036.<ref name="Asher">{{cite web |title=Taurid swarm years |publisher=University of Cambridge |author=David J. Asher |url=http://www.cantab.net/users/davidasher/taurid/swarmyears.html |accessdate=2019-06-16}}</ref> ==Northern Taurids== [[Image:Taurid Meteor Shower - Joshua Tree, California - 6 Nov. 2015.jpg|right|thumb|300px|11/6/2015 - Image shows the Taurid Meteor Shower - Joshua Tree , CA. Credit: [https://www.flickr.com/people/14376841@N06 Channone Arif].{{tlx|free media}}]] Parent body = 2004 TG₁₀<ref name=JenniskensTG10>Meteor showers and their parent comets [https://books.google.com/books?id=QpajMuyXG8AC&pg=PA470&dq=TG10 pg 470] by Peter Jenniskens</ref><ref name=moore_rees2011>{{citation | first1=Patrick | last1=Moore | first2=Robin | last2=Rees | title=Patrick Moore's Data Book of Astronomy | edition=2nd | publisher=Cambridge University Press | date=2011 | isbn=0-521-89935-4 | page=275 | url=https://books.google.com/books?id=2FNfjWKBZx8C&pg=PA275 }}</ref> Radiant point = RA {{RA|03|52}} Dec = {{DEC|+22}}.<ref name="IMO2015">{{cite web |title=IMO Meteor Shower Calendar 2015 (Working list of visual meteor showers) |publisher=International Meteor Organization |url=https://www.imo.net/resources/calendar/meteor-shower-calendar-archive/english2015/#table5 |accessdate=2019-06-20}}</ref> Occurs during October 20 – December 10, with a peak at 12 November.<ref name=IMO2015/> Velocity = 29 km/s.<ref name=IMO2015/> Zenithal hourly rate is 5.<ref name=IMO2015/> The Northern Taurids originated from the asteroid 2004 TG₁₀.<ref name="Taurids 2017">{{cite web|url=https://patch.com/us/across-america/taurids-meteor-shower-fireballs-peak-dates-what-expect|title=Taurids Meteor Shower Fireballs: Peak Dates, What To Expect|author=Beth Dalbey|date=24 October 2017|accessdate=11 November 2017}}</ref> The Taurids are also made up of weightier material, pebbles instead of dust grains.<ref name="NASA2005">{{cite web |date=2005-11-03 |title=Earth is orbiting through a swarm of space debris that may be producing an unusual number of nighttime fireballs |publisher=NASA Science News |author=Dr. Tony Phillips |url=https://science.nasa.gov/science-news/science-at-nasa/2005/03nov_taurids/}}</ref> Typically, Taurids appear at a rate of about 5 per hour, moving slowly across the sky at about 28 km/s (17 mi/s), or 100,800 km/h (65,000 mph).<ref name="NASA2005"/> If larger than a pebble, these meteors may become bolides as bright as the moon and leave behind smoke trails.<ref name="NASA2005"/> The Beta Taurids could be the cause of the Tunguska event of June 30, 1908.<ref name=NG>[http://news.nationalgeographic.com/news/2003/11/1107_031107_taurids.html Meteor Shower Promises Seven Shooting Stars an Hour] – National Geographic News (November 7, 2003)</ref> In 1962 and 1963, the Mars 1 probe recorded one micrometeorite strike every two minutes at altitudes ranging from {{convert|6000|to|40000|km|mi|abbr=on}} from Earth's surface due to the Taurids meteor shower, and also recorded similar densities at distances from {{convert|20|to|40|e6km|mi}} from Earth.<ref name='Sputnik 23'>{{cite web | url = http://jtgnew.sjrdesign.net/exploration_space_planetary_mars.html | title = "Journey Through the Galaxy" Mars Program: Mars ~ 1960-1974 | accessdate = 2014-01-26 | last = Robbins | first = Stuart | date = 2008 | publisher = SJR Design}}</ref><ref name='Mihos 2006'>{{cite web | url = http://burro.astr.cwru.edu/stu/advanced/20th_soviet_mars.html | title = Mars (1960-1974): Mars 1 | accessdate = 2014-01-26 | last = Mihos | first = Chris | date = 11 January 2006 | work = Department of Astronomy, Case Western Reserve University. | archiveurl = https://web.archive.org/web/20131013211415/http://burro.astr.cwru.edu/stu/advanced/20th_soviet_mars.html | archivedate = 13 October 2013 | df = }}</ref> The Taurid stream has a cycle of activity that peaks roughly every 2,500 to 3,000 years,<ref name=NG/> when the core of the stream passes nearer to Earth and produces more intense showers. In fact, because of the separate "branches" (night-time in one part of the year and daytime in another; and Northern/Southern in each case) there are two (possibly overlapping) peaks separated by a few centuries, every 3000 years. The next peak is expected around 3000 AD.<ref name=NG/> Over Poland in 1995, all-sky cameras imaged an absolute magnitude –17 Taurid bolide that was estimated to be 900 kg and perhaps a meter in diameter.<ref name=Jenniskens-pg467>Meteor showers and their parent comets [https://books.google.com/books?id=QpajMuyXG8AC&pg=PA467&dq=meter pg 467] by Peter Jenniskens</ref> In 1993, it was predicted that there would be a swarm of activity in 2005.<ref name="NASA2005"/> Around Halloween in 2005, many fireballs were witnessed that affected people's night vision.<ref name="NASA2005"/> Astronomers have taken to calling these the "Halloween fireballs."<ref name="NASA2005"/> The Tunguska event may have been caused by a Beta Taurid.<ref name="TunguskaTaurid">{{cite news |author=Joel Achenbach |date=2018-12-25 |title=Incoming! A June meteor swarm could be loaded with surprises. |url=https://www.washingtonpost.com/science/2018/12/25/incoming-june-meteor-swarm-could-be-loaded-with-surprises/?utm_term=.3d66234e3b12 |website=www.washingtonpost.com |location=Washington DC, US |accessdate=2019-05-04}}</ref> A brief flash of light from a lunar impact event was recorded on November 7, 2005, while testing a new 250 mm (10 in) telescope and video camera built to monitor the Moon for meteor strikes.<ref>[http://news.bbc.co.uk/1/hi/sci/tech/4570730.stm BBC News: Nasa team sees explosion on Moon] (3 January 2006)</ref> This may be the first photographic record of such a strike.<ref>[http://neo.jpl.nasa.gov/news/news118.html An Eyewitness Impact Debunked]{{Dead link|date=June 2019}}</ref> {{clear}} ==Southern Taurids== During the Southern Taurid meteor shower in 2013, fireball sightings were spotted over southern California, Arizona, Nevada, and Utah.<ref name="BSkyB">{{cite news |author=Sky News US Team |date=2013-11-07 |title=Meteor 'Fireball' Lights Up California Sky |url=http://news.sky.com/story/1165152/meteor-fireball-lights-up-california-sky |website=news.sky.com |location=London, UK |publisher=BSkyB |accessdate=2013-11-07}}</ref> The Southern Taurids originated from Comet Encke, while the Northern Taurids originated from the asteroid 2004 TG₁₀.<ref name=Dalbey>{{cite web|url=https://patch.com/us/across-america/taurids-meteor-shower-fireballs-peak-dates-what-expect|title=Taurids Meteor Shower Fireballs: Peak Dates, What To Expect|author=Beth Dalbey|date=24 October 2017|accessdate=11 November 2017}}</ref> Encke and the Taurids are believed to be remnants of a much larger comet, which has disintegrated over the past 20,000 to 30,000 years.<ref name=Babadzhanov2008>{{cite journal |last1=Babadzhanov |first1=P. B. |last2=Williams |first2=I. P. |last3=Kokhirova |first3=G. I. |title=Near-Earth Objects in the Taurid complex |journal=Monthly Notices of the Royal Astronomical Society |volume=386 |issue=3 |pages=1436–1442 |year=2008 |doi=10.1111/j.1365-2966.2008.13096.x|bibcode = 2008MNRAS.386.1436B }}</ref> ==Gamma Ursae Minorids== Dates of occurrence 15 Jan - 25 Jan, with a peak (as of 2017) on 20 Jan, Celestial longitude of 299 <math>\lambda (^\circ)</math>, RA 15.2 Dec +67, speed = 31 km/s speed, ZHR = 3.<ref name=ListMeteorShowers>{{ cite web |author=[[w:User:Lasunncty|Lasunncty]] |title=List of meteor showers |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=2 November 2017 |url=https://en.wikipedia.org/wiki/List_of_meteor_showers |accessdate=29 June 2019 }}</ref> ==Ursids== The Ursids were probably discovered by William F. Denning who observed them for several years around the start of the 20th century.<ref name="KronkU">{{cite web |title=Observing the Ursids |publisher=Meteor Showers Online |author=Gary W. Kronk |url=http://meteorshowersonline.com/showers/ursids.html |accessdate=2012-11-17}}</ref> While there were sporadic observations after, the first coordinated studies of shower didn't begin until Dr. A. Bečvář observed an outburst of 169 per hour in 1945.<ref name="Kronk"/> Further observations in the 1970s and ongoing to current have established a relationship with comet 8P/Tuttle.<ref name="KronkU"/> Parent body = 8P/Tuttle.<ref name="KronkU"/> Right ascension = {{RA|14|28}}<ref name=moore_rees2011/>, Declination = {{DEC|+78}}<ref name=moore_rees2011/> Constellation = Ursa Minor (near Kochab) Occurs during December 17 – December 26.<ref name="KronkU"/> Date of peak = December&nbsp;22.<ref name="KronkU"/> Velocity = 33 km/s.<ref name=IMO2012P>[http://www.imo.net/calendar/2012#urs IMO Meteor Shower Calendar 2012: Ursids (URS)]</ref> Zenithal hourly rate = 10.<ref name="KronkU"/> ==Alpha Virginids== The Alpha Virginids occur between March 10 and May 6, peaking between April 7 and April 18, with five to ten meteors per hour. They were first detected in 1895.<ref>{{cite web |title=Alpha Virginids|publisher=Meteor Showers Online|pages=1|accessdate=2008-08-05|url=https://web.archive.org/web/20080827202359/http://meteorshowersonline.com/showers/alpha_virginids.html }}</ref> ==Eta Virginids== The Eta Virginids occur between February 24 and March 27, peaking around March 18 with only one to two meteors per hour. The shower was first detected in 1961.<ref>{{cite web |title=Eta Virginids|publisher=Meteor Showers Online|pages=1|accessdate=2008-08-05|url=https://web.archive.org/web/20080807161455/http://meteorshowersonline.com/showers/eta_virginids.html }}</ref> Parent body: D/1766 G1 (Helfenzrieder)?<ref name=Lindblad1971>{{ cite web |author=B. A. Lindblad |title=A computerized stream search among 2401 photographic meteor orbits |journal=Smithson. Contrib. Astrophys. |issue=12 |pages=14-24 |date=1971 |bibcode=1971SCoA...12...14L |url=https://www.ta3.sk/IAUC22DB/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=00011&colecimy=1&kodmin=00001&kodmax=00569&sortowanie= |accessdate=30 June 2019 }}</ref> ==Gamma Virginids== The North and South Gamma Virginids are a slow-moving minor meteor shower stream,<ref name="gamma1">{{cite web |title=North Gamma Virginids|last=Jopek|first=T.J.|date=Nov 15, 2007|publisher=IAU Meteor Data Center|accessdate=2008-08-07 |url=https://web.archive.org/web/20110614170520/http://www.astro.amu.edu.pl/~jopek/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=134&colecimy=0 }}</ref><ref name="gamma2">{{cite web |title=South Gamma Virginids|last=Jopek|first=T.J.|date=Nov 15, 2007|publisher=IAU Meteor Data Center|accessdate=2008-08-07 |url=https://web.archive.org/web/20110614170612/http://www.astro.amu.edu.pl/~jopek/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=135&colecimy=0 }}</ref> although the May Gamma Virginids and Daytime Gamma Virginids are faster-moving.<ref name="gamma3">{{cite web |title=May Gamma Virginids|last=Jopek|first=T.J.|date=Nov 15, 2007|publisher=IAU Meteor Data Center|accessdate=2008-08-07 |url=https://web.archive.org/web/20110614170653/http://www.astro.amu.edu.pl/~jopek/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=148&colecimy=0 }}</ref><ref name="gamma4">{{cite web |title=Dayt. Gamma Virginids|last=Jopek|first=T.J.|date=Nov 15, 2007|publisher=IAU Meteor Data Center|accessdate=2008-08-07 |url=https://web.archive.org/web/20110614170804/http://www.astro.amu.edu.pl/~jopek/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=223&colecimy=0 }}</ref> The source of the North and South Gamma Virginid streams are thought to be 2002 FC and 2003 BD44, respectively.<ref name="gamma1"/><ref name="gamma2"/> It usually spans from April 5 to April 21, peaking on April 14 and April 15, with less than five meteors per hour. It was first discovered in 1895.<ref>{{cite web |title=Gamma Virginids|publisher=Meteor Showers Online|accessdate=2008-08-07|url=https://web.archive.org/web/20080827201148/http://meteorshowersonline.com/showers/gamma_virginids.html }}</ref> Collectively, the shower normally lasts from late January to mid-April<ref name="pop">{{cite web title=SPA Meteor Section|date=2006|publisher=Pop astro|pages=1|accessdate=2008-08-05 |url=https://web.archive.org/web/20080509163726/http://www.popastro.com/sections/meteor/meteor-activity.htm }}</ref> and into early May,<ref name="complex"/> peaking in March and April,<ref name="complex"/><ref name="pop"/> with one to two meteors per hour on average.<ref name="pop"/> The main radiant shifts southeastwards from central Leo in late January to central Virgo near Spica in mid-May.<ref>{{cite web|url=http://www.nightskyinfo.com/maps_images/html/shower1_map.htm|title=Virginids Radiant Position|publisher=nightskyinfo.com|pages=1|accessdate=2008-08-05}}</ref> ==Iota Virginids== The Iota Virginids are a minor daytime meteor shower stream.<ref name="iota">{{cite web |title=Iota Virginids|last=Jopek|first=T. J.|date=Nov 15, 2007|publisher=IAU Meteor Data Center|accessdate=2008-08-07 |url=https://web.archive.org/web/20110614170904/http://www.astro.amu.edu.pl/~jopek/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=251&colecimy=0 }}</ref> ==Lambda Virginids== The Lambda Virginids are a minor meteor shower stream.<ref name="lambda">{{cite web |title=Lambda Virginids|last=Jopek|first=T. J.|date=Nov 15, 2007|publisher=IAU Meteor Data Center|accessdate=2008-08-07 |url=https://web.archive.org/web/20110614170924/http://www.astro.amu.edu.pl/~jopek/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=49&colecimy=2 }}</ref> ==March Virginids== The March Virginids are a minor meteor shower stream; the source of the Northern March Virginids is thought to be 1998 SJ70.<ref name="march">{{cite web|title=Northern<!--[sic]--> March Virginids |last=Jopek|first=T. J.|date=Nov 15, 2007|publisher=IAU Meteor Data Center|accessdate=2008-08-07 |url=https://web.archive.org/web/20110614171104/http://www.astro.amu.edu.pl/~jopek/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=123&colecimy=0 }}</ref><ref>{{cite web|title=South. March Virginids|last=Jopek|first=T. J.|date=Nov 15, 2007|publisher=IAU Meteor Data Center|accessdate=2008-08-07 |url=https://web.archive.org/web/20110614171110/http://www.astro.amu.edu.pl/~jopek/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=124&colecimy=0 }}</ref> The Beta Leonids, lasting from February 14 to April 25, peaking around March 20 with three to four meteors per hour, were also referred to as the "March Virginids".<ref>{{cite web |title=Beta Leonids|publisher=Meteor Showers Online|accessdate=2008-08-07|url=https://web.archive.org/web/20080516071922/http://meteorshowersonline.com/showers/beta_leonids.html }}</ref> ==Mu Virginids== The Mu virginids<ref>{{cite web |title=Mu Virginids|last=Jopek|first=T. J.|date=Nov 15, 2007|publisher=IAU Meteor Data Center|accessdate=2008-08-07 |url=https://web.archive.org/web/20110614170941/http://www.astro.amu.edu.pl/~jopek/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=47&colecimy=0 }}</ref> are a minor meteor shower stream, visible in April<ref>{{cite web |title=(meteorbs) NAMN Notes: May 2001|last=Davis|first=Mark|date=3 May 2001|publisher=meteorbs|accessdate=2008-08-07 |url=https://web.archive.org/web/20080516093352/http://www.meteorobs.org/maillist/msg21552.html }}</ref> and early May. The shower typically lasts from April 1 to May 12, peaking around April 24<ref>{{cite web|url=http://www.imo.net/files/data/vmdb/vmdbrad.txt|title=meteor shower list|publisher=International Meteor Organization|accessdate=2008-08-07}}</ref> to April 25. Its radiant is near Libra, peaking with seven to ten meteors per hour.<ref>{{cite web|url=http://www.projectpluto.com/gloss/help_17.htm|title=glossary|publisher=Project Pluto|accessdate=2008-08-07}}</ref> ==Pi Virginids== The Pi Virginids occur between February 13 and April 8, peaking between March 3 and March 9, with two to five meteors per hour. The shower was first observed in 1908 and identified in 1948.<ref>{{cite web |title=Pi Virginids|publisher=Meteor Showers Online|pages=1|accessdate=2008-08-05|url=https://web.archive.org/web/20080807161559/http://meteorshowersonline.com/showers/pi_virginids.html }}</ref> ==Psi Virginids== The Psi Virginids are a minor daytime meteor shower stream.<ref name="psi">{{cite web|title=Psi Virginids |last=Jopek|first=T. J.|date=Nov 15, 2007|publisher=IAU Meteor Data Center|accessdate=2008-08-07 |url=https://web.archive.org/web/20110614171022/http://www.astro.amu.edu.pl/~jopek/MDC2007/Roje/pojedynczy_obiekt.php?kodstrumienia=240&colecimy=0 }}</ref> ==Theta Virginids== The Theta Virginids occur between March 10 and April 21, peaking around March 20, with only one to three meteors per hour. The shower was first observed in 1850 and identified in 1948.<ref>{{cite web |title=Theta Virginids|publisher=Meteor Showers Online|pages=1|accessdate=2008-08-05|url=https://web.archive.org/web/20080807160910/http://meteorshowersonline.com/showers/theta_virginids.html }}</ref> ==Virginids== The Virginids are a meteor shower. There are many major and minor meteor shower streams that occur during the Virginid Complex.<ref name="complex">{{cite web |title=The Virginids Complex|publisher=Meteor Showers Online|pages=1|accessdate=2008-08-05|url=https://web.archive.org/web/20081120190444/http://meteorshowersonline.com/showers/virginid_complex.html }}</ref> ==Mercury== Meteors or meteor showers have been discussed for most of the objects in the Solar System with an atmosphere: Mercury.<ref>{{cite journal | author1=Rosemary M. Killen | author2=Joseph M. Hahn | title =Impact Vaporization as a Possible Source of Mercury's Calcium Exosphere | journal=Icarus | date=December 10, 2014 | doi=10.1016/j.icarus.2014.11.035 |bibcode = 2015Icar..250..230K | volume=250 | pages=230–237| hdl=2060/20150010116}}</ref> ==Venus== Venus can receive meteor showers from P/Halley Stream.<ref>[https://link.springer.com/chapter/10.1007%2F978-0-387-78419-9_19#page-1 The P/Halley Stream: Meteor Showers on Earth, Venus and Mars], by Apoistolos A. Christou, Geremie Vaubaillon and Paul Withers, ''Earth, Moon, and Planets'', vol 102, # 1–4, {{doi|10.1007/s11038-007-9201-3}}</ref> ==Earth crossers== [[Image:2008VK184-year2014.png|thumb|right|250px|The close approach of apollo asteroid 2007 VK184 was in May 2014. Credit: Osamu Ajiki (AstroArts) and Ron Baalke (JPL).]] EC denotes Earth-crossing.<ref name=Michel>{{ cite journal |author=Patrick Michel, Fabbio Migliorini, Alessandro Morbidelli, Vincenzo Zappalà |title=The Population of Mars-Crossers: Classification and Dynamical Evolution |journal=Icarus |month=June |year=2000 |volume=145 |issue=2 |pages=332-47 |url=http://www.obs-nice.fr/morby/papers/6358a.pdf |bibcode= |doi=10.1006/icar.2000.6358 |pmid= |accessdate=2011-10-06 }}</ref> "50 % of the MB Mars-crossers [MCs] become ECs within 59.9 Myr and [this] contribution ... dominates the production of ECs".<ref name= Michel /> [[Image:SmallAsteroidImpacts-Frequency-Bolide-20141114.jpg|thumb|upright=2|center|500px|This diagram maps the data gathered from 1994-2013 on small asteroids impacting Earth's atmosphere. Credit: NASA/Planetary Science.]] "This diagram [center] maps the data gathered from 1994-2013 on small asteroids impacting Earth's atmosphere to create very bright meteors, technically called "bolides" and commonly referred to as "fireballs". Sizes of red dots (daytime impacts) and blue dots (nighttime impacts) are proportional to the optical radiated energy of impacts measured in billions of Joules (GJ) of energy, and show the location of impacts from objects about 1 meter (3 feet) to almost 20 meters (60 feet) in size."<ref name=Agle>{{ cite book |author=DC Agle |title=New Map Shows Frequency of Small Asteroid Impacts, Provides Clues on Larger Asteroid Population |publisher=NASA's Jet Propulsion Laboratory |location=Pasadena, California |date=14 November 2014 |editor= |pages= |url=https://www.jpl.nasa.gov/news/news.php?release=2014-397 |arxiv= |bibcode= |doi= |pmid= |isbn= |accessdate=2018-04-01 }}</ref> "A map released [...] by NASA's Near Earth Object (NEO) Program reveals that small asteroids frequently enter and disintegrate in the Earth's atmosphere with random distribution around the globe. Released to the scientific community, the map visualizes data gathered by U.S. government sensors from 1994 to 2013. The data indicate that Earth's atmosphere was impacted by small asteroids, resulting in a bolide (or fireball), on 556 separate occasions in a 20-year period. Almost all asteroids of this size disintegrate in the atmosphere and are usually harmless. The notable exception was the Chelyabinsk event which was the largest asteroid to hit Earth in this period."<ref name=Agle/> {{clear}} ==Apollo asteroids== [[Image:Minor Planets - Apollo.svg|thumb|right|250px|This a diagram showing the Apollo asteroids, compared to the orbits of the terrestrial planets of the [[Solar System]]. {| style="width: 100%; margin-bottom: 8px; margin-left: 12px;" |- | valign=top width=45% | {{legend2|#e61d30|border=1px solid #770D18|Mars}}&nbsp;(M)<br> {{legend2|#000000|border=1px solid black|Venus}}&nbsp;(V) {{legend2|#aaa|border=1px solid #444|Mercury}}&nbsp;(H) | valign=top | {{legend2|#FFFF00|border=1px solid #B3B300|[[Stars/Sun|Sun]]}}<br> {{legend2|#00CC00|border=1px solid #003300|'''Apollo asteroids'''}}<br> {{legend2|#1708a6|border=1px solid black|Earth}}&nbsp;(E) |} Credit: [[commons:User:AndrewBuck|AndrewBuck]].]] [[Image:162173 Ryugu.jpg|left|thumb|300px|Photograph of the full disc of the asteroid 162173 Ryugu, as it appeared to the Hayabusa2 spacecraft's Optical Navigation Camera – Telescopic (ONC-T) at a distance of 20 kilometres (12 miles) at 03:50 UTC on 26 June 2018. Credit: JAXA, University of Tokyo, Kochi University, Rikkyo University, Nagoya University, Chiba Institute of Technology, Meiji University, University of Aizu, AIST.{{tlx|fairuse}}]] [[Image:Asteroid-Bennu-OSIRIS-RExArrival-GifAnimation-20181203.gif|right|thumb|300px|Asteroid Bennu imaged by the OSIRIS-REx probe on arrival 3 December 2018. Credit: NASA/Goddard/University of Arizona.{{tlx|free media}}]] [[Image:Bennuarrival.png|thumb|left|250px|Photo of 101955 Bennu was taken by the OSIRIS-REx probe on 3 December 2018. Credit: NASA/Goddard/University of Arizona.]] Note that sizes and distances of bodies and orbits are ''not'' to scale in the image on the right. As of 2015, the Apollo asteroid group includes a total of 6,923 known objects of which 991 are numbered ([http://ssd.jpl.nasa.gov/sbdb_query.cgi JPL SBDB]). Ryugu shown on the left was discovered on 10 May 1999 by astronomers with the Lincoln Near-Earth Asteroid Research at the Lincoln Laboratory's Experimental Test Site near Socorro, New Mexico, in the United States.<ref name="MPC-object">{{cite web | title = 162173 Ryugu (1999 JU3) | url = https://www.minorplanetcenter.net/db_search/show_object?object_id=162173 | accessdate= 22 June 2018}}</ref> The asteroid was officially named "Ryugu" by the Minor Planet Center on 28 September 2015.<ref name="MPC-Circulars-Archive">{{cite book | title = MPC/MPO/MPS Archive | url = https://www.minorplanetcenter.net/iau/ECS/MPCArchive/MPCArchive_TBL.html | accessdate= 22 June 2018}}</ref> Initial images taken by the ''Hayabusa-2'' spacecraft on approach at a distance of 700&nbsp;km were released on 14 June 2018 and revealed a diamond shaped body and confirmed its retrograde rotation.<ref>{{cite web|title=From a distance of about 700km, Ryugu's rotation was observed|url=http://www.hayabusa2.jaxa.jp/topics/20180616je/index_e.html|publisher=JAXA|accessdate= 18 June 2018}}</ref> Between 17 and 18 June 2018, Hayabusa 2 went from 330&nbsp;km to 240&nbsp;km from Ryugu and captured a series of additional images from the closer approach.<ref>Plait, Phil, [http://www.syfy.com/syfywire/asteroid-ryugu-starts-to-come-into-focus "Asteroid Ryugu Starts to Come Into Focus"], SyFy Wire, 20 June 2018. Accessed 20 June 2018.</ref> On 21 September 2018, the first two of these rovers, which will hop around the surface of the asteroid, were released from ''Hayabusa2''.<ref>{{cite book |last1=Wall |first1=Mike |title=Japanese Probe Deploys Tiny Hopping Robots Toward Big Asteroid Ryugu |url=https://www.space.com/41898-hayabusa2-deploys-hopping-robots-asteroid-ryugu.html |accessdate=21 September 2018}}</ref> On September 22, 2018, JAXA confirmed the two rovers had successfully touched down on Ryugu's surface which marks the first time a mission has completed a successful landing on a fast-moving asteroid body.<ref name="MINERVA-II-1 Landings">{{cite book |last1=Yoshimitsu |first1=Tetsuo |last2=Kubota |first2=Takashi |last3=Tsuda |first3=Yuichi |last4=Yoshikawa |first4=Makoto |title=MINERVA-II1: Successful image capture, landing on Ryugu and hop! |url=http://www.hayabusa2.jaxa.jp/en/topics/20180922e/ |publisher=JAXA |accessdate=24 September 2018}}</ref> "This series of images [second down on the right] taken by the OSIRIS-REx spacecraft shows Bennu in one full rotation from a distance of around 50 miles (80 km). The spacecraft’s PolyCam camera obtained the 36 2.2-millisecond frames over a period of four hours and 18 minutes."<ref name=Drake>{{ cite book |author=Michael J. Drake |title=Bennu Full Rotation at a Distance of 50 Miles |publisher=NASA |location=Washington, DC USA |date=November 25, 2018 |url=https://www.asteroidmission.org/?attachment_id=12182 |accessdate=5 December 2018 }}</ref> 101955 Bennu (provisional designation 1999 RQ₃₆<ref name="contest">{{ cite book |author=Diane Murphy |title=Nine-Year-Old Names Asteroid Target of NASA Mission in Competition Run By The Planetary Society |url=http://www.planetary.org/press-room/releases/2013/nine-year-old-names-asteroid.html |accessdate=20 August 2016 |publisher=The Planetary Society |date=1 May 2013 }}</ref>, a C-type carbonaceous asteroid in the Apollo group discovered by the Lincoln Near-Earth Asteroid Research (LINEAR) Project on September 11, 1999, is a potentially hazardous object that is listed on the Sentry monitoring system, Sentry Risk Table, with the second-highest cumulative rating on the Palermo Technical Impact Hazard Scale.<ref name="risk-table">{{ cite book | title=Sentry Risk Table | publisher=NASA/JPL Near-Earth Object Program Office | accessdate=2018-03-20 |url=https://web.archive.org/web/20160911172609/http://neo.jpl.nasa.gov/risk/ }}</ref> It has a cumulative 1-in-2,700 chance of impacting Earth between 2175 and 2199.<ref name="Sentry">{{cite book | url=https://cneos.jpl.nasa.gov/sentry/details.html#?des=101955 | title=101955 1999 RQ36: Earth Impact Risk Summary | publisher=Jet Propulsion Laboratory | date=25 March 2016 | accessdate=20 March 2018 }}</ref><ref name="Meteoritics">{{ cite journal |last1=Lauretta|first1=D. S.|last2=Bartels|first2=A. E.|last3=Barucci|first3=M. A.|last4=Bierhaus|first4=E. B.|last5=Binzel|first5=R. P.|last6=Bottke|first6=W. F.|last7=Campins|first7=H.|last8=Chesley|first8=S. R.|last9=Clark|first9=B. C.|last10=Clark|first10=B. E.|last11=Cloutis|first11=E. A.|last12=Connolly|first12=H. C.|last13=Crombie|first13=M. K.|last14=Delbó|first14=M.|last15=Dworkin|first15=J. P.|last16=Emery|first16=J. P.|last17=Glavin|first17=D. P.|last18=Hamilton|first18=V. E.|last19=Hergenrother|first19=C. W.|last20=Johnson|first20=C. L.|last21=Keller|first21=L. P.|last22=Michel|first22=P.|last23=Nolan|first23=M. C.|last24=Sandford|first24=S. A.|last25=Scheeres|first25=D. J.|last26=Simon|first26=A. A.|last27=Sutter|first27=B. M.|last28=Vokrouhlický|first28=D.|last29=Walsh|first29=K. J. |title=The OSIRIS-REx target asteroid (101955) Bennu: Constraints on its physical, geological, and dynamical nature from astronomical observations |journal=Meteoritics & Planetary Science |date=April 2015 |volume=50 |issue=4 |pages=834–849 |doi=10.1111/maps.12353 |bibcode=2015M&PS...50..834L |citeseerx=10.1.1.723.9955 }}</ref> 101955 Bennu has a mean diameter of approximately {{convert|492|m|ft mi|abbr=on}} and has been observed extensively with the Arecibo Observatory planetary radar and the Goldstone Deep Space Communications Complex NASA Deep Space Network.<ref name="Nolan2013">{{cite journal |last1=Nolan|first1=M. C.|last2=Magri|first2=C.|last3=Howell|first3=E. S.|last4=Benner|first4=L. A. M.|last5=Giorgini|first5=J. D.|last6=Hergenrother|first6=C. W.|last7=Hudson|first7=R. S.|last8=Lauretta|first8=D. S.|last9=Margot|first9=J. L.|last10=Ostro|first10=S. J.|last11=Scheeres|first11=D. J. |title=Shape model and surface properties of the OSIRIS-REx target Asteroid (101955) Bennu from radar and lightcurve observations |journal=Icarus |volume=226 |issue=1 |year=2013 |pages=629–640 |issn=0019-1035 |doi=10.1016/j.icarus.2013.05.028 |bibcode=2013Icar..226..629N | url=http://www.escholarship.org/uc/item/6mh8b2t5|format=Submitted manuscript }}</ref><ref name="GDSCC-img">{{cite book | url= http://echo.jpl.nasa.gov/~lance/1999RQ36/1999RQ36.html | title=Goldstone Delay-Doppler Images of 1999 RQ36 | publisher=Jet Propulsion Laboratory }}</ref><ref name="Hudson et al. 2000">{{cite journal |author=Hudson, R. S. |author2=Ostro, S. J. |author3=Benner, L. A. M. |bibcode=2000DPS....32.0710H |title=Recent Delay-Doppler Radar Asteroid Modeling Results: 1999 RQ36 and Craters on Toutatis |journal=Bulletin of the American Astronomical Society |volume=32 |page=1001 | year=2000}}</ref> Asteroid Bennu has a roughly spheroidal shape, resembling a spinning top, with the direction of rotation about its axis retrograde with respect to its orbit and a fairly smooth shape with one prominent 10–20 m boulder on its surface, in the southern hemisphere.<ref name="Meteoritics" /> There is a well-defined ridge along the equator of asteroid Bennu that suggests that fine-grained regolith particles have accumulated in this area, possibly because of its low gravity and fast rotation.<ref name="Meteoritics" /> Observations of this minor planet by the Spitzer Space Telescope in 2007 gave an effective diameter of {{Val|484|10|u=m}}, which is in line with other studies. It has a low visible geometric albedo of {{Val|0.046|0.005}}. The thermal inertia was measured and found to vary by ±19% during each rotational period suggesting that the regolith grain size is moderate, ranging from several millimeters up to a centimeter, and evenly distributed. No emission from a potential dust coma has been detected around asteroid Bennu, which puts a limit of 10⁶&nbsp;g of dust within a radius of 4750&nbsp;km.<ref name=Emery2014>{{ cite book | title=Thermal infrared observations and thermophysical characterization of the OSIRIS-REx target asteroid (101955) Bennu, In: Conference Proceedings Asteroids, Comets, Meteors 2014 |last1=Emery | first1=J. | last2=Fernandez | first2=Y. | last3=Kelley | first3=M. | last4=Warden | first4=K. | last5=Hergenrother | first5=C. | last6=Lauretta | first6=D. | last7=Drake | first7=M. | last8=Campins | first8=H. | last9=Ziffer | first9=J. | pages=148 | editor=K. Muinonen | date=July 2014 | bibcode=2014acm..conf..148E }}</ref> Astrometric observations between 1999 and 2013 have demonstrated that 101955 Bennu is influenced by the Yarkovsky effect, causing the semimajor axis to drift on average by {{Val|284|1.5}}&nbsp;meters/year; analysis of the gravitational and thermal effects give a bulk density of ρ = {{Val|1260|70}}&nbsp;kg/m³, which is only slightly denser than water, the predicted macroporosity is {{Val|40|10}}%, suggesting that the interior has a rubble pile structure, with an estimated mass is {{Val|7.8|0.9|e=10|u=kg}}.<ref name="ChesleyFarnocchia2014">{{cite journal|last1=Chesley|first1=Steven R.|last2=Farnocchia|first2=Davide|last3=Nolan|first3=Michael C.|last4=Vokrouhlický|first4=David|last5=Chodas|first5=Paul W.|last6=Milani|first6=Andrea|last7=Spoto|first7=Federica|last8=Rozitis|first8=Benjamin|last9=Benner|first9=Lance A.M.|last10=Bottke|first10=William F.|last11=Busch|first11=Michael W.|last12=Emery|first12=Joshua P.|last13=Howell|first13=Ellen S.|last14=Lauretta|first14=Dante S.|last15=Margot|first15=Jean-Luc|last16=Taylor|first16=Patrick A.|title=Orbit and bulk density of the OSIRIS-REx target Asteroid (101955) Bennu|journal=Icarus|volume=235|date=2014|pages=5–22|issn=0019-1035|doi=10.1016/j.icarus.2014.02.020 | bibcode=2014Icar..235....5C|arxiv=1402.5573 }}</ref> Photometric observations of Bennu in 2005 yielded a synodic rotation period of {{Val|4.2905|0.0065|u=h}}, a B-type asteroid classification, which is a sub-category of C-type asteroid or carbonaceous asteroids. Polarimetric observations show that Bennu belongs to the rare F-type asteroid or F subclass of carbonaceous asteroids, which is usually associated with cometary features.<ref name="Cellino2018" /> Measurements over a range of phase angles show a phase function slope of 0.040 magnitudes per degree, which is similar to other near-Earth asteroids with low albedo.<ref name=Hergenrother2013>{{ cite journal | title=Lightcurve, Color and Phase Function Photometry of the OSIRIS-REx Target Asteroid (101955) Bennu | last1=Hergenrother | first1=Carl W. | last2=Nolan | first2=Michael C. | last3=Binzel | first3=Richard P. | last4=Cloutis | first4=Edward A. | last5=Barucci | first5=Maria Antonietta | last6=Michel | first6=Patrick | last7=Scheeres | first7=Daniel J. | last8=d'Aubigny | first8=Christian Drouet | last9=Lazzaro | first9=Daniela | last10=Pinilla-Alonso | first10=Noemi | last11=Campins | first11=Humberto | last12=Licandro | first12=Javier | last13=Clark | first13=Beth E. | last14=Rizk | first14=Bashar | last15=Beshore | first15=Edward C. | last16=Lauretta | first16=Dante S. | journal=Icarus | volume=226 | issue=1 | pages=663–670 | date=September 2013 | doi=10.1016/j.icarus.2013.05.044 | bibcode=2013Icar..226..663H }}</ref> Asteroid Bennu's basic [[mineralogy]] and chemical nature would have been established during the first 10 million years of the Solar System's formation, where the carbonaceous material underwent some geologic heating and chemical transformation into more complex minerals.<ref name="Meteoritics" /> Bennu probably began in the inner asteroid belt as a fragment from a larger body with a diameter of 100&nbsp;km, where simulations suggest a 70% chance it came from the Polana family and a 30% chance it derived from the 495 Eulalia (Eulalia family).<ref name=Bottke2015>{{cite journal | title=In search of the source of asteroid (101955) Bennu: Applications of the stochastic YORP model | last1=Bottke | first1=William F. | last2=Vokrouhlický | first2=David | last3=Walsh | first3=Kevin J. | last4=Delbo | first4=Marco | last5=Michel | first5=Patrick | last6=Lauretta | first6=Dante S. | last7=Campins | first7=Humberto | last8=Connolly | first8=Harold C. | last9=Scheeres | first9=Daniel J. | last10=Chelsey | first10=Steven R. | journal=Icarus | volume=247 | pages=191–217 | date=February 2015 | doi=10.1016/j.icarus.2014.09.046 | bibcode=2015Icar..247..191B }}</ref> Subsequently, the orbit drifted as a result of the Yarkovsky effect and mean motion resonances with the giant planets, such as Jupiter and Saturn modified the asteroid, possibly changing its spin, shape, and surface features.<ref name=Lauretta2015>{{ cite journal | title=The OSIRIS-REx target asteroid (101955) Bennu: Constraints on its physical, geological, and dynamical nature from astronomical observations | last1=Lauretta | first1=D. S. | last2=Bartels | first2=A. E. | last3=Barucci | first3=M. A. | last4=Bierhaus | first4=E. B. | last5=Binzel | first5=R. P. | last6=Bottke | first6=W. F. | last7=Campins | first7=H. | last8=Chesley | first8=S. R. | last9=Clark | first9=B. C. | last10=Clark | first10=B. E. | last11=Cloutis | first11=E. A. | last12=Connolly | first12=H. C. | last13=Crombie | first13=M. K. | last14=Delbó | first14=M. | last15=Dworkin | first15=J. P. | last16=Emery | first16=J. P. | last17=Glavin | first17=D. P. | last18=Hamilton | first18=V. E. | last19=Hergenrother | first19=C. W. | last20=Johnson | first20=C. L. | last21=Keller | first21=L. P. | last22=Michel | first22=P. | last23=Nolan | first23=M. C. | last24=Sandford | first24=S. A. | last25=Scheeres | first25=D. J. | last26=Simon | first26=A. A. | last27=Sutter | first27=B. M. | last28=Vokrouhlický | first28=D. | last29=Walsh | first29=K. J. | journal=Meteoritics & Planetary Science | volume=50 | issue=4 | pages=834–849 | date=April 2015 | doi=10.1111/maps.12353 | bibcode=2015M&PS...50..834L | citeseerx=10.1.1.723.9955}}</ref> A possible cometary origin for Bennu, based on similarities of its spectroscopic properties with known comets, with the estimated fraction of comets in the population of Near Earth asteroids is {{val|8|5|u=%}}.<ref name="Cellino2018">{{Cite arXiv | title=Unusual polarimetric properties of (101955) Bennu: similarities with F-class asteroids and cometary bodies | eprint=1808.07812 | last1=Hergenrother | first1=Carl W | author2=Maria Antonietta Barucci | last3=Barnouin | first3=Olivier | last4=Bierhaus | first4=Beau | last5=Binzel | first5=Richard P | last6=Bottke | first6=William F | last7=Chesley | first7=Steve | last8=Clark | first8=Ben C | last9=Clark | first9=Beth E | last10=Cloutis | first10=Ed | author11=Christian Drouet d'Aubigny | last12=Delbo | first12=Marco | last13=Emery | first13=Josh | last14=Gaskell | first14=Bob | last15=Howell | first15=Ellen | last16=Keller | first16=Lindsay | last17=Kelley | first17=Michael | last18=Marshall | first18=John | last19=Michel | first19=Patrick | last20=Nolan | first20=Michael | last21=Rizk | first21=Bashar | last22=Scheeres | first22=Dan | last23=Takir | first23=Driss | last24=Vokrouhlický | first24=David D | last25=Beshore | first25=Ed | last26=Lauretta | first26=Dante S | class=astro-ph.EP | year=2018 }}</ref> {{clear}} ==Moon== As the Moon is in the neighborhood of Earth it can experience the same showers, but will have its own phenomena due to its lack of an atmosphere ''per se'', such as vastly increasing its sodium tail.<ref>{{cite journal|doi=10.1029/91GL02543 | bibcode=1991GeoRL..18.2101H | volume=18 | issue=11 | title=A possible meteor shower on the Moon | year=1991 | journal=Geophysical Research Letters | pages=2101–2104 | last1 = Hunten | first1 = D. M.}}</ref> NASA now maintains an ongoing database of observed impacts on the moon<ref>[http://www.nasa.gov/centers/marshall/news/lunar/ Lunar Impacts]</ref> maintained by the Marshall Space Flight Center whether from a shower or not. ==Mars== [[Image:Earth Sol63A UFO-A067R1.jpg|thumb|right|250px|Mars meteor is photographed by the MER ''Spirit'' rover. Credit: .]] Mars, and thus its moons, is known to have meteor showers.<ref>[http://www.arm.ac.uk/~aac/meteors_on_mars.html Meteor showers at Mars]</ref> Only the relatively slower motion of the meteoroids due to increased distance from the sun should marginally decrease meteor brightness. This is somewhat balanced in that the slower descent means that Martian meteors have more time in which to ablate.<ref>[http://star.arm.ac.uk/~aac/atmos2.jpg Can Meteors Exist at Mars?]</ref> On March 7, 2004, the panoramic camera on Mars Exploration Rover ''Spirit'' recorded a streak shown in he image on the right which is now believed to have been caused by a meteor from a Martian meteor shower associated with comet 114P/Wiseman-Skiff. A strong display from this shower was expected on December 20, 2007. Other showers speculated about are a "Lambda Geminid" shower associated with the Eta Aquariids of Earth (''i.e.'', both associated with Comet 1P/Halley), a "Beta Canis Major" shower associated with Comet 13P/Olbers, and "Draconids" from 5335 Damocles.<ref>[http://star.arm.ac.uk/~aac/showers.jpg Meteor Showers and their Parent Bodies]</ref> {{clear}} ==Hypotheses== {{main|Hypotheses}} # All Zodiac constellations should have meteor showers. # All constellations some 80-90° away from the Zodiac should not have meteor showers. ==See also== {{div col|colwidth=20em}} * [[Radiation astronomy/Asteroids|Asteroid astronomy]] * [[Radiation astronomy/Clouds|Cloud astronomy]] * [[Radiation astronomy/Comets|Cometary astronomy]] * [[Radiation astronomy/Cryometeors|Cryometeor astronomy]] * [[Radiation astronomy/Fieries|Fiery meteor astronomy]] * [[Radiation astronomy/Kuiper belts|Kuiper belt astronomy]] * [[Radiation astronomy/Lithometeors|Lithometeor astronomy]] * [[Radiation astronomy/Meteoroids|Meteoroid radiation astronomy]] * [[Radiation astronomy/Oort clouds|Oort cloud astronomy]] * [[Radiation astronomy/Planets|Planetary astronomy]] {{Div col end}} ==References== {{reflist|2}} ==External links== * [https://www.ta3.sk/IAUC22DB/MDC2007/Roje/sortowanie_roje.php?corobic_roje=1 Established meteor showers] * [[w:List of meteor showers|List of meteor showers]] <!-- footer templates --> {{Radiation astronomy resources}}{{Sisterlinks|Meteor showers}} <!-- footer categories --> [[Category:Radiation astronomy/Lectures]] 3wyl5e3w8yj3ll9qvu1roxix5zdzzbg 0 234531 2416236 2398989 2022-08-18T17:51:44Z Marshallsumter 311529 wikitext text/x-wiki [[Image:Upperatmoslight1.jpg|300px|thumb|right|Representation of upper-atmospheric lightning and electrical-discharge phenomena are displayed. Credit: [[c:User:Abestrobi|Abestrobi]].{{tlx|free media}}]] [[Image:Lightning sprites2.png|left|thumb|300px|An illustration of different kinds of transient luminous events (TLEs) is displayed. Credit: Carlos Miralles (AeroVironment) and Tom Nelson (FMA).{{tlx|fairuse}}]] Lightning is more than ground-to-cloud electron transfer. "Cloud flashes sometimes have visible channels that extend out into the air around the storm (cloud-to-air or CA), but do not strike the ground. The terms sheet lightning or intra-cloud lightning (IC) refers to lightning embedded within a cloud that lights up as a sheet of luminosity during the flash. A related term, heat lightning, is lightning or lightning-induced illumination that is too far away for thunder to be heard. Lightning can also travel from cloud-to-cloud (CC). Spider lightning refers to long, horizontally traveling flashes often seen on the underside of stratiform clouds."<ref name=Miralles>{{ cite book |author=Carlos Miralles (AeroVironment) |author2=Tom Nelson (FMA) |title=SEVERE WEATHER 101 Lightning Types |publisher=NSSL, NOAA |location= |date=25 June 2019 |url=https://www.nssl.noaa.gov/education/svrwx101/lightning/types/ |accessdate=25 June 2019 }}</ref> "Large thunderstorms are capable of producing other kinds of electrical phenomena called transient luminous events (TLEs) that occur high in the atmosphere. They are rarely observed visually and not well understood. The most common TLEs include red sprites, blue jets, and elves."<ref name=Miralles/> The illustration on the left labels Elves, Sprites, and tendrils, a stratiform region producing positive cloud-to-ground flashes with spider lightning on the right to conventional cloud-to-air discharge, upward superbolt, blue jets and negative cloud-to-ground flash near convective core. Approximate altitudes in the Earth's atmosphere and ionosphere are indicated. {{clear}} ==Natural electric field of the Earth== [[Image:ISS-40 Thunderheads near Borneo.jpg|thumb|right|250px|Thunderheads near Borneo, Indonesia are featured in this image photographed by an Expedition 40 crew member on the International Space Station. Credit: M. Justin Wilkinson, Jacobs and Michael Trenchard, Barrios Technology at NASA-JSC.{{tlx|free media}}]] The '''natural electric field of the Earth''' refers to the planet Earth having a natural direct current (DC) electric field or potential gradient from the ground upwards to the ionosphere. The static fair-weather electric field in the atmosphere is ~150 volts per meter (V/m) near the Earth's surface, but it drops exponentially with height to under 1 V/m at 30 km altitude, as the conductivity of the atmosphere increases. "In this image [on the right], late afternoon sunlight casts long shadows from thunderhead anvils down onto southern Borneo. Near the horizon (image top center), more than 1000 kilometers away from the space station, storm formation is assisted by air currents rising over the central mountains of Borneo."<ref name=Wilkinson>{{ cite book |author=M. Justin Wilkinson |author2=Jacobs |author3=Michael Trenchard |title=Thunderstorms over Borneo |publisher=NASA |location=Washington, DC USA |date=5 August 2014 |url=http://earthobservatory.nasa.gov/IOTD/view.php?id=84308&src=ve |accessdate=2016-01-06 }}</ref> "Winds usually blow in different directions at different altitudes. At the time of this photo, high-altitude winds were clearly sweeping the tops off the many tallest thunderclouds, generating long anvils of diffuse cirrus plumes that trail south. At lower levels of the atmosphere, “streets” of white dots—fair-weather cumulus clouds—are aligned with west-moving winds. Small smoke plumes from forest fires in Borneo are also aligned west."<ref name=Wilkinson/> The surface of the Earth is negatively charged, carrying 500,000 Coulombs (C) of electric charge (500 kC),<ref name=Labrador>{{ cite book |author=Lorenzo Labrador |title=Sensitivity of Tropospheric Chemistry to the Source of NO_x from Lightning: Simulations with the Global 3D Chemistry-Transport Model MATCH-MPIC |publisher=University of Heidelberg |location=Heidelberg, Germany |date=April 5, 2005 |url=http://archiv.ub.uni-heidelberg.de/volltextserver/5870/1/diss.ps |accessdate=2014-12-20 }}</ref> and is at 300,000 volts (V), 300 kV,<ref name=Polk>{{ cite book |author=C Polk |title=Relation of ELF Noise and Schumann Resonances to Thunderstorm Activity, In: ''Planetary Electrodynamics'' |publisher=Gordon and Breach Science Publishers |location=New York, NY, USA |date=1969 |editor=Samuel C. Coroniti, J. Hughes |pages=55-83 |url=https://books.google.com/books?id=wDXzhyskXOgC&printsec=frontcover&hl=en&sa=X&ei=1zeMVLYFh_nJBPHagdAN&ved=0CBQQ6AEwAA#v=onepage&f=false |arxiv= |bibcode= |doi= |pmid= |{{isbn|0677136005}} |accessdate=2014-12-13 }}</ref> relative to the positively charged ionosphere. There is a constant flow of electricity, at around 1350 amperes (A) [approximately 1100 A]<ref name=Polk/>, and resistance of the Earth's atmosphere is around 220 Ohms.[3] This gives a power of around 400 megawatts (MW). There are several theoretical proposals to harvest this power that Nikola Tesla elaborated (around 1901) and his followers. The main principle is to mimic the conditions when charged particles from space interact with the higher and lower layers of Earth's atmosphere. In any case averages a mere 0.8 microwatts per square metre of the earth's surface (compare solar radiation which is one billion times as large). The charge is maintained by the stream of charged particles from the Sun. This process affects the ionosphere, as well as the troposphere, possibly causing thunderstorms. The electrical energy stored in the Earth's atmosphere is around 150 gigajoules (GJ). The Earth-ionosphere system acts as a giant capacitor, of capacity 1.8 Farads. The Earth's surface carries around -1 nC of electric charge per square meter. {{clear}} ==Lightning theory== [[Image:Russia’s Soyuz-2 rocket was hit by lightning.png|right|thumb|300px|Russia’s Soyuz-2 rocket was hit by lightning shortly after its launch on 27 May 2019. Credit: Roscosmos/Ministry of Defense of the Russian Federation/Dmitry Rogozin.{{tlx|fairuse}}]] '''Def.''' a "flash of light produced by short-duration, high-voltage discharge of electricity within a cloud, between clouds, or between a cloud and the earth"<ref name=LightningWikt>{{ cite book |title=lightning |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=9 June 2014 |url=https://en.wiktionary.org/wiki/lightning |accessdate=2014-06-09 }}</ref> is called '''lightning'''. A typical cloud-to-ground lightning flash culminates in the formation of an electrically conducting plasma channel through the air in excess of {{convert|5|km|mi|abbr=on}} tall, from within the cloud to the ground's surface. The actual discharge is the final stage of a very complex process.<ref>Uman (1986) p. 81.</ref> At its peak, a typical thunderstorm produces three or more ''strikes'' to the Earth per minute.<ref>Uman (1986) p. 55.</ref> Lightning primarily occurs when warm air is mixed with colder air masses.<ref>{{Cite book |title=Sprites, Elves and Intense Lightning Discharges |last=Füllekrug |first=Martin |last2=Mareev |first2=Eugene A. |last3=Rycroft |first3=Michael J. |date=2006-05-01 |publisher=Springer Science & Business Media |{{isbn=9781402046285}} |language=en |url=https://web.archive.org/web/20171104190958/https://books.google.com/books?id=g1foWWN5odwC&pg=PA90&hl=en&sa=X&ved=0ahUKEwiV44XT9uXUAhVJ7mMKHb3pBUoQ6AEIMDAC#v=onepage&f=false |bibcode=2006seil.book.....F }}</ref> However, it can also occur during dust storms, forest fires, tornadoes, volcanic eruptions, and even in the cold of winter, where the lightning is known as thundersnow.<ref>New Lightning Type Found Over Volcano? [https://web.archive.org/web/20100209015048/http://news.nationalgeographic.com/news/2010/02/100203-volcanoes-lightning/ February 9, 2010 News.nationalgeographic.com (February 2010). Retrieved on June 23, 2012.</ref><ref>{{cite book |title=Bench collapse sparks lightning, roiling clouds |accessdate=October 7, 2012 |publisher=United States Geological Survey |date=June 11, 1998 |work=Volcano Watch |url=https://web.archive.org/web/20120114172155/http://hvo.wr.usgs.gov/volcanowatch/1998/98_06_11.html }}</ref> Hurricanes typically generate some lightning, mainly in the rainbands as much as {{convert|160|km|mi|abbr=on}} from the center.<ref>Pardo-Rodriguez, Lumari (Summer 2009) Lightning Activity in Atlantic Tropical Cyclones: Using the Long-Range Lightning Detection Network (LLDN) [url=https://web.archive.org/web/20130309085405/http://nldr.library.ucar.edu/repository/assets/soars/SOARS-000-000-000-193.pdf] March 9, 2013. MA Climate and Society, Columbia University Significant Opportunities in Atmospheric Research and Science Program.</ref><ref> Hurricane Lightning [url=https://web.archive.org/web/20170815013425/https://science.nasa.gov/science-news/science-at-nasa/2006/09jan_electrichurricanes/] August 15, 2017 NASA, January 9, 2006.</ref><ref>The Promise of Long-Range Lightning Detection in Better Understanding, Nowcasting, and Forecasting of Maritime Storms [url=https://web.archive.org/web/20130309085405/http://www.unidata.ucar.edu/committees/polcom/2009spring/statusreports/BusingerS09.pdf] March 9, 2013. Long Range Lightning Detection Network</ref> The image on the right demonstrates that proper electrical conduction during a rocket launch can protect the rocket, payload and prevent fuel explosions. {{clear}} ==Magnetism== [[Image:LIRM anomaly with archaeological hearths.jpg|thumb|upright|right|250px|Lightning-induced remanent magnetization (LIRM) is mapped during a magnetic field gradient survey of an archaeological site located in Wyoming, United States. Credit: [[w:user:Pöhönen|Pöhönen]].{{tlx|fairuse}}.]] The movement of electrical charges produces a magnetic field, where the intense currents of a lightning discharge create a fleeting but very strong magnetic field and the lightning current path passes through rock, soil, or metal these materials can become permanently magnetized, known as lightning-induced remanent magnetism, or LIRM which follow the least resistive path, often horizontally near the surface<ref>{{cite journal |title=The Re-magnetization of a Surface Outcrop by Lightning Currents |doi=10.1111/j.1365-246X.1961.tb02963.x |date=1961 |last1=Graham |first1=K.W.T. |journal=Geophysical Journal International |volume=6 |issue=1 |page=85 |bibcode = 1961GeoJ....6...85G }}</ref><ref>Cox A. (1961). Anomalous Remanent Magnetization of Basalt [url=https://web.archive.org/web/20130529011301/http://pubs.usgs.gov/bul/1083e/report.pdf] May 29, 2013 . U.S. Geological Survey Bulletin 1038-E, pp. 131–160.</ref> but sometimes vertically, where faults, ore bodies, or ground water offers a less resistive path.<ref>Bevan B. (1995). "Magnetic Surveys and Lightning". ''Near Surface Views'' (newsletter of the Near Surface Geophysics section of the Society of Exploration Geophysics). October 1995, pp. 7–8.</ref> One theory suggests that lodestones, natural magnets encountered in ancient times, were created in this manner.<ref>{{cite journal | doi=10.1029/1999GL900496 | first=Peter | last=Wasilewski |author2=Günther Kletetschka | title=Lodestone: Nature's only permanent magnet – What it is and how it gets charged | url=http://lep694.gsfc.nasa.gov/gunther/gunther/Wasilewski1999.pdf | archiveurl=https://web.archive.org/web/20061003193325/http://lep694.gsfc.nasa.gov/gunther/gunther/Wasilewski1999.pdf | archivedate=October 3, 2006 | journal=Geophysical Research Letters | volume=26 | issue=15 | pages=2275–78 | date=1999 | accessdate=July 13, 2009|bibcode = 1999GeoRL..26.2275W }}</ref> Lightning-induced magnetic anomalies can be mapped in the ground,<ref>{{Cite journal |last=Sakai |first=H. S. |last2=Sunada |first2=S. |last3=Sakurano |first3=H. |date=1998 |title=Study of Lightning Current by Remanent Magnetization |journal=Electrical Engineering in Japan |volume=123 |issue=4 |pages=41–47 |doi=10.1002/(SICI)1520-6416(199806)123:4<41::AID-EEJ6>3.0.CO;2-O }}</ref><ref> Archaeo-Physics, LLC Lightning-induced magnetic anomalies on archaeological sites [https://web.archive.org/web/20071012080847/http://www.archaeophysics.com/pubs/LIRM.html] October 12, 2007. Archaeophysics.com. Retrieved on June 23, 2012.</ref> and analysis of magnetized materials can confirm lightning was the source of the magnetization<ref>{{Cite journal |last=Maki |first=David |date=2005 |title=Lightning strikes and prehistoric ovens: Determining the source of magnetic anomalies using techniques of environmental magnetism |journal=Geoarchaeology |volume=20 |issue=5 |pages=449–459 |doi=10.1002/gea.20059 |url=http://www.archaeophysics.com/pubs/lightning-oven.pdf |archiveurl=https://web.archive.org/web/20130515025335/http://www.archaeophysics.com/pubs/lightning-oven.pdf |archivedate=May 15, 2013 |df=mdy-all |citeseerx=10.1.1.536.5980 }}</ref> and provide an estimate of the peak current of the lightning discharge.<ref>{{Cite journal |last=Verrier |first=V. |last2=Rochette |first2=P. |date=2002 |title=Estimating Peak Currents at Ground Lightning Impacts Using Remanent Magnetization |journal=Geophysical Research Letters |volume=29 |issue=18 |page=1867 |doi=10.1029/2002GL015207 |bibcode = 2002GeoRL..29.1867V }}</ref> Magnetic fields generated by plasma may induce hallucinations in subjects located within 200 meters of a severe lightning storm.<ref>{{cite book | url=https://www.technologyreview.com/s/418887/magnetically-induced-hallucinations-explain-ball-lightning-say-physicists/ |title = Magnetically Induced Hallucinations Explain Ball Lightning, Say Physicists }}</ref> {{clear}} ==Negative lightning== Lightning is a naturally occurring electrostatic discharge during which two electrically charged regions in the atmosphere or ground temporarily equalize themselves, causing the instantaneous release of as much as one billion joules of energy.<ref>{{Cite journal |last=Maggio |first=Christopher R. |last2=Marshall |first2=Thomas C. |last3=Stolzenburg |first3=Maribeth |date=2009 |title=Estimations of charge transferred and energy released by lightning flashes |url=https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2008JD011506 |journal=Journal of Geophysical Research: Atmospheres |language=en |volume=114 |issue=D14 |doi=10.1029/2008JD011506 |issn=2156-2202 }}</ref> The negative end of the bidirectional leader fills a positive charge region, also called a well, inside the cloud while the positive end fills a negative charge well. Leaders often split, forming branches in a tree-like pattern.<ref>Ultraslow-motion video of stepped leader propagation: [https://web.archive.org/web/20100413125231/http://www.ztresearch.com/] April 13, 2010 </ref> Electricity takes every path available to it.<ref>{{cite book |title=The Path of Least Resistance |url=http://ecmweb.com/content/path-least-resistance |archiveurl=https://web.archive.org/web/20160104215214/http://ecmweb.com/content/path-least-resistance |archivedate=January 4, 2016 |df=mdy-all |date=July 2001 }}</ref> The rate at which the return stroke current travels has been found to be around 100,000&nbsp;km/s.<ref>{{Cite journal | last1 = Idone | first1 = V. P. | last2 = Orville | first2 = R. E. | last3 = Mach | first3 = D. M. | last4 = Rust | first4 = W. D. | title = The propagation speed of a positive lightning return stroke | doi = 10.1029/GL014i011p01150 | journal = Geophysical Research Letters | volume = 14 | issue = 11 | page = 1150 | year = 1987 | pmid = | pmc = |bibcode = 1987GeoRL..14.1150I }}</ref> High-speed videos (examined frame-by-frame) show that most negative CG lightning flashes are made up of 3 or 4 individual strokes, though there may be as many as 30.<ref>[[#Uman|Uman (1986)]] Ch. 5, p. 41.</ref> The dart leaders making connections with the ground is what causes a majority of subsequent return strokes.<ref name=Warner>{{cite book |url=https://ztresearch.blog/education/ground-flashes/ |title=Ground Flashes |last=Warner |first=Tom |website=ZT Research |accessdate=2017-11-09 |date=2017-05-06 }}</ref> Large bolts of negative lightning can carry up to 120 kA and 350 coulombs.<ref>Hasbrouck, Richard. Mitigating Lightning Hazards [https://web.archive.org/web/20131005005230/https://www.llnl.gov/str/pdfs/05_96.1.pdf] October 5, 2013, Science & Technology Review May 1996. Retrieved on April 26, 2009.</ref> Cloud-to-ground lightning, where lightning appears to exit from the cumulonimbus cloud and propagate a considerable distance through clear air before veering towards, and striking, the ground, are known as "bolts from the blue", which are in fact negative flashes that begin as intracloud flashes within the cloud, the negative leader then exits the cloud from the positive charge region before propagating through clear air and striking the ground some distance away.<ref name="ReferenceA">{{cite journal |doi=10.1029/2011JD016890 |title=Lightning morphology and impulse charge moment change of high peak current negative strokes |journal=Journal of Geophysical Research: Atmospheres |volume=117 |issue=D4 |pages=n/a |year=2012 |last1=Lu |first1=Gaopeng |last2=Cummer |first2=Steven A |last3=Blakeslee |first3=Richard J |last4=Weiss |first4=Stephanie |last5=Beasley |first5=William H |bibcode=2012JGRD..117.4212L |citeseerx=10.1.1.308.9842 }}</ref><ref name="ReferenceB">{{cite journal |doi=10.1038/ngeo162 |title=Upward electrical discharges from&nbsp;thunderstorms |journal=Nature Geoscience |volume=1 |issue=4 |page=233 |year=2008 |last1=Krehbiel |first1=Paul R |last2=Riousset |first2=Jeremy A |last3=Pasko |first3=Victor P |last4=Thomas |first4=Ronald J |last5=Rison |first5=William |last6=Stanley |first6=Mark A |last7=Edens |first7=Harald E |bibcode=2008NatGe...1..233K }}</ref> Clear-air lightning describes lightning that occurs with no apparent cloud close enough to have produced it: in the U.S. and Canadian Rockies, a thunderstorm can be in an adjacent valley and not observable from the valley where the lightning bolt strikes, either visually or audibly; European and Asian mountainous areas experience similar events; in areas such as sounds, large lakes or open plains, when the storm cell is on the near horizon (within {{convert|26|km|mi|abbr=on}}) there may be some distant activity, a strike can occur and as the storm is so far away, the strike is referred to as '''a bolt from the blue'''.<ref>{{cite book |url=http://news.discovery.com/earth/weather-extreme-events/lightning-on-clear-day-110729.htm |title=When Lightning Strikes Out of a Blue Sky |work=DNews |accessdate=October 15, 2015 |archiveurl=https://web.archive.org/web/20151101020717/http://news.discovery.com/earth/weather-extreme-events/lightning-on-clear-day-110729.htm |archivedate=November 1, 2015 |df=mdy-all }}</ref> These flashes usually begin as normal intracloud lightning flashes before the negative leader exits the cloud and strikes the ground a considerable distance away.<ref name="ReferenceA"/><ref name="ReferenceB"/> Positive clear-air strikes can occur in highly sheared environments where the upper positive charge region becomes horizontally displaced from the precipitation area.<ref>{{cite book | url = http://www.crh.noaa.gov/gid/Web_Stories/2004/other/lightningsafety/intro/introduction.php | archiveurl = https://web.archive.org/web/20090514211951/http://www.crh.noaa.gov/gid/Web_Stories/2004/other/lightningsafety/intro/introduction.php | archivedate = May 14, 2009 | title = Bolt from the Blue | publisher = National Oceanic and Atmospheric Administration | accessdate = August 20, 2009 | last = Lawrence | first = D | date = November 1, 2005 }}</ref> ==Positive lightning== The reverse happens in a positive CG flash, where electrons travel upward along the lightning channel and a positive charge is transferred to the ground. Positive lightning is less common than negative lightning, and on average makes up less than 5% of all lightning strikes.<ref>{{cite book | title = NWS JetStream – The Positive and Negative Side of Lightning | publisher = National Oceanic and Atmospheric Administration | url = http://www.srh.noaa.gov/jetstream/lightning/positive.htm | accessdate = September 25, 2007 | archiveurl = https://web.archive.org/web/20070705205815/http://www.srh.noaa.gov/jetstream/lightning/positive.htm | archivedate = July 5, 2007 | df = mdy-all }}</ref> There are six different mechanisms theorized to result in the formation of downward positive lightning.<ref>{{cite journal |doi=10.1029/2012JD017545 |title=Positive lightning: An overview, new observations, and inferences |journal=Journal of Geophysical Research: Atmospheres |volume=117 |issue=D8 |pages=n/a |year=2012 |last1=Nag |first1=Amitabh |last2=Rakov |first2=Vladimir A |bibcode=2012JGRD..117.8109N }}</ref> * Vertical wind shear displacing the upper positive charge region of a thundercloud, exposing it to the ground below. * The loss of lower charge regions in the dissipating stage of a thunderstorm, leaving the primary positive charge region. * A complex arrangement of charge regions in a thundercloud, effectively resulting in an '''inverted dipole''' or '''inverted tripole''' in which the main negative charge region is above the main positive charge region instead of beneath it. * An unusually large lower positive charge region in the thundercloud. * Cutoff of an extended negative leader from its origin which creates a new bidirectional leader in which the positive end strikes the ground, commonly seen in anvil-crawler spider flashes. * The initiation of a downward positive branch from an intracloud lightning flash. The average positive ground flash has roughly double the peak current of a typical negative flash, and can produce peak currents up to 400,000 amperes (400 kA) and charges of several hundred coulombs.<ref>V.A. Rakov, M.A. Uman, Positive and bipolar lightning discharges to ground, in: Light. Phys. Eff., Cambridge University Press, 2003: pp. 214–240</ref><ref name="U.A.BakshiM.V.Bakshi2009">{{cite book |author1=U.A.Bakshi |author2=M.V.Bakshi |title=Power System – II |url=https://books.google.com/books?id=oOj4NjQ8xGQC&pg=SA12-PA5 |date=1 January 2009 |publisher=Technical Publications |isbn=978-81-8431-536-3 |page=12 |archiveurl=https://web.archive.org/web/20170312135216/https://books.google.com/books?id=oOj4NjQ8xGQC&pg=SA12-PA5 |archivedate=March 12, 2017 |df=mdy-all }}</ref> Furthermore, positive ground flashes with high peak currents are commonly followed by long continuing currents, a correlation not seen in negative ground flashes.<ref>{{cite journal |doi=10.1029/2010JD014330 |title=High-speed video observations of positive lightning flashes to ground |journal=Journal of Geophysical Research: Atmospheres |volume=115 |issue=D24 |pages=D24201 |year=2010 |last1=Saba |first1=Marcelo M. F |last2=Schulz |first2=Wolfgang |last3=Warner |first3=Tom A |last4=Campos |first4=Leandro Z. S |last5=Schumann |first5=Carina |last6=Krider |first6=E. Philip |last7=Cummins |first7=Kenneth L |last8=Orville |first8=Richard E |bibcode=2010JGRD..11524201S }}</ref> Positive lightning tends to occur more frequently in winter storms, as with thundersnow, during intense tornadoes<ref name="Tornadic lightning">{{cite journal |author1=Antony H. Perez |author2=Louis J. Wicker |author3=Richard E. Orville |last-author-amp=yes |title=Characteristics of Cloud-to-Ground Lightning Associated with Violent Tornadoes |journal=Weather Forecast. |volume=12 |issue=3 |pages=428–37 |doi=10.1175/1520-0434(1997)012<0428:COCTGL>2.0.CO;2 |bibcode = 1997WtFor..12..428P |year=1997 }}</ref> and in the dissipation stage of a [[thunderstorm]].<ref name="GHCC primer2">{{cite book | url = http://thunder.nsstc.nasa.gov/primer/primer2.html|title = A Lightning Primer – Characteristics of a Storm |accessdate = February 8, 2009 |author1=Christian, Hugh J. |author2=McCook, Melanie A. |work =NASA |archive-url=https://web.archive.org/web/20160305002214/http://thunder.nsstc.nasa.gov/primer/primer2.html |archive-date=March 5, 2016}}</ref> Huge quantities of extremely low frequency (ELF) and very low frequency (VLF) radio waves are also generated.<ref name="boccippio">{{cite journal | last = Boccippio | first = DJ | title = Sprites, ELF Transients, and Positive Ground Strokes | journal = Science | volume = 269 | pages = 1088–1091 |date=August 1995 | doi = 10.1126/science.269.5227.1088 | pmid = 17755531 | last2 = Williams | first2 = ER | last3 = Heckman | first3 = SJ | last4 = Lyons | first4 = WA | last5 = Baker | first5 = IT | last6 = Boldi | first6 = R | issue = 5227 | bibcode = 1995Sci...269.1088B }}</ref> The lightning shows that during the Southern Ontario tornado outbreak of 2005 there was a strong peak during the F2 tornadoes, dominated by the positive cloud-to-ground strikes, that this is often the case in tornadic storms, followed by a sharp drop and then another peak, but this one dominated by negative lightning during the flooding phase of the storm.<ref name=cstar>{{cite book |url=http://cstar.cestm.albany.edu/nrow/nrow7/Hamilton%5CTornadoes%20to%20Torrents.ppt |title=The Southern Ontario Severe Weather Event of Aug 19, 2005 |author1=Collaborative Science, Technology |author2=Applied Research Program |publisher=State University of New York at Albany |accessdate = June 15, 2010}}</ref> The dangers of more powerful positive lightning were not understood until the destruction of a glider in 1999.<ref name=schleicher>{{cite book |url=http://www.aaib.gov.uk/publications/bulletins/december_1999/schleicher_500699.cfm |title=Schleicher ASK 21 two seat glider, 17 April 1999 - GOV.UK |publisher= |accessdate=21 March 2018 }}</ref> The 1985 standard in force in the US at the time of the glider crash, Advisory Circular AC 20-53A,<ref name=schleicher /> was replaced by Advisory Circular AC 20-53B in 2006.<ref>{{cite book |url=http://www.airweb.faa.gov/Regulatory_and_Guidance_Library/rgAdvisoryCircular.nsf/MainFrame?OpenFrameSet&CFID=11301165& |title=FAA Advisory Circulars |publisher= |accessdate=21 March 2018 }}</ref> However, it is unclear whether adequate protection against positive lightning was incorporated.<ref>[http://www.nolan-law.com/hiding-requirements-suspicion-theyre-inadequate/ Hiding requirements = suspicion they're inadequate], Nolan Law Group, January 18, 2010</ref><ref>[http://www.lightningtech.com/pdfs/A_PROPOSED_ADDITION_TO_LIGHTNING_ENVIRONMENT_STDS.pdf A Proposed Addition to the Lightning Environment Standards Applicable to Aircraft]. J. Anderson Plumer. Lightning Technologies, Inc. published 2005-09-27.</ref> In 2010 a lightning measurement station was installed atop a {{convert|120|m|ft|abbr=on}} tall telecommunications tower on the mountain Säntis by the Electromagnetic Compatibility Lab of the EPFL in Lausanne, where in the first nine months of operation it recorded about 50 strikes, including 7 positive lightning strikes.<ref>{{Cite news |url=https://actu.epfl.ch/news/lightning-does-actually-strike-in-the-same-place-t/ |title=Lightning does actually strike in the same place twice! |last=Barraud |first=Emmanuel |date=7 February 2011 |accessdate=29 May 2018 }}</ref> Typical speed of a positive lightning return stroke (''cf.'' stepped leader above) is 100,000,000 m/s, 360,000,000 km/h, 220,000,000 mph, 0.3 c.<ref>{{Cite journal | doi=10.1029/GL014i011p01150 |title = The propagation speed of a positive lightning return stroke | journal=Geophysical Research Letters | volume=14 | issue=11 | pages=1150–1153 |year = 1987 |last1 = Idone |first1 = Vincent P. | last2=Orville | first2=Richard E. | last3=Mach | first3=Douglas M. | last4=Rust | first4=W. David }}</ref> ==Thunderstorms== [[Image:Thunderstorm formation.jpg|thumb|upright=2.0|right|250px|Stages of a thunderstorm's life. Credit: [[c:User:The High Fin Sperm Whale|The High Fin Sperm Whale]].{{tlx|free media}}]] [[Image:Anvil shaped cumulus panorama edit crop.jpg|thumb|right|250px|Anvil-shaped thundercloud in the mature stage. Credit: [[c:user:Fir0002|Fir0002]].{{tlx|free media}}]] [[Image:Lightning in Spring Creek, Nevada.JPG|thumb|left|Lightning, which is responsible for the majority of fires in the American West. Credit: [[c:user:Jrmichae|Jrmichae]].{{tlx|free media}}]] A thunderstorm, also known as an electrical storm or a lightning storm, is a storm characterized by the presence of lightning and its acoustic effect on the Earth's atmosphere, known as thunder.<ref>{{cite book |date=21 April 2005 |url=http://www.weather.gov/glossary/index.php?letter=t |title=Weather Glossary – T |publisher=National Weather Service |accessdate=2006-08-23 }}</ref> Relatively weak thunderstorms are sometimes called '''thundershowers'''.<ref>{{cite book |url=https://www.weather.gov/jetstream/lightning_faq |title=NWS JetStream |website=National Weather Service |accessdate=26 January 2019 }}</ref> Thunderstorms can form and develop in any geographic location but most frequently within the mid-latitude, where warm, moist air from tropical latitudes collides with cooler air from polar latitudes.<ref name="nssl.noaa.gov">{{cite book |publisher=National Oceanic and Atmospheric Administration |author=National Severe Storms Laboratory |title=tornadoes...''Nature's Most Violent Storms'' |url=http://www.nssl.noaa.gov/edu/safety/tornadoguide.html |date=September 1992 |work=A PREPAREDNESS GUIDE |accessdate=2008-08-03 }}</ref> Warm air has a lower density than cool air, so warmer air rises upwards and cooler air will settle at the bottom<ref>{{cite book |url=https://books.google.com/?id=PDtIAAAAIAAJ&pg=PA462 |title=Civil engineers' pocket book: a reference-book for engineers, contractors |author=Albert Irvin Frye |page=462 |publisher=D. Van Nostrand Company |date=1913 |accessdate=2009-08-31 }}</ref> (this effect can be seen with a hot air balloon).<ref>{{cite book|url= https://books.google.com/?id=ssO_19TRQ9AC&pg=PA112 |title=Ancient Chinese Inventions | author = Yikne Deng | publisher = Chinese International Press | isbn=978-7-5085-0837-5 | date=2005 | pages = 112–13|accessdate=2009-06-18}}</ref> Clouds form as relatively warmer air, carrying moisture, rises within cooler air. The moist air rises, and, as it does so, it cools and some of the water vapor in that rising air condenses.<ref>{{cite book |author=FMI |date=2007 |url=http://www.zamg.ac.at/docu/Manual/SatManu/main.htm?/docu/Manual/SatManu/CMs/FgStr/backgr.htm |title=Fog And Stratus – Meteorological Physical Background |publisher=Zentralanstalt für Meteorologie und Geodynamik |accessdate=2009-02-07 }}</ref> When the moisture condenses, it releases energy known as latent heat of condensation, which allows the rising packet of air to cool less than the cooler surrounding air<ref>{{cite book |url=https://books.google.com/?id=RRSzR4NQdGkC&pg=PA20 |title=Storm world: hurricanes, politics, and the battle over global warming |author=Chris C. Mooney |page=20 |isbn=978-0-15-101287-9 |publisher=Houghton Mifflin Harcourt |date=2007 |accessdate=2009-08-31 }}</ref> continuing the cloud's ascension. If enough instability is present in the atmosphere, this process will continue long enough for cumulonimbus clouds to form and produce lightning and thunder. Meteorological indices such as convective available potential energy (CAPE) and the lifted index can be used to assist in determining potential upward vertical development of clouds.<ref name="CAPE">{{cite journal |author=David O. Blanchard |title=Assessing the Vertical Distribution of Convective Available Potential Energy |journal=Weather and Forecasting |volume=13 |issue=3 |pages=870–7 |publisher=American Meteorological Society |date=September 1998 |doi= 10.1175/1520-0434(1998)013<0870:ATVDOC>2.0.CO;2 |bibcode= 1998WtFor..13..870B }}</ref> Generally, thunderstorms require three conditions to form: # Moisture # An unstable airmass # A lifting force (heat) All thunderstorms, regardless of type, go through three stages: the '''developing stage''', the '''mature stage''', and the '''dissipation stage'''.<ref name="Extreme Weather">{{cite book |title=Extreme Weather |author=Michael H. Mogil |date=2007 |publisher=Black Dog & Leventhal Publisher |location=New York |isbn=978-1-57912-743-5 |pages=210–211 }}</ref> The average thunderstorm has a {{convert|24|km|mi|abbr=on}} diameter. Depending on the conditions present in the atmosphere, each of these three stages take an average of 30 minutes.<ref name="tsbasics">{{cite book |url=http://www.nssl.noaa.gov/primer/tstorm/tst_basics.html |title=A Severe Weather Primer: Questions and Answers about Thunderstorms |author=National Severe Storms Laboratory |publisher=National Oceanic and Atmospheric Administration |date=2006-10-15 |accessdate=2009-09-01 }}</ref> A "thunderstorm supplies a negative charge to the Earth. The net positive space charge in the air between the ground and a height of ~ 10 km is nearly equal to the negative charge on the Earth's surface".<ref name=Stansbery>{{ cite book |author=Eileen K. Stansbery |title=A global model of thunderstorm electricity and the prediction of whistler duct formation |publisher=Rice University |location=Houston, Texas USA |date=March 1989 |editor= |pages=174 |url=http://scholarship.rice.edu/bitstream/handle/1911/16298/9012871.PDF?sequence=1 |arxiv= |bibcode= |doi= |pmid= |isbn= |accessdate=2015-01-03 }}</ref> 'Giant' "thunderclouds can produce transverse electric fields of tens of microvolts per meter in the equatorial plane of the midlatitude magnetosphere."<ref name=Park>{{ cite journal |author=C. G. Park |author2=M. Dejnakarintra |title=Penetration of thundercloud electric fields into the ionosphere and magnetosphere: 1. Middle and subauroral latitudes |journal=Journal of Geophysical Research Space Physics |month=1 October |year=1973 |volume=78 |issue=28 |pages=6623-33 |url=http://onlinelibrary.wiley.com/doi/10.1029/JA078i028p06623/abstract |arxiv= |bibcode= |doi=10.1029/JA078i028p06623 |pmid= |accessdate=2015-01-06 }}</ref> The "contribution to global thunderstorm activity by oceanic thunderstorms should be regarded as itself having a diurnal variation of some 18% in amplitude."<ref name=Muir>{{ cite journal |author=M.S. Muir |author2=C.A. Smart |title=Diurnal variations in the atmospheric electric field on the South Polar ice-cap |journal=Journal of Atmospheric and Terrestrial Physics |month=February |year=1981 |volume=43 |issue=2 |pages=171-7 |url=http://www.sciencedirect.com/science/article/pii/0021916981900775 |arxiv= |bibcode= |doi=10.1016/0021-9169(81)90077-5 |pmid= |accessdate=2015-01-06 }}</ref> Pyrocumulonimbus are cumuliform clouds that can form over a large fire and that are particularly dry.<ref>{{cite book | url=http://glossary.ametsoc.org/wiki/Pyrocumulonimbus |title=Pyrocumulonimbus |work= AMS Glossary |accessdate=December 16, 2015 }}</ref> There is "a decrease in thunderstorms at the time of high cosmic rays and an increase in thunderstorms 2-4 days later."<ref name="Lethbridge">{{cite journal |author=Mae Devoe Lethbridge |title=Thunderstorms, cosmic rays, and solar-lunar influences |journal=Journal of Geophysical Research |month= |year=1990 |volume=95 |issue=D9 |pages=13,645-9 |url=http://www.agu.org/pubs/crossref/1990/JD095iD09p13645.shtml |arxiv= |bibcode= |doi=10.1029/JD095iD09p13645 |pmid= |accessdate=2012-08-22 }}</ref> {{clear}} ==Elves== "A third phenomenon was discovered in video from the STS-41 mission (October 1990) in the lower ionosphere directly above an active thunderstorm. It consisted of a large horizontal brightening several hundred kilometers across at the altitude of the airglow layer. In 1995, Lyons and associates confirmed the existence of this type of very brief brightening which they named Emissions of Light and Very Low Frequency Perturbations From Electromagnetic Pulse Sources (ELVES)."<ref name=Boeck>{{ cite book |author= William L. Boeck |author2=Otha H. Vaughan Jr. |author3=Richard J. Blakeslee |author4=Bernard Vonnegut |author5=Marx Brook |title=The Role of the Space Shuttle Videotapes in the Discovery of Sprites, Jets, and Elves |publisher=Global Hydrology Resource Center |location=Huntsville, Alabama USA |date= 1994 |url=http://thunder.nsstc.nasa.gov/bookshelf/pubs/sprites.html |accessdate=2015-04-10 }}</ref> "When the lightning phenomena were viewed from a different vantage point - from above the thunderstorms (e.g., from space, aircraft or mountaintop) - new discoveries were made and insights gained into the upper atmospheric optical flashes now commonly referred to as sprites, jets, starters, and ELVES."<ref name=Boeck/> "There are no historical reports from eyewitnesses describing the phenomenon that is now called "Emission of Light and Very Low Frequency Perturbations From Electro-magnetic Pulse Sources" or ELVES (Lyons and Nelson, 1995). The one millisecond lifetime of this phenomenon explains why there have been no eyewitness accounts describing a brief flash that would fill the entire night sky for any observer within a 100 km radius from the causative lightning flash. Inan (1990) and Inan et al. (1991) predicted the existence of strong Joule heating of the base of the ionosphere by the electromagnetic pulses of natural lightning."<ref name=Boeck/> "After fifteen sprites and one jet had been identified in the shuttle video, a distinctively different event was discovered in shuttle video acquired on October 7, 1990 directly above an active thunderstorm off the coast of French Guyana (Boeck et al., 1992). A large horizontal flash appeared at the altitude of the airglow layer. It occurred in the video field before the appearance of main lightning flash in a thunderstorm that was near the limb of the Earth. They concluded that the causative lightning flash occurred slightly after the video scan passed the location of the storm image. There was a clear view of the mesosphere below the airglow layer, but there was no indication of a sprite in the video sequence (although a sprite event was captured 4½ hours earlier under similar moonlight conditions). A search of the shuttle video failed to produce a second example of this type of horizontal flash. Since it was clear that this was not an example of a sprite or a jet, the observations were published on the basis of this single example. The video was presented at the 1991 Spring AGU meeting (Boeck et al., 1991a) as well as at the Aerospace Lightning Conference (Boeck et al., 1991b). To promote a better understanding of these new phenomena as seen from space, Vaughan distributed a number of video tapes to various researchers who had express an interest in the phenomena. Seminars based on the video tape observations were given at Los Alamos National Laboratory and Stanford University. Researchers at both of these institutions have made major contributions to the theory of Sprite and ELVES phenomena."<ref name=Boeck/> "Several years passed before there was a second successful measurement of the ELVES phenomenon. On June 23, 1995 Lyons et al. (1995) and Fukunishi et al. (1996) confirmed the existence of a flash similar to the airglow flash seen earlier in the shuttle data, and Lyons et al. (1995) gave it the name Emissions of Light and Very Low Frequency Perturbations From Electromagnetic Pulse Sources (ELVES). Lyons presented video images captured by a low-light-level TV camera sited near Ft. Collins, Colorado. The ELVES phenomenon has a characteristic event duration of one millisecond (Fukunishi, 1996)."<ref name=Boeck/> ==Sprites== [[Image:BigRed-Sprite.jpg|thumb|right|250px|This is the first color image of a sprite. Credit: [[c:User:Eastview|Eastview]].{{tlx|free media}}]] [[Image:Jellyfish lightning sprite.jpg|thumb|right|250px|The image shows an example of jellyfish lightning or sprites. Credit: H. H. C. Stenbaek-Nielsen.{{tlx|fairuse}}]] "The phenomenon, now known as a sprite, was first accidently documented on ground based videotape recordings on the night of July 6, 1989. Video observations from the space shuttle acquired from 1989 through 1991 provided 17 additional examples to confirm the existence of the sprites phenomenon."<ref name=Boeck/> "Throughout the historical scientific literature, there are sprinklings of eyewitness accounts of unusual "lightning" observed in the clear air above nighttime thunderstorms. The descriptions use phases such as "continuous darts of light... ascended to a considerable altitude, resembling rockets more than lightning." (MacKenzie and Toynbee, 1886), "a luminous trail shot up to 15 degrees or so, about as fast as, or faster than, a rocket" (Everett, 1903), "a long weak streamer of a reddish hue" (Malan, 1937), "flames appearing to rise from the top of the cloud" (Ashmore, 1950), or "the discharge assumed a shape similar to roots of a tree in an inverted position" (Wood, 1951). Partly because these eyewitness reports of unusual "lightning" appearing above thunderstorms were never captured on film, the lightning science community generally ignored them. The lack of an established vocabulary and the existence of several distinctive phenomena contributed to the variation in the verbal descriptions."<ref name=Boeck/> "[Boccippio et al., 1994] has shown that these bright discharges are associated with large amplitude return strokes bringing positive charge downward. In fact, a positive return stroke accompanied the only MLE sprite recorded within range of a ground based lightning detection network. The videos showed that additional discharges continued in the clouds after a sprite for a total mean time of a second, which can be interpreted as evidence for a continuing current. All together, this was strong evidence that the sprite above the thunderstorm was caused, directly or indirectly, by an energetic lightning discharge."<ref name=Boeck/> The "range to the sprites was well over 1000 kilometers [...] The width of the sprites varied considerably from very thin or even several thin filaments to broad columns some kilometers across, while the bright "head" (when visible) had dimensions on the order of kilometers."<ref name=Boeck/> "The optical and RF measurements collected during the 1994 field campaign rapidly uncovered the basic properties of sprites (Lyons, 1994; Lyons and Williams, 1994; Lyons et al., 1994; Sentman et al., 1994; Sentman et al., 1995; Wescott et al., 1994; Lyons et al., 1995a,b). Other workers (e.g., Boccippio, 1994) established the causal association of sprites with positive cloud-to-ground lightning discharges."<ref name=Boeck/> Sprites "are typically associated with low flash rate cells [...] The shuttle videos established that lightning directly or indirectly causes sprites."<ref name=Boeck/> The first "color image of a sprite [...] was obtained during a 1994 NASA/University of Alaska aircraft campaign to study sprites. The event was captured using an intensified color TV camera. The red color was subsequently determined to be from nitrogen fluorescent emissions excited by a lightning stroke in the underlying thunderstorm."<ref name=Eastview>{{ cite book |author=[[c:User:Eastview|Eastview]] |title=''File:BigRed-Sprite.jpg'' |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=10 March 2009 |url=https://commons.wikimedia.org/wiki/File:BigRed-Sprite.jpg |accessdate=2015-04-11 }}</ref> "During a thunderstorm, high in the ionosphere, you’ll find an odd variety of lightning that is far above the thunderstorm itself. Jellyfish lightning, also known as sprites, are red flashes of light that last for a few seconds. They can have a wide bell-shaped top and tentacle-like wisps of light at the bottom, resembling a jellyfish."<ref name=Winter>{{ cite book |author=Lisa Winter |title=Origins Of Mysterious "Sprite" Lightning Discovered |publisher=IFLScience |location= |date=May 12, 2014 |url=http://www.iflscience.com/environment/origins-mysterious-sprite-lightning-discovered |accessdate=2014-08-30 }}</ref> "In high-speed videos we can see the dynamics of sprite formation and then use that information to model and to reproduce the dynamics."<ref name=Qin>{{ cite book |author=Jianqi Qin |title=Origins Of Mysterious "Sprite" Lightning Discovered |publisher=IFLScience |location= |date=May 12, 2014 |url=http://www.iflscience.com/environment/origins-mysterious-sprite-lightning-discovered |accessdate=2014-08-30 }}</ref> "These sprites only occur during thunderstorms, though sprites are about three times higher up than storms. [...] the storms are necessary for sprites to occur, they aren’t quite sufficient enough to cause them on their own [...] The tentacle-like tendrils at the bottom were shown to form much faster than the bell-shaped top. [...] localized plasma irregularities can spark the formation of a sprite."<ref name=Winter/> Sprites are large-scale electrical discharges which occur high above a thunderstorm cloud, or cumulonimbus, giving rise to a quite varied range of visual shapes, triggered by the discharges of positive lightning between the thundercloud and the ground.<ref name="boccippio"/> The phenomena were named after the mischievous sprite, e.g., Shakespeare's Ariel or Puck,<ref>From [https://books.google.com/books?id=dtBrs114-NEC&pg=PA128 page 128] of: John Friedman, ''Out of the Blue: A History of Lightning'' (New York, New York: Random House, Inc., 2008):<br/> "Dr. Davis Sentman of the University of Alaska, one of the few scientists studying these luminous, ghostlike phenomena [i.e., sprites], named the eerie flashes of colored lights after Shakespeare's mischievous spirits of the air — Ariel in ''The Tempest'' and Puck in "A Midsummer Night's Dream."</ref> and is also an acronym for '''S'''tratospheric/mesospheric '''P'''erturbations '''R'''esulting from '''I'''ntense '''T'''hunderstorm '''E'''lectrification.<ref>{{cite book | url=http://news.psu.edu/story/140845/1997/09/01/research/sprites-and-elves-atmosphere | title=Sprites and Elves in the Atmosphere &#124; Penn State University }}</ref> They normally are colored reddish-orange or greenish-blue, with hanging tendrils below and arcing branches above. They can also be preceded by a reddish halo. They often occur in clusters, lying {{convert|50|km|mi}} to {{convert|90|km|mi}} above the Earth's surface. Sprites have been witnessed thousands of times.<ref>Walter A. Lyons and Michey D. Schmidt (2003). [http://ams.confex.com/ams/pdfpapers/65547.pdf P1.39 The Discovery of Red Sprites as an Opportunity For Informal Science Education.] American Meteorological Society. Retrieved on 2009-02-18.</ref> Sprites have been held responsible for otherwise unexplained accidents involving high altitude vehicular operations above thunderstorms.<ref>{{cite book |title=Full report on the uncontrolled free fall of a stratospheric balloon payload provoked by a Sprite |url=http://stratocat.com.ar/fichas-e/1989/PAL-19890609.htm |author=STRATOCAT – Stratospheric balloons history and present }}</ref> {{clear}} ==Blue jets== [[Image:Gigantic jet NOIRLab.jpg|thumb|right|250px|Blue jets appear to initiate between the upper positive charge region and a negative screening layer directly above this region. Credit: International Gemini Observatory/NOIRLab/NSF/AURA/A. Smith.{{tlx|free media}}]] "In the summer of 1994, Wescott and associates (Wescott et al., 1994; Wescott et al., 1995a,b) confirmed the existence of jets and named the phenomena blue jets when they recorded a very active thunderstorm in Arkansas, USA using both a low-light-level monochrome and a color video cameras. The video was collected during a nighttime research flight using two aircraft that were flying around the thunderstorm. During this flight, color video imagery established that jets are blue in color and sprites are red. A total of 52 jets were seen during a 20-minute time span. The jets developed over several video frames, with a characteristic time of the order of 100 ms and propagation speeds similar to that of a step leader process (i.e., ~ 105 m/s). The video released after this flight proved to be a turning point in establishing wide interest in these phenomena. The spectacular multiple close-up images of these jets completely overshadowed the single, poorly resolved jet observation from the space shuttle."<ref name=Boeck/> Blue jets differ from sprites in that they project from the top of the cumulonimbus above a thunderstorm, typically in a narrow cone, to the lowest levels of the ionosphere 40 to 50&nbsp;km (25 to 30 miles) above the earth. In addition, whereas red sprites tend to be associated with significant lightning strikes, blue jets do not appear to be directly triggered by lightning (they do, however, appear to relate to strong hail activity in thunderstorms).<ref name=Sprites>{{ cite book |url=http://www.psu.edu/ur/2001/bluejets.html |title=Fractal Models of Blue Jets, Blue Starters Show Similarity, Differences to Red Sprites }}</ref> They are also brighter than sprites and, as implied by their name, are blue in color. The color is believed to be due to a set of blue and near-ultraviolet emission lines from neutral and ionized molecular nitrogen. {{clear}} ==Blue starters== Blue "starters (Wescott et al., 1995) [are] an upward moving luminous phenomenon closely related to blue jets."<ref name=Boeck/> "Blue starters appear to be blue jets that never quite make it."<ref name="50oaK">[https://archive.today/20120630142356/http://www.spaceref.ca/news/viewpr.html?pid=6878 Fractal models of blue jets, blue starters show similarity, differences to red sprites]</ref> ==Gigantic jets== [[Image:Sprites-Argentina-2014-02-02.png|thumb|right|250px|This is a gigantic jet observed over a thunderstorm. Credit: [[c:User:Welias|Welias]].{{tlx|free media}}]] [[Image:Gigantic jet near Phillipines.png|thumb|left|250px|This is an image of a gigantic jet above a thunderstorm near the Philippines. Credit: H. T. Su, R. R. Hsu, A. B. Chen, Y. C. Wang, W. S. Hsiao, W. C. Lai, L. C. Lee, M. Sato & H. Fukunishi.{{tlx|free media}}]] "On February 02, 2014, the Oro Verde Observatory (República Argentina) reported 10 or more gigantic jet event[s] observed over a thunderstorm in Entre Ríos south. Storm center [is] located at 33°S, 60°W, near the Rosario city."<ref name=Welias>{{ cite book |author=[[commons:User:Welias|Welias]] |title=''File:Sprites-Argentina-2014-02-02.png'' |publisher=Wikimedia Foundation, Inc |location=San Francisco, California |date=17 July 2014 |url=https://commons.wikimedia.org/wiki/File:Sprites-Argentina-2014-02-02.png |accessdate=2015-04-11 }}</ref> Each gigantic "jet could transfer 30 coulombs of negative charge from the clouds to the ionosphere (H T Su ''et al.'' 2003 Nature 423 974)."<ref name=Pennicott>{{ cite book |author=Katie Pennicott |title=Giant jets caught on camera |publisher=Institute of Physics |location= |date=25 June 2003 |url=http://physicsworld.com/cws/article/news/2003/jun/25/giant-jets-caught-on-camera |accessdate=2015-04-11 }}</ref> "During a thunderstorm in the South China Sea in July 2002, Su and co-workers used low-light-level cameras to photograph the clouds every 17 milliseconds. The five jets they observed - dubbed carrot-jets or tree-jets according to their shapes - were visible for some tens of milliseconds. But crucially, the team also detected simultaneous bursts of radio waves in four of the five cases, which indicates that the jets had transferred significant amounts of charge. The thunderclouds were at an altitude of 16 km."<ref name=Pennicott/> "Such electromagnetic bursts have only previously been linked with powerful lightning strikes, which are known to transfer large quantities of charge. But [lightning may not have] triggered the radio waves they detected, since the local lightning detection network registered no strikes at the times of the jets."<ref name=Pennicott/> On the left is an image of a fully developed gigantic jet above a thunderstorm near the Philippine. {{clear}} ==Gamma rays== {{main|Radiation astronomy/Gamma rays}} [[Image:Antimatter Explosions.ogv|thumb|right|200px|The red dots show some of the ~500 terrestrial gamma-ray flashes daily detected by the Fermi Gamma-ray Space Telescope through 2010. Credit: NASA/Goddard Space Flight Center.{{tlx|free media}}]] "A number of observations by space-based telescopes have revealed ... gamma ray emissions ... terrestrial gamma-ray flashes (TGFs). These observations pose a challenge to current theories of lightning, especially with the discovery of the clear signatures of antimatter produced in lightning.<ref>[http://www.sciencenews.org/view/generic/id/49288/title/Signature_of_antimatter_detected_in_lightning Signature Of Antimatter Detected In Lightning - Science News]</ref> A TGF has been linked to an individual lightning stroke occurring within 1.5&nbsp;ms of the TGF event,<ref name=Inan>U.S. Inan, S.C. Reising, G.J. Fishman, and J.M. Horack. On the association of terrestrial gamma-ray bursts with lightning and implications for sprites. ''Geophysical Research Letters'', 23(9):1017-20, May 1996. As quoted by [http://elf.gi.alaska.edu/spr20010406.html#InanUS:theatg elf.gi.alaska.edu] Retrieved 2007-03-06.</ref> proving for the first time that the TGF was of atmospheric origin and associated with lightning strikes. The Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) spacecraft, as reported by David Smith of University of California (UC) Santa Cruz, has been observing TGFs at a much higher rate, indicating that these occur about 50 times per day globally (still a very small fraction of the total lightning on the planet). The energy levels recorded exceed 20 MeV. Apparently, the gamma radiation fountains upward from starting points at surprisingly low altitudes in thunderclouds. Steven Cummer, from Duke University's Pratt School of Engineering, said, "These are higher energy gamma rays than come from the sun. And yet here they are coming from the kind of terrestrial thunderstorm that we see here all the time." In 2009, the Fermi Gamma Ray Telescope in Earth orbit observed an intense burst of gamma rays corresponding to positron annihilations coming out of a storm formation. Scientists wouldn't have been surprised to see a few positrons accompanying any intense gamma ray burst, but the lightning flash detected by Fermi appeared to have produced about 100 trillion positrons. This has been reported by media in January 2011, it is an effect, never considered to happen before.<ref>http://news.nationalgeographic.com/news/2011/01/110111-thunderstorms-antimatter-beams-fermi-radiation-science-space/</ref> {{clear}} ==Atmospheres== {{main|Radiation astronomy/Atmospheres}} Charge is apparently replaced by the action of thunderstorms including lightning. The thunderstorm system acts as a type of battery to keep the fine weather system charged."<ref name=Uman>{{ cite book |author=Martin A. Uman |title=The Lightning Discharge |publisher=Academic Press, Inc. |location=Orlando, Florida USA |date=1987 |editor=William L. Donn |pages=375 |url= |arxiv= |bibcode= |doi= |pmid= |isbn= }}</ref> ==Solar winds== [[Image:1024px--D3- LighningStormPanorama.jpg|thumb|right|300px|A lightning storm over England is imaged. Credit: Catalin Fatu.{{tlx|fairuse}}]] "The sun spits out charged particles that hit our atmosphere two to four days later at 1.5-2.7 million kph, but it does not do so evenly."<ref name=Luntz>{{ cite book |author=Stephen Luntz |title=Solar Activity Could Cause Lightning Storms On Earth |location= |date=May 16, 2014 |url=http://www.iflscience.com/space/solar-activity-could-cause-lightning-storms-earth |accessdate=2014-08-31 }}</ref> "The solar wind is not continuous, it has slow and fast streams. Because the Sun rotates, these streams can be sent out behind each other - so if you have a fast solar wind catching up with a slow solar wind, it causes a concentration to occur."<ref name=Scott>{{ cite book |author=Christ Scott |title=Solar Activity Could Cause Lightning Storms On Earth |location= |date=May 16, 2014 |url=http://www.iflscience.com/space/solar-activity-could-cause-lightning-storms-earth |accessdate=2014-08-31 }}</ref> "The slow phase is composed similarly to the solar corona while fast particles have a composition close to that of the photosphere, the outer layer of the sun that produces the light."<ref name=Luntz/> There is "a 31% increase in average lightning strikes over central England (422 to 321) in the 40 days after major solar wind events compared to the days beforehand. Lightning peaked 12-18 days after the wind's arrival. A matching increase in thundery days provided supporting evidence."<ref name=Luntz/> "It's unexpected, because these streams of particles bring with them an enhanced magnetic field - and this shields Earth from the very high-energy cosmic rays from outside of the Solar System.”<ref name=Scott/> "The reduction in cosmic rays is only around 1%, but still noticeable. Cosmic rays emitted by supernovae are thought to trigger lighting strikes, and it was expected that the shielding effect of the solar wind would cause a reduction in lightning, rather than an increase."<ref name=Luntz/> Sunspot "numbers negatively correlate with thunder days in other parts of the world."<ref name=Luntz/> "High speed streams were found to occur after periods when the sun was putting out less light, but sunspot numbers increased. Scott and his fellow authors attribute this to the streams coming from an active region appearing on the eastern side of the sun."<ref name=Luntz/> "We propose that these particles, while not having sufficient energies to reach the ground and be detected there, nevertheless electrify the atmosphere as they collide with it, altering the electrical properties of the air and thus influencing the rate or intensity at which lightning occurs."<ref name=Scott/> {{clear}} ==Venus== During the Soviet Venera program, the Venera 11 and Venera 12 probes detected a constant stream of lightning, and Venera 12 recorded a powerful clap of thunder soon after it landed. The European Space Agency's Venus Express recorded abundant lightning in the high atmosphere.<ref name="Venus Express">{{ cite book |url=http://web.archive.org/web/20071130201237/http://www.cnn.com/2007/TECH/space/11/28/venus.lightning.ap/index.html |title=Venus also zapped by lightning |publisher=CNN |date=29 November 2007 |accessdate=2007-11-29 }}</ref> The clouds of Venus are capable of producing [[lightning]] much like the clouds on Earth.<ref name="Russell_2007" /> The existence of lightning had been controversial since the first suspected bursts were detected by the Soviet [[w:Venera|Venera probe]]s. The lightning rate is at least half of that on Earth.<ref name="Russell_2007">{{ cite journal | doi = 10.1038/nature05930 | author = S. T. Russell |author2=T. L. Zhang |author3=M. Delva |author4=W. Magnes |author5=R. J. Strangeway |author6=H. Y. Wei | year = 2007 | title = Lightning on Venus inferred from whistler-mode waves in the ionosphere |journal=Nature | volume = 450 | issue = 7170 | pages = 661–662 | pmid = 18046401 | bibcode = 2007Natur.450..661R }}</ref> In 2007 the Venus Express probe discovered that a huge double atmospheric vortex exists at the south pole of the planet.<ref>{{ cite journal |first=Eric|last=Hand |year=2007 |title=European mission reports from Venus |date=November|journal=Nature|issue=450 |pages=633–660|doi=10.1038/news.2007.297}}</ref><ref>{{cite book|author=Staff|url=http://news.bbc.co.uk/1/hi/sci/tech/7117303.stm |title=Venus offers Earth climate clues |accessdate=2007-11-29 |date=28 November 2007}}</ref> During the Soviet Venera program, the ''Venera 9'' orbiter obtained spectroscopic evidence of [[lightning]] on Venus,<ref>{{cite journal |last1=Kranopol'skii |first1=V. A. |title=Lightning on Venus according to Information Obtained by the Satellites ''Venera 9'' and ''10'' |journal=Cosmic Research |date=1980 |volume=18 |issue=3 |pages=325–330 |bibcode=1980CosRe..18..325K }}</ref> and the ''Venera 12'' descent probe obtained additional evidence of lightning and thunder.<ref name="Russell, Philips">{{cite journal |url=http://www-ssc.igpp.ucla.edu/personnel/russell/papers/ashen/ |title=The Ashen Light |journal=Advances in Space Research |last1=Russell |first1=C. T. |last2=Phillips |first2=J. L. |date=1990 |pages=137–141 |volume=10 |issue=5 |bibcode=1990AdSpR..10..137R |doi=10.1016/0273-1177(90)90174-X }}</ref><ref>{{cite book |url=http://nssdc.gsfc.nasa.gov/nmc/spacecraftDisplay.do?id=1978-086C |title=''Venera 12'' Descent Craft |work=National Space Science Data Center |publisher=NASA |accessdate=10 September 2015 }}</ref> The European Space Agency's ''Venus Express'' in 2007 detected whistler waves further confirming the occurrence of lightning on Venus.<ref name="Russell_2007"/><ref name="Venus Express"/> One possibility is that ash from a volcanic eruption was generating the lightning. Another piece of evidence comes from measurements of sulfur dioxide concentrations in the atmosphere, which dropped by a factor of 10 between 1978 and 1986, jumped in 2006, and again declined 10-fold.<ref name = "ESA_2012-12-03">{{cite book |url=http://www.esa.int/Our_Activities/Space_Science/Venus_Express/Have_Venusian_volcanoes_been_caught_in_the_act |title=Have Venusian volcanoes been caught in the act? |publisher=European Space Agency |first=Markus |last=Bauer |date=3 December 2012 |accessdate=20 June 2015 |archiveurl=https://web.archive.org/web/20131103183610/http://www.esa.int/Our_Activities/Space_Science/Venus_Express/Have_Venusian_volcanoes_been_caught_in_the_act |archivedate=3 November 2013 }}</ref> This may mean that levels had been boosted several times by large volcanic eruptions.<ref>{{cite journal |title=Transport of {{chem |SO |2}} by explosive volcanism on Venus |journal=Journal of Geophysical Research |first=Lori S. |last=Glaze |volume=104 |issue=E8 |pages=18899–18906 |date=August 1999 |doi=10.1029/1998JE000619 |bibcode=1999JGR...10418899G }}</ref><ref name="Marcq2012">{{cite journal |title=Variations of sulphur dioxide at the cloud top of Venus's dynamic atmosphere |journal=Nature Geoscience |first1=Emmanuel |last1=Marcq |first2=Jean-Loup |last2=Bertaux |first3=Franck |last3=Montmessin |first4=Denis |last4=Belyaev |volume=6 |issue=1 |pages=25–28 |date=January 2013 |doi=10.1038/ngeo1650 |bibcode=2013NatGe...6...25M }}</ref> ==Earth== [[Image:-D3- LighningStormPanorama.jpg|thumb|center|400px|This is a panorama photograph taken during a lightning storm over Bucharest, Romania. Credit: [[w:User:Catalin.Fatu|Catalin.Fatu]].{{tlx|free media}}]] [[Image:Blesk.jpg|thumb|right|250px|The image shows multiple lightning discharges. Credit: [[c:user:EL Caballero|EL Caballero]].{{tlx|free media}}]] [[Image:Lightning striking the Eiffel Tower - NOAA.jpg|left|thumb|250px|Lightning striking the Eiffel Tower, June 3, 1902, at 9:20 P.M. This is one of the earliest photographs of lightning in an urban setting In:"Thunder and Lightning", Camille Flammarion, translated by Walter Mostyn Published in 1906. Credit: M. G. Loppé.{{tlx|free media}}]] [[Image:Relazione dell' ultima eruzione del Vesuvio - accaduta nel mese di agosto di questo anno 1779 = Rélation de la dernière èruption du Vésuve - arrivée au mois d'Août de cette année 1779 (1779) (14580004709).jpg|thumb|right|200px|This painting of the 1779 eruption of Vesuvius depicts lightning occurring during the eruption. Credit: Michele Torcia, Pierre-Jacques Antoine Volaire, Louis Boilly.{{tlx|free media}}]] [[Image:Port and lighthouse overnight storm with lightning in Port-la-Nouvelle.jpg|left|thumb|300px|Port and lighthouse overnight storm has lightning in Port-la-Nouvelle in the Aude department in southern France. Credit: [https://www.flickr.com/people/108892623@N02 Maxime Raynal from France].{{tlx|free media}}]] Lightning is an atmospheric discharge of electricity, which typically occurs during thunderstorms, and sometimes during volcanic eruptions or dust storms. Lightning strikes or bolts across the Earth's sky emit X-rays.<ref name=Newitz>{{ cite journal |author=Newitz, A. |year=2007 |title=Educated Destruction 101 |journal=Popular Science |month=September |page=61 }}</ref> A superbolt, more powerful than an ordinary lightning bolt, struck a cornfield near Leland, Illinois, leaving a crater one foot deep, and breaking windows in homes almost a mile away.<ref name=Burt>Christopher C. Burt, ''Extreme Weather: A Guide & Record Book'' (W. W. Norton & Company, 2007), p149</ref> During the 1779 eruption of Vesuvius as depicted in the painting of the right, lightning was observed to strike at or near the column of eruption. {{clear}} ===Spatial distribution=== [[Image:Global lightning strikes.png|thumb|right|300px|World map showing frequency of lightning strikes, in flashes per km² per year (equal-area projection), from combined 1995–2003 data from the Optical Transient Detector and 1998–2003 data from the Lightning Imaging Sensor. Credit: [[w:user:Citynoise|Citynoise]].{{tlx|free media}}]] On Earth, the lightning frequency is approximately 44 (± 5) times per second, or nearly 1.4 billion flashes per year<ref name="EncyWorldClim freq">{{cite book |url=https://books.google.com/?id=-mwbAsxpRr0C&pg=PA452 |title=Encyclopedia of World Climatology |accessdate=February 8, 2009 |publisher=National Oceanic and Atmospheric Administration |author=Oliver, John E. | isbn=978-1-4020-3264-6 |date=2005 }}</ref> and the average duration is 0.2 seconds made up from a number of much shorter flashes (strokes) of around 60 to 70 microseconds.<ref>{{cite book |url=http://hyperphysics.phy-astr.gsu.edu/hbase/electric/lightning2.html |title=Lightning |work=gsu.edu |accessdate=December 30, 2015 |archiveurl=https://web.archive.org/web/20160115043319/http://hyperphysics.phy-astr.gsu.edu/hbase/electric/lightning2.html |archivedate=January 15, 2016 |df=mdy-all }}</ref> '''Lightning hotspots''': The place on Earth where lightning occurs most often is near the small village of Kifuka in the mountains of the eastern Democratic Republic of the Congo,<ref>{{cite book |url=http://www.wondermondo.com/Countries/Af/CongoDR/SudKivu/Kifuka.htm |title=Kifuka – place where lightning strikes most often |accessdate=November 21, 2010 |publisher=Wondermondo |archiveurl=https://web.archive.org/web/20111001201900/http://www.wondermondo.com/Countries/Af/CongoDR/SudKivu/Kifuka.htm |archivedate=October 1, 2011 |df=mdy-all |date=2010-11-07 }}</ref> where the elevation is around {{convert|975|m|ft|-2|abbr=on}}. On average, this region receives 158 lightning strikes per 1 square kilometer (0.39 sq mi) per year.<ref name="NOAA freq">{{cite book |url=http://sos.noaa.gov/datasets/Atmosphere/lightning.html |archiveurl=https://web.archive.org/web/20080330025304/http://sos.noaa.gov/datasets/Atmosphere/lightning.html |archivedate=March 30, 2008 |title=Annual Lightning Flash Rate |accessdate=February 8, 2009 |publisher=National Oceanic and Atmospheric Administration }}</ref> Lake Maracaibo in Venezuela averages 297 days per year with lightning activity.<ref>Fischetti, M. (2016) [http://www.nature.com/scientificamerican/journal/v314/n5/full/scientificamerican0516-76.html Lightning Hotspots], Scientific American 314: 76 (May 2016)</ref> Other lightning hotspots include Catatumbo in Venezuela, Singapore,<ref name="nea">{{cite book |url=http://app.nea.gov.sg/cms/htdocs/article.asp?pid=1203 |archiveurl=https://web.archive.org/web/20070927224804/http://app.nea.gov.sg/cms/htdocs/article.asp?pid=1203 |archivedate=2007-09-27 |title=Lightning Activity in Singapore |accessdate=September 24, 2007 |publisher=National Environmental Agency |date=2002 }}</ref> and Lightning Alley in Central Florida.<ref name="alley">{{cite book |url=http://www.nasa.gov/centers/kennedy/news/lightning_alley.html |title=Staying Safe in Lightning Alley |accessdate=September 24, 2007 |publisher=NASA |date=January 3, 2007 |archiveurl=https://web.archive.org/web/20070713041430/http://www.nasa.gov/centers/kennedy/news/lightning_alley.html |archivedate=July 13, 2007 |df=mdy-all }}</ref><ref name="fe">{{cite book |url=http://www.floridaenvironment.com/programs/fe00703.htm |title=Summer Lightning Ahead |accessdate=September 24, 2007 |publisher=Florida Environment.com |date=2000 |author=Pierce, Kevin |archiveurl=https://web.archive.org/web/20071012160959/http://floridaenvironment.com/programs/fe00703.htm |archivedate=October 12, 2007 |df= }}</ref> {{clear}} ===Intra-cloud lightning=== [[Image:396078main horizontal bolt.jpg|right|thumb|300px|Flashes of intracloud lightning, the horizontal type shown here, usually occur within clouds high in the atmosphere. Credit: Christoph Laidig, NASA Earth Science Picture of the Day.{{tlx|fairuse}}]] Lightning discharges may occur between areas of cloud without contacting the ground. When it occurs between two separate clouds it is known as ''inter-cloud lightning'', and when it occurs between areas of differing electric potential within a single cloud it is known as ''intra-cloud lightning''. Intra-cloud lightning is the most frequently occurring type.<ref name="GHCC primer2" /> {{clear}} ===Cloud to cloud lightning=== [[Image:Lightning cloud to cloud (aka).jpg|right|thumb|250px|This image shows a cloud to cloud lightning in a very stormy and rainy night in Zwickau, Germany. Credit: [[c:user:Aka|André Karwath]].{{tlx|free media}}]] [[Image:Ligtning new delhi view 1.GIF|thumb|left|250px|Branching of cloud to cloud lightning, New Delhi, India. Credit: [[c:user:Karthik Easvur|Karthik Easvur]].{{tlx|free media}}]] [[Image:Cloud to cloud lightning strike.jpg|thumb|center|250px|Multiple paths of cloud-to-cloud lightning, Swifts Creek, Victoria, Australia. Credit: [[c:user:Fir0002|Fir0002]].{{tlx|free media}}]] [[Image:Cloud to cloud lightning strike nov08.jpg|thumb|right|250px|Cloud-to-cloud lightning, Victoria, Australia. Credit: [[c:user:Fir0002|Fir0002]].{{tlx|free media}}]] [[Image:Lightningcloudtocloud.png|thumb|left|250px|Cloud-to-cloud lightning seen in Gresham, Oregon. Credit: [[w:user:JoJoes123|JoJoes123]].{{tlx|free media}}]] Lightning discharges may occur between areas of cloud without contacting the ground. When it occurs between two separate clouds it is known as ''inter-cloud lightning'', and when it occurs between areas of differing electric potential within a single cloud it is known as ''intra-cloud lightning''. Intra-cloud lightning is the most frequently occurring type.<ref name="GHCC primer2">{{cite book | url = http://thunder.nsstc.nasa.gov/primer/primer2.html|title = A Lightning Primer – Characteristics of a Storm |accessdate = February 8, 2009 |author1=Christian, Hugh J. |author2=McCook, Melanie A. |work =NASA |archive-url=https://web.archive.org/web/20160305002214/http://thunder.nsstc.nasa.gov/primer/primer2.html |archive-date=March 5, 2016}}</ref> {{clear}} ===Anvil crawler lightning=== [[Image:Anvil Crawler over Lake Wright Patman south of Redwater, Texas..JPG|thumb|left|250px|Anvil Crawler over Lake Wright Patman south of Redwater, Texas on the backside of a large area of rain associated with a cold-front. Credit: [[c:user:Griffinstorm|Griffinstorm]].{{tlx|free media}}]] [[Image:Largest lightning flash.jpg|right|thumb|300px|This map shows the largest lightning flash spotted in the satellite data. Credit: Michael Peterson, Los Alamos National Laboratory.{{tlx|fairuse}}]] '''Anvil crawler lightning''', sometimes called '''Spider lightning''', when leaders propagate through horizontally-extensive charge regions in mature thunderstorms, usually the stratiform regions of mesoscale convective systems, beginning as intracloud discharges originating within the convective region; the negative leader end then propagates well into the aforementioned charge regions in the stratiform area. If the leader becomes too long, it may separate into multiple bidirectional leaders then, the positive end of the separated leader may strike the ground as a positive CG flash or crawl on the underside of the cloud, creating a spectacular display of lightning crawling across the sky. Ground flashes produced in this manner tend to transfer high amounts of charge, and this can trigger upward lightning flashes and upper-atmospheric lightning.<ref name=Warner/> The map on the right "shows the largest lightning flash spotted in the satellite data. Its footprint extended across 44,400 square miles across multiple states. The bar along the bottom shows how the energies and sizes of individual pulses change over time."<ref name=Peterson>{{ cite book |author=Michael Peterson |title=Record-breaking lightning as long as Kansas spotted |publisher=National Geographic |location= |date=22 August 2019 |url=https://www.nationalgeographic.com/science/2019/08/record-breaking-lightning-as-long-as-kansas-spotted/ |accessdate=25 August 2019 }}</ref> Two "record-breaking lightning flashes, the longest by length and by duration [were found in the satellite observations]. One stretched over Brazil some 418 miles from tip to tail—slightly longer than Kansas is across. The second lit up skies for 13.5 seconds over the central United States. A third lightning flash over the southern United States sprawled some 44,400 square miles—nearly the area of Ohio. (Official data aren’t kept for the flash with the largest area, so it's not possible to determine if it set a record.)"<ref name=Haas>{{ cite book |author=Maya Wei-Haas |title=Record-breaking lightning as long as Kansas spotted |publisher=National Geographic |location= |date=22 August 2019 |url=https://www.nationalgeographic.com/science/2019/08/record-breaking-lightning-as-long-as-kansas-spotted/ |accessdate=25 August 2019 }}</ref> The previous record-holding flashes "called into question our typical view of lightning." But these latest mega-flashes "are now essentially pushing the boundary further for what lightning can be."<ref name=Peterson/> The "identification of such large flashes of light demonstrates the power of NOAA’s newest weather satellites, GOES-16 and GOES-17. And the data is a proving ground for [a] new automated processing system, [...] which tackles the most complex lightning data beamed back from space."<ref name=Haas/> "Lightning science is relatively new, and we’re moving basically as quickly as we can get instruments to detect."<ref name=Calhoun/> The data from these satellites will "give us an opportunity to study lightning in a way we’ve never been able to see before."<ref name=Calhoun>{{ cite book |author=Kristin Calhoun |title=Record-breaking lightning as long as Kansas spotted |publisher=National Geographic |location= |date=22 August 2019 |url=https://www.nationalgeographic.com/science/2019/08/record-breaking-lightning-as-long-as-kansas-spotted/ |accessdate=25 August 2019 }}</ref> {{clear}} ===Cloud-to-ground lightning=== [[Image:Lightning over Oradea Romania 3.jpg|thumb|right|250px|Strokes of cloud-to-ground lightning occurred during a thunderstorm. Credit: Mircea Madau.{{tlx|free media}}]] [[Image:Rayo cielo a tierra en Maracaibo-Venezuela.jpg|thumb|left|250px|Cloud-to-ground lightning in Maracaibo, Venezuela. Credit: [[c:user:Fox221|Fox221]].{{tlx|free media}}]] [[Image:Desert Electric.jpg|thumb|center|250px|A lightning strike from '''cloud to ground''' in the California, Mojave Desert. Credit: [[c:user:Pocxketthis|Jessie Eastland]].{{tlx|free media}}]] [[Image:Lightning strikes mountain top.jpg|thumb|right|250px|''Cloud to ground'' lightning is shown. Credit: Yintan.{{tlx|free media}}]] Lightning is a naturally occurring electrostatic discharge during which two electrically charged regions in the atmosphere or ground temporarily equalize themselves, causing the instantaneous release of as much as one billion joules of energy.<ref>{{Cite journal |last=Maggio |first=Christopher R. |last2=Marshall |first2=Thomas C. |last3=Stolzenburg |first3=Maribeth |date=2009 |title=Estimations of charge transferred and energy released by lightning flashes |url=https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2008JD011506 |journal=Journal of Geophysical Research: Atmospheres |language=en |volume=114 |issue=D14 |doi=10.1029/2008JD011506 |issn=2156-2202 }}</ref> {{clear}} ===Heat lightning=== [[Image:Heat Lightning - 100613.jpg|right|thumb|300px|Heat Lightning occurs near Louisville, Kentucky. Credit: [[c:user:Bbadgett|Bbadgett]].{{tlx|free media}}]] Heat lightning, sometimes known as silent lightning, 'summer lightning, or dry lightning (mainly used in the American south; not to be confused with dry thunderstorms, which are also often called dry lightning), is a misnomer<ref>{{cite book |title=What Is Heat Lightning? |url=https://weather.com/science/weather-explainers/news/heat-lightning-explainer |publisher=Weather.com }}</ref> used for the faint flashes of lightning on the horizon or other clouds from distant thunderstorms that do not appear to have accompanying sounds of thunder. ''Heat lightning'' is a lightning flash that appears to produce no discernible because it occurs too far away for the thunder to be heard; the sound waves dissipate before they reach the observer.<ref>{{cite book |author=Haby, Jeff |url=http://www.theweatherprediction.com/habyhints/274/ |title=What is heat lightning? |work=theweatherprediction.com |archiveurl=https://web.archive.org/web/20161104174721/http://www.theweatherprediction.com/habyhints/274/ |archivedate=November 4, 2016 |df=mdy-all }}</ref> {{clear}} ===Dry lightning=== [[Image:Cloud to ground lightning strikes south-west of Wagga Wagga.jpg|thumb|right|250px|Cloud to ground lightning strike occurs during a dry thunderstorm near Wagga Wagga, Australia. Credit: [[c:user:Bidgee|Bidgee]].{{tlx|free media}}]] A dry thunderstorm is a thunderstorm that produces thunder and lightning, but most or all of its precipitation evaporates before reaching the ground.<ref>{{cite book |url=http://www.srh.noaa.gov/abq/?n=clifeature2010drythunderstorms |title=Dry thunderstorms |author=Albuquerque Office |work=Fire Weather Topics |publisher=National Weather Service |accessdate=December 16, 2015 }}</ref> Dry lightning refers to lightning strikes occurring in this situation. Both are so common in the Western United States that they are sometimes used interchangeably.<ref name=drycb>{{cite book |url=http://weather.about.com/od/typesofprecipitation/f/dry_storms.htm |title=What Is a Dry Thunderstorm? |author= |publisher=weather.about.com |accessdate=December 16, 2015 }}</ref> The latter term is a technical misnomer since lightning itself is neither wet nor dry.<ref name=drycb/> Dry thunderstorms occur essentially in dry conditions, and their lightning is a major cause of wildfires.<ref name=NSSL/><ref>{{cite book | url= http://www.wfas.net/index.php/dry-lightning-experimental-products-97 | title = Dry Lightning | author= | work = Wild Fire Assessment System | publisher= US Forest Service | accessdate= December 16, 2015 }}</ref> Because of that, the National Weather Service, and other agencies around the world, issue forecasts for its likelihood over large areas.<ref name=NSSL>{{cite book | url= http://www.nssl.noaa.gov/education/svrwx101/lightning/faq/ | title = Frequently Asked Questions About Lightning | work = Severe Weather 101 | publisher= NSSL |author= | accessdate= December 16, 2015 }}</ref><ref>{{ cite conference | url= http://ams.confex.com/ams/pdfpapers/65549.pdf | format= pdf | title= Forecasting Dry Lightning in the Western United States | author1= Miriam Rorig | author2= Sue Ferguson |author3= Steven McKay | conference = The 5th Symposium on Fire and Forest Meteorology and the 2nd International Wildland Fire Ecology and Fire Management Congress | location = Orlando, FL |date = 17 November 2003 | accessdate= December 16, 2015 }}</ref> ''Dry lightning'' is used in Australia, Canada and the United States for lightning that occurs with no precipitation at the surface and is the most common natural cause of wildfires.<ref name='Scott2000'>{{Cite journal | doi =10.1016/S0031-0182(00)00192-9 | title =The Pre-Quaternary history of fire | date =2000 | author =Scott, A | journal = Palaeogeography, Palaeoclimatology, Palaeoecology | volume =164 | issue =1–4 | page =281 | bibcode =2000PPP...164..281S }}</ref> {{clear}} ===Ball lightning=== [[Image:Ball lightning spectrum.svg|thumb|upright=1.4|right|250px|The emission spectrum (intensity vs. wavelength) is from a natural ball lightning. Credit: [[c:user:Olli Niemitalo|Olli Niemitalo]].{{tlx|free media}}]] Ball lightning may be an atmospheric electrical phenomenon, that refers to reports of luminous, usually spherical objects which vary from pea-sized to several meters in diameter.<ref>{{cite book | last = Singer | first = Stanley | title = The Nature of Ball Lightning | publisher = Plenum Press | date = 1971 | location = New York | isbn = 978-0-306-30494-1 }}</ref> Ball lightning has been described by eyewitnesses but rarely recorded by meteorologists.<ref>{{cite journal |url=http://physics.aps.org/articles/v7/5 |title=Focus:First Spectrum of Ball Lightning |volume=7 |last=Ball |first=Philip |date=January 17, 2014 |journal=Focus |accessdate=January 18, 2014 |archiveurl=https://web.archive.org/web/20140118064740/http://physics.aps.org/articles/v7/5 |archivedate=January 18, 2014 }}</ref><ref name="gsu1">{{cite book |title=Ball Lightning |accessdate=September 21, 2007 |publisher=Georgia State University |date=2007 |author=Tennakone, Kirthi |url=https://web.archive.org/web/20080212161614/http://www.phy-astr.gsu.edu/seminar/ST070612_Tennakone_abstract.html }}</ref> A wildlife ranger reported taking a photo at Queensland of Australia in 1987.<ref>{{cite news | last=Porter | first=Brett | title=Brett Porter, Photo in 1987, BBC:Ball lightning baffles scientists, day, 21 December, 2001, 00:26 GMT | url=http://news.bbc.co.uk/2/hi/sci/tech/1721473.stm | date=1987 | archiveurl=https://web.archive.org/web/20160420194815/http://news.bbc.co.uk/2/hi/sci/tech/1721473.stm | archivedate=April 20, 2016 | df=mdy-all }}</ref> Recordings were made in July 2012 of the optical spectrum of natural ball lightning during the study of ordinary cloud–ground lightning on the Tibetan Plateau.<ref name="BLspectrum" /><ref name="BLspectrumvideo">{{cite journal |title=Focus: First Spectrum of Ball Lightning |last1=Ball |first1=Philip |date=17 January 2014 |journal=Focus |doi=10.1103/Physics.7.5 |volume=7 |page=5 |bibcode = 2014PhyOJ...7....5B }}</ref> The observed data are consistent with vaporization of soil as well as with ball lightning's sensitivity to electric fields.<ref name="BLspectrum"> {{cite journal |last1= Cen |first1= Jianyong |first2= Ping |last2= Yuan |first3= Simin |last3= Xue |date= 17 January 2014 |title= Observation of the Optical and Spectral Characteristics of Ball Lightning |journal= Physical Review Letters |volume= 112 |issue= 3 |page = 035001 |doi= 10.1103/PhysRevLett.112.035001 |bibcode= 2014PhRvL.112c5001C |pmid= 24484145 }}</ref><ref name="BLspectrumvideo" /><ref>{{cite journal |url= https://www.newscientist.com/article/dn24886-natural-ball-lightning-probed-for-the-first-time.html |title= Natural ball lightning probed for the first time |last= Slezak |first= Michael |journal= New Scientist |volume= 221 |issue= 2953 |page= 17 |date= 16 January 2014 |accessdate= 22 January 2014 |bibcode= 2014NewSc.221...17S |doi= 10.1016/S0262-4079(14)60173-1 }}</ref> {{clear}} ===Ribbon lightning=== [[Image:Ribbon lightning.png|thumb|right|250px|Ribbon lightning has been photographed where the channel has shifted slightly. Credit: [[c:user:Christos Doudoulakis|Christos Doudoulakis]].{{tlx|free media}}]] Ribbon lightning occurs in thunderstorms with high cross winds and multiple return strokes, where the wind will blow each successive return stroke slightly to one side of the previous return stroke, causing a ribbon effect.<ref>{{cite book |title=Lightning Types and Classifications |accessdate=2017-10-26 |url=https://web.archive.org/web/20171026111249/http://stormhighway.com/types.php }}</ref> {{clear}} ===Rocket lightning=== Rocket lightning is a form of cloud discharge, generally horizontal and at cloud base, with a luminous channel appearing to advance through the air with visually resolvable speed, often intermittently.<ref name="AMS Glossary">{{cite book |url = http://amsglossary.allenpress.com/glossary/search?id=rocket-lightning1 |title = Definition of Rocket Lightning, AMS Glossary of Meteorology |accessdate = July 5, 2007 |archiveurl = https://web.archive.org/web/20070817211059/http://amsglossary.allenpress.com/glossary/search?id=rocket-lightning1 |archivedate = August 17, 2007 |df = mdy-all }}</ref> ===Staccato lightning=== [[Image:Staccoto Lightning.jpg|thumb|upright|right|250px|This CG was of very short duration, exhibited highly branched channels and was very bright indicating that it was staccato lightning near New Boston, Texas. Credit: [[c:user:Griffinstorm|Griffinstorm]].{{tlx|free media}}]] Staccato lightning is a cloud-to-ground lightning (CG) strike which is a short-duration stroke that (often but not always) appears as a single very bright flash and often has considerable branching,<ref>{{cite book | title = Glossary | work = National Oceanic and Atmospheric Administration | publisher = National Weather Service | url = http://www.weather.gov/glossary/index.php?letter=s | accessdate = September 2, 2008 | archiveurl = https://web.archive.org/web/20080915070315/http://www.weather.gov/glossary/index.php?letter=s | archivedate = September 15, 2008 | df = mdy-all }}</ref> often found in the visual vault area near the mesocyclone of rotating thunderstorms and coincides with intensification of thunderstorm updrafts, where there is similar cloud-to-cloud strike consisting of a brief flash over a small area, appearing like a blip, also occurs in a similar area of rotating updrafts.<ref name="Storm Talk">{{cite book |last=Marshall |first=Timothy P. |author2=David Hoadley (illustrator) |title=Storm Talk |date=May 1995 |location=Texas }}</ref> {{clear}} ===Superbolts=== Superbolts are rather loosely defined as strikes with a peak source power of more than 100 gigawatts (most lightning strikes come in at around 1 gigawatt), which occur about as frequently as one in 240 strikes, not categorically distinct from ordinary lightning strikes, and simply represent the uppermost edge of a continuum, can be either positively or negatively charged, and the charge ratio is comparable to that of "ordinary" lightning. <ref>{{cite journal |title=Detection of lightning superbolts |doi=10.1029/JC082i018p02566 |bibcode=1977JGR....82.2566T |volume=82 |issue=18 |journal=Journal of Geophysical Research |pages=2566–2568 |year=1977 |last1=Turman |first1=B. N. |df=mdy-all }}</ref><ref>{{cite book |url=http://www.forte.lanl.gov/science/publications/1999/Kirkland_1999_1_Examination.pdf |title=An examination of superbolt-class lightning events observed by the FORTE satellite |accessdate=2015-12-27 |archiveurl=https://web.archive.org/web/20160304142728/http://www.forte.lanl.gov/science/publications/1999/Kirkland_1999_1_Examination.pdf |archivedate=March 4, 2016 |df=mdy-all }}</ref> ===Upward lightning=== Upward lightning or ground-to-cloud lightning is a lightning flash which originates from the top of a grounded object and propagates upward from this point, can be triggered by a preceding lightning flash, or it may initiate entirely on its own; where the former is generally found in regions where spider lightning occurs, and may involve multiple grounded objects simultaneously.<ref>Saba, M. M. F., C. Schumann, T. A. Warner, M. A. S. Ferro, A. R. de Paiva, J. Helsdon Jr, and R. E. Orville (2016), Upward lightning flashes characteristics from high-speed videos, J. Geophys. Res. Atmos., 121, doi:10.1002/2016JD025137</ref> The latter usually occurs during the cold season and may be the dominant lightning type in thundersnow events.<ref>Warner, T. A., T. J. Lang, and W. A. Lyons (2014), Synoptic scale outbreak of self-initiated upward lightning (SIUL) from tall structures during the central U.S. blizzard of 1–2 February 2011, J. Geophys. Res. Atmospheres, 119, 9530–9548, doi:10.1002/2014JD021691</ref> ===Bead lightning=== Beading of a lightning channel is usually a small-scale feature, and therefore is often only apparent when the observer/camera is close to the lightning.<ref>{{cite book |last=Robinson |first=Dan |title=Weather Library: Lightning Types & Classifications |url=http://stormhighway.com/types.shtml |accessdate=March 17, 2013 |archiveurl=https://web.archive.org/web/20130215041135/http://stormhighway.com/types.shtml |archivedate=February 15, 2013 |df=mdy-all }}</ref> ==Jupiter== {{main|Jupiter}} [[Image:Jupiter xray ill.jpg|thumb|right|250px|Jupiter shows intense X-ray emission associated with auroras in its polar regions (Chandra observatory X-ray image on the left). The accompanying schematic illustrates how Jupiter's unusually frequent and spectacular auroral activity is produced. Observation period: 17 hrs, February 24-26, 2003. Credit: X-ray: NASA/CXC/MSFC/R.Elsner et al.; Illustration: CXC/M.Weiss.{{tlx|free media}}]] In the image at right is a diagram describing interaction with the local magnetic field. Jupiter's strong, rapidly rotating magnetic field (light blue lines in the figure) generates strong electric fields in the space around the planet. Charged particles (white dots), "trapped in Jupiter's magnetic field, are continually being accelerated (gold particles) down into the atmosphere above the polar regions, so auroras are almost always active on Jupiter. Electric voltages of about 10 million volts, and currents of 10 million amps - a hundred times greater than the most powerful lightning bolts - are required to explain the auroras at Jupiter's poles, which are a thousand times more powerful than those on Earth. On Earth, auroras are triggered by solar storms of energetic particles, which disturb Earth's magnetic field. As shown by the swept-back appearance in the illustration, gusts of particles from the Sun also distort Jupiter's magnetic field, and on occasion produce auroras."<ref name=Elsner>{{ cite book |author=X-ray: NASA/CXC/MSFC/R.Elsner |author2=Illustration: CXC/M.Weiss |title=Jupiter: Chandra Probes High-Voltage Auroras on Jupiter |publisher=Harvard-Smithsonian Center for Astrophysics |location=Cambridge, Massachusetts |date=March 2, 2005 |url=http://chandra.harvard.edu/photo/2005/jupiter/ |accessdate=2012-07-11 }}</ref> Supporting the idea of water clouds are the flashes of lightning detected in the atmosphere of Jupiter that can be up to a thousand times as powerful as lightning on Earth.<ref>{{cite book |editor=Watanabe, Susan |date=February 25, 2006 |url=http://www.nasa.gov/vision/universe/solarsystem/galileo_end.html |title=Surprising Jupiter: Busy Galileo spacecraft showed jovian system is full of surprises |publisher=NASA |accessdate=February 20, 2007 }}</ref> The water clouds are assumed to generate thunderstorms in the same way as terrestrial thunderstorms, driven by the heat rising from the interior.<ref>{{cite journal |last=Kerr |first=Richard A. |title=Deep, Moist Heat Drives Jovian Weather |journal=Science |date=2000 |volume=287 |issue=5455 |pages=946–947 |url=http://www.sciencemag.org/cgi/content/full/287/5455/946b |doi=10.1126/science.287.5455.946b |accessdate=February 24, 2007 }}</ref> As the spacecraft passed behind the planet, it observed flashes of lightning in the night side atmosphere.<ref name="burgess">{{cite book |first=Eric |last=Burgess |date=1982 |title=By Jupiter: Odysseys to a Giant |publisher=Columbia University Press |location=New York |isbn=978-0-231-05176-7 }}</ref><ref name="voyager1">{{cite book |date=January 14, 2003 |url=http://voyager.jpl.nasa.gov/science/jupiter.html |title=Jupiter |publisher=NASA/JPL |accessdate=November 28, 2006 }}</ref> {{clear}} ==Saturn== [[Image:Saturn Lightning.ogv|thumb|right|250px|Video shows ligthtning storms on Saturn. Credit: NASA.{{tlx|free media}}]] Starting in early 2005, scientists used ''Cassini'' to track lightning on Saturn. The power of the lightning is approximately 1,000&nbsp;times that of lightning on Earth.<ref>{{cite book |title=Astronomers Find Giant Lightning Storm At Saturn |date=2007 |accessdate=27 July 2007 |publisher=ScienceDaily LLC |url=https://www.webcitation.org/616W9ngSD?url=http://www.sciencedaily.com/releases/2006/02/060215090726.htm }}</ref> The Radio and Plasma Wave Science instrument (RPWS) on ''Cassini-Huygens'' studied the configuration of Saturn's magnetic field and its relationship to Saturn Kilometric Radiation (SKR), as well as monitoring and mapping Saturn's ionosphere, plasma, and lightning from Saturn's (and possibly Titan's) atmosphere.<ref name="instruments-cassini">{{cite book |url=http://saturn.jpl.nasa.gov/spacecraft/cassiniorbiterinstruments/ |title=Cassini Orbiter Instruments |publisher=Saturn.jpl.nasa.gov |accessdate=August 20, 2011 }}</ref><ref>{{cite book |url=http://caps.space.swri.edu/ |title=CAPS team page |publisher=Caps.space.swri.edu |accessdate=August 20, 2011 }}</ref> {{clear}} ==Uranus== The extreme pressure and temperature deep within Uranus may break up the methane molecules, with the carbon atoms condensing into crystals of diamond that rain down through the mantle like hailstones.<ref>{{cite book |url=http://www.spacedaily.com/news/carbon-99d.html |title=Is It Raining Diamonds on Uranus |publisher=SpaceDaily.com |date=1 October 1999 |accessdate=17 May 2013 }}</ref><ref name="WP-20170825">{{cite news |last=Kaplan |first=Sarah |title=It rains solid diamonds on Uranus and Neptune |url=https://www.washingtonpost.com/news/speaking-of-science/wp/2017/08/25/it-rains-solid-diamonds-on-uranus-and-neptune/ |date=25 August 2017 |work=The Washington Post |accessdate=27 August 2017 }}</ref><ref>{{Cite journal |title=Formation of diamonds in laser-compressed hydrocarbons at planetary interior conditions |journal=Nature Astronomy |last=Kraus |first=D. |last2=Vorberger |first2=J. |last3=Pak |first3=A. |last4=Hartley |first4=N. J. |last5=Fletcher |first5=L. B. |last6=Frydrych |first6=S. |last7=Galtier |first7=E. |last8=Gamboa |first8=E. J. |last9=Gericke |first9=D. O. |last10=Glenzer |first10=S. H. |last11=Granados |first11=E. |last12=MacDonald |first12=M. J. |last13=MacKinnon |first13=A. J. |last14=McBride |first14=E. E. |last15=Nam |first15=I. |last16=Neumayer |first16=P. |last17=Roth |first17=M. |last18=Saunders |first18=A. M. |last19=Schuster |first19=A. K. |last20=Sun |first20=P. |last21=van Driel |first21=T. |last22=Döppner |first22=T. |last23=Falcone |first23=R. W. |display-authors=1 |volume=1 |issue=9 |pages=606–611 |date=September 2017 |doi=10.1038/s41550-017-0219-9 |bibcode=2017NatAs...1..606K |url=http://www.escholarship.org/uc/item/1cf3b8v4 }}</ref> Very-high-pressure experiments at the Lawrence Livermore National Laboratory suggest that the base of the mantle may comprise an ocean of liquid diamond, with floating solid 'diamond-bergs'.<ref name="Bland, Eric">{{cite news |last1=Bland |first1=Eric |title=Outer planets may have oceans of diamond |url=http://www.abc.net.au/science/articles/2010/01/18/2794635.htm |accessdate=9 October 2017 |work=ABC Science |date=18 January 2010 |language=en-AU }}</ref><ref>{{cite journal |url=http://www.astronomynow.com/news/n1001/21diamond/ |title=Oceans of diamond possible on Uranus and Neptune |last1=Baldwin |first1=Emily |journal=Astronomy Now |date=21 January 2010 |accessdate=6 February 2014 |archiveurl=https://web.archive.org/web/20131203065217/http://www.astronomynow.com/news/n1001/21diamond/ |archivedate=3 December 2013 |df=dmy-all }}</ref> Scientists also believe that lightning storms make rainfalls of solid diamonds occur on Uranus, as well as on [[Jupiter]], [[Gases/Gaseous objects/Saturn|Saturn]], and [[Neptune]].<ref>{{cite book |url=https://www.businessinsider.com/diamond-rain-saturn-jupiter-2016-4 |title=Lightning storms make it rain diamonds on Saturn and Jupiter |author=Sean Kane |website=Business Insider |date=April 29, 2016 |accessdate=May 22, 2019 }}</ref><ref>{{cite news |url=https://www.washingtonpost.com/news/speaking-of-science/wp/2017/08/25/it-rains-solid-diamonds-on-uranus-and-neptune/ |title=It rains solid diamonds on Uranus and Neptune |author=Sarah Kaplan |newspaper=The Washington Post |date=March 25, 2017 |accessdate=May 22, 2019 }}</ref> ==Volcanoes== {{main|Lightning}} [[Image:Villarrica lightning 2015.jpg|thumb|right|250px|The Villarica volcano erupts near Pucon, Chile, March 3, 2015, with lightning. Credit: Aton Chile/AP Photo.{{tlx|fairuse}}]] [[Image:Rinjani 1994.jpg|thumb|right|250px|The 1995 eruption of Mount Rinjani in Indonesia exhibits volcanic lightning. Credit: [[commons:User:Spolloman|Oliver Spalt]].{{tlx|free media}}]] [[Image:Galunggung.jpg|thumb|left|250px|The slide depicts a spectacular view of lightning strikes during a third eruption on December 3, 1982. Credit: R. Hadian, U.S. Geological Survey.{{tlx|free media}}]] On the right is an image of the Villarrica volcano erupting near Pucon, Chile, March 3, 2015, with lightning striking apparently the lower back of the eruption cloud with few leaders. Many volcanic eruptions put on impressive lightning displays such as during the 1995 eruption of Mount Rinjani in Indonesia shown in the image on the right which exhibits many leaders. The image on the left shows spectacular lightning strikes around Galunggung, including multiple leaders apparently involved in cloud to cloud lightning. "This stratovolcano with a lava dome is located in western Java. Its first eruption in 1822 produced a 22-km-long mudflow that killed 4,000 people. The second eruption in 1894 caused extensive property loss. The photo depicts a spectacular view of lightning strikes during a third eruption on December 3, 1982, which resulted in 68 deaths. A fourth eruption occurred in 1984."<ref name=Hadian>{{ cite book |author=R. Hadian |title=Galunggung, Indonesia |publisher=NOAA National Geophysical Data Center |location= |date=3 December 1982 |url=http://www.ngdc.noaa.gov/nndc/servlet/ShowDatasets?EQ_0=603&bt_0=&st_0=&EQ_1=&bt_1=&st_1=&query=&dataset=101634&search_look=2&group_id=null&display_look=4,44&submit_all=Select+Data |accessdate=2015-03-24 }}</ref> Volcanic lightning arises from colliding, fragmenting particles of volcanic ash (and sometimes ice),<ref>{{Cite news |url=https://www.washingtonpost.com/news/capital-weather-gang/wp/2016/04/13/scientists-think-theyve-solved-the-mystery-of-how-volcanic-lightning-forms/?utm_term=.80462ad8051d |title=Scientists think they've solved the mystery of how volcanic lightning forms |last=Fritz |first=Angela |date=2016 |work=The Washington Post |accessdate= }}</ref><ref>{{Cite news |url=https://www.seeker.com/mystery-of-volcano-lightning-explained-1771209774.html |title=Mystery of Volcano Lightning Explained |last=Mulvaney |first=Kieran |date=2016 |work=Seeker |accessdate= }}</ref> which generate [[static electricity]] within the volcanic plume.<ref>{{Cite news |url=https://blogs.agu.org/geospace/2016/04/12/new-studies-uncover-mysterious-processes-generate-volcanic-lightning/ |title=New studies uncover mysterious processes that generate volcanic lightning |last=Lipuma |first=Lauren |date=2016 |work=American Geophysical Union GeoSpace Blog |accessdate= }}</ref> Volcanic eruptions have been referred to as '''dirty thunderstorms'''<ref>{{Cite journal |last=Hoblitt |first=Richard P. |date=2000 |title=Was the 18 May 1980 lateral blast at Mt St Helens the product of two explosions? |journal=Philosophical Transactions of the Royal Society of London. Series A. Mathematical, Physical and Engineering Sciences |volume=358 |issue=1770 |pages=1639–1661 |doi=10.1098/rsta.2000.0608 }}</ref><ref name=":6">{{Cite journal |last=Bennett |first=A J |last2=Odams |first2=P |last3=Edwards |first3=D |last4=Arason |first4=Þ |date=2010-10-01 |title=Monitoring of lightning from the April–May 2010 Eyjafjallajökull volcanic eruption using a very low frequency lightning location network |journal=Environmental Research Letters |volume=5 |issue=4 |pages=044013 |doi=10.1088/1748-9326/5/4/044013 |doi-access=free |issn=1748-9326 }}</ref> due to [http://glossary.ametsoc.org/wiki/Moist_convection moist convection] and ice formation that drive the eruption plume dynamics<ref name=":7">{{Cite journal |last=Woods |first=Andrew W. |date=1993 |title=Moist convection and the injection of volcanic ash into the atmosphere |journal=Journal of Geophysical Research: Solid Earth |volume=98 |pages=17627–17636 |doi=10.1029/93JB00718 |via= }}</ref><ref name=":8">{{Cite journal |last=Van Eaton |first=Alexa R. |last2=Mastin |first2=Larry G. |last3=Herzog |first3=Michael |last4=Schwaiger |first4=Hans F. |last5=Schneider |first5=David J. |last6=Wallace |first6=Kristi L. |last7=Clarke |first7=Amanda B. |date=2015-08-03 |title=Hail formation triggers rapid ash aggregation in volcanic plumes |journal=Nature Communications |volume=6 |issue=1 |doi=10.1038/ncomms8860 |doi-access=free |issn=2041-1723 }}</ref> and can trigger volcanic lightning.<ref>{{Cite journal |last=Williams |first=Earl R. |last2=McNutt |first2=Stephen R. |date=2005 |title=Total water contents in volcanic eruption clouds and implications for electrification and lightning |url=http://www.giseis.alaska.edu/Input/steve/PUBS/williams-mcn-signpost.PDF |journal=Proceedings of the 2nd International Conference on Volcanic Ash and Aviation Safety |volume= |pages=67–71 |via= }}</ref><ref name=":9">{{Cite journal |last=Van Eaton |first=Alexa R. |last2=Amigo |first2=Álvaro |last3=Bertin |first3=Daniel |last4=Mastin |first4=Larry G. |last5=Giacosa |first5=Raúl E. |last6=González |first6=Jerónimo |last7=Valderrama |first7=Oscar |last8=Fontijn |first8=Karen |last9=Behnke |first9=Sonja A. |date=2016-04-12 |title=Volcanic lightning and plume behavior reveal evolving hazards during the April 2015 eruption of Calbuco volcano, Chile |journal=Geophysical Research Letters |volume=43 |issue=7 |pages=3563–3571 |doi=10.1002/2016gl068076 |doi-access=free |issn=0094-8276 |via= }}</ref> But unlike ordinary thunderstorms, volcanic lightning can also occur before any ice crystals have formed in the ash cloud.<ref>{{Cite journal |last=Cimarelli |first=C. |last2=Alatorre-Ibargüengoitia |first2=M.A. |last3=Kueppers |first3=U. |last4=Scheu |first4=B. |last5=Dingwell |first5=D.B. |date=2014 |title=Experimental generation of volcanic lightning |journal=Geology |volume=42 |issue=1 |pages=79–82 |doi=10.1130/g34802.1 |doi-access=free |issn=1943-2682 }}</ref><ref>{{Cite journal |last=Cimarelli |first=C. |last2=Alatorre-Ibargüengoitia |first2=M. A. |last3=Aizawa |first3=K. |last4=Yokoo |first4=A. |last5=Díaz-Marina |first5=A. |last6=Iguchi |first6=M. |last7=Dingwell |first7=D. B. |date=2016-05-06 |title=Multiparametric observation of volcanic lightning: Sakurajima Volcano, Japan |journal=Geophysical Research Letters |volume=43 |issue=9 |pages=4221–4228 |doi=10.1002/2015gl067445 |doi-access=free |issn=0094-8276 }}</ref> The earliest recorded observations of volcanic lightning<ref name=":4">{{cite journal |last=Mather |first=T. A. |last2=Harrison |first2=R. G. |date=July 2006 |title=Electrification of volcanic plumes |journal=Surveys in Geophysics |volume=27 |issue=4 |pages=387–432 |doi=10.1007/s10712-006-9007-2 |issn=0169-3298 }}</ref> are from Pliny the Younger, describing the eruption of Mount Vesuvius in 79 AD, “There was a most intense darkness rendered more appalling by the fitful gleam of torches at intervals obscured by the transient blaze of lightning.”<ref name="volcano.oregonstate.edu">{{cite book |url=http://volcano.oregonstate.edu/history-volcanic-lightning |title=History of Volcanic Lightning {{!}} Volcano World {{!}} Oregon State University |website=volcano.oregonstate.edu |accessdate=2018-05-09 }}</ref> The first studies of volcanic lightning were also conducted at Mount Vesuvius by Professor Palmieri who observed the eruptions of 1858, 1861, 1868, and 1872 from the Vesuvius Observatory. These eruptions often included lightning activity.<ref name="volcano.oregonstate.edu"/> A famous image of the phenomenon was photographed by Carlos Gutierrez and occurred in Chile above the Chaiten Volcano.<ref>{{cite book |url=http://news.nationalgeographic.com/news/2008/05/photogalleries/volcano-photos/ |title=Chile Volcano Erupts with Ash and Lightning |date=May 6, 2008 |publisher=National Geographic |accessdate=2009-01-09 |archive-url=https://web.archive.org/web/20090106184308/http://news.nationalgeographic.com/news/2008/05/photogalleries/volcano-photos/ |archive-date=2009-01-06 }}</ref> It circulated widely on the internet. Another notable image of this phenomenon is "The Power of Nature",<ref>{{cite news |url=https://www.worldpressphoto.org/collection/photo/2016/nature/sergio-tapiro |title=The Power of Nature |newspaper=World Press Photo |accessdate=2017-01-19 }}</ref> taken by Mexican photographer Sergio Tapiro<ref>{{cite book |url=https://about.me/tapiro |title=Sergio Tapiro Velasco on about.me |last=Velasco |first=Sergio Tapiro |website=about.me |accessdate=2017-01-19 }}</ref> in Colima, Mexico, which won third place (Nature category) in the 2016 World Press Photo Contest.<ref>{{cite book |url=https://www.theguardian.com/media/gallery/2016/feb/18/world-press-photo-2016-winners-in-pictures |title=World Press Photo 2016 winners - in pictures |author=World Press Photo |date=2016-02-18 |website=the Guardian |accessdate=2017-01-19 }}</ref> Other instances have been reported above Alaska's Mount Augustine volcano,<ref name="Handwerk">{{cite book |url=http://news.nationalgeographic.com/news/2007/02/070222-volcano-lightning.html |title=Volcanic Lightning Sparked by "Dirty Thunderstorms" |last=Handwerk |first=Brian |date=February 22, 2007 |publisher=National Geographic |accessdate=2009-01-09 }}</ref> Iceland's 2010 eruptions of Eyjafjallajökull volcano<ref>{{cite book |url=http://news.nationalgeographic.com/news/2010/04/photogalleries/100419-iceland-volcano-lightning-ash-pictures/ |title=Iceland Volcano Pictures: Lightning Adds Flash to Ash |date=April 19, 2010 |publisher=National Geographic |accessdate=2010-04-20 }}</ref> and Mount Etna in Sicily, Italy.<ref>{{cite book |title = Sky lights up over Sicily as Mount Etna's Voragine crater erupts |url = https://www.theguardian.com/world/2015/dec/03/sicily-mount-etna-voragine-crater-erupts-lightning |website = the Guardian |accessdate = 2015-12-03 |editor = Ian Sample Science }}</ref> {{clear}} ==See also== {{div col|colwidth=20em}} * [[Radiation astronomy/Aerometeors|Aerometeor astronomy]] * [[Radiation astronomy/Electrons|Electron astronomy]] * [[Radiation astronomy/Fieries|Fiery meteor astronomy]] * [[Radiation astronomy/Gamma rays|Gamma-ray astronomy]] * [[Radiation astronomy/Meteoroids|Meteoroid astronomy]] * [[Radiation astronomy/Showers|Meteor-shower astronomy]] * [[Radiation astronomy/Protons|Proton astronomy]] * [[Radiation astronomy/Ultraviolets|Ultraviolet astronomy]] * [[Radiation astronomy/X-rays|X-ray astronomy]] {{Div col end}} ==References== {{reflist|2}} ==External links== {{Radiation astronomy resources}} [[Category:Radiation astronomy/Lectures]] sgrpqegyapqjmmnzdzm3070w20zpdae 0 250749 2416449 2146677 2022-08-19T06:55:01Z Marshallsumter 311529 wikitext text/x-wiki [[Image:1024px--D3- LighningStormPanorama.jpg|thumb|right|250px|A lightning storm over England is imaged. Credit: Catalin Fatu.{{tlx|fairuse}}]] '''[[Radiation astronomy/Lightnings|Lightning radiation astronomy]]''' is a lecture from the [[Portal:Radiation astronomy|radiation astronomy department]] on the origin, constitution and occurrence of lightning. You are free to take this quiz based on [[Radiation astronomy/Lightnings|lightning radiation astronomy]] at any time. To improve your score, read and study the lecture, the links contained within, listed under [[Radiation astronomy/Lightnings/Quiz#See also|'''See also''']], [[Radiation astronomy/Lightnings/Quiz#External links|'''External links''']], or in the {{tlx|Principles of radiation astronomy}} template. This should give you adequate background to get 100 %. As a "learning by doing" resource, this quiz helps you to assess your knowledge and understanding of the information, and it is a quiz you may take over and over as a learning resource to improve your knowledge, understanding, test-taking skills, and your score. '''Suggestion:''' Have the lecture available in a separate window. To master the information and use only your memory while taking the quiz, try rewriting the information from more familiar points of view, or be creative with association. Enjoy learning by doing! {{clear}} ==Quiz== <quiz> {Complete the text: |type="{}"} A composite { electron density (i) } spectrum is approximately a { power (i) } law over at least the ≈ 5 decade wavenumber range 10^-13 m^-1 < wavenumber < 10^-8 m^-1 and that may extend to { higher (i) } wavenumbers. {True or False, Some lightning that occurs is actually positive charges moving downward to the Earth's surface. |type="()"} + TRUE - FALSE {Which of the following are associated with electromagnetics? |type="[]"} + angular momentum transfer + solar wind + protons + electrons - the baryon neutrino + charge neutralization {True or False, Like all elementary particles, the electron has a corresponding antiparticle of opposite spin but equal mass and charge (-1). |type="()"} - TRUE + FALSE {Complete the text: |type="{}"} Blue jets differ from sprites in that they project from the { top (i) } of the cumulonimbus above a { thunderstorm (i) }, typically in a narrow cone, to the lowest levels of the { ionosphere (i) }. {True or False, Cloud-to-cloud lightning occurs between charged portions of clouds with opposite sign. |type="()"} + TRUE - FALSE {Yes or No, Blue jets differ from sprites in that they project from the top of the cumulonimbus above a thunderstorm to the lowest levels of the ionosphere. |type="()"} - No + Yes {True or False, Some lightning strikes may be originated by cosmic rays. |type="()"} + TRUE - FALSE {Yes or No, Red sprites tend to be associated with significant lightning strikes. |type="()"} - No + Yes {Which of the following is associated with an atmosphere? |type="[]"} + clouds + jet streams + rain + wind - faults + lightning {True or False, Antilightning is lightning spinning backward in time. |type="()"} - TRUE + FALSE </quiz> ==Hypotheses== {{main|Hypotheses}} # Lightning is an effort by the Earth to balance charges impinging on the ionosphere from the solar wind. ==See also== {{div col|colwidth=20em}} * [[Radiation astronomy/Acoustics/Quiz|Acoustic astronomy quiz]] * [[Radiation astronomy/Aerometeors/Quiz|Aerometeor astronomy quiz]] * [[Radiation astronomy/Alpha particles/Quiz|Alpha-particle astronomy quiz]] * [[Astronomical observatories/Quiz]] * [[Radiation astronomy/Astronomy|Astronomy/Quiz]] * [[Astrophysics/Quiz]] * [[Radiation astronomy/Atomics/Quiz|Atomic astronomy quiz]] * [[Radiation astronomy/Beta particles/Quiz|Beta-particles astronomy/Quiz]] * [[Blue astronomy/Quiz]] * [[Radiation astronomy/Clouds/Quiz|Cloud astronomy quiz]] * [[Cosmic-ray astronomy/Quiz]] * [[Crater astronomy/Quiz]] * [[Radiation astronomy/Cryometeors/Quiz|Cryometeor astronomy quiz]] * [[Cyan astronomy/Quiz]] * [[Radiation astronomy/Electromagnetics/Quiz|Electromagnetic astronomy/Quiz]] * [[Radiation/Electromagnetics/Quiz|Electromagnetic radiation/Quiz]] * [[Electron astronomy/Quiz]] * [[Radiation astronomy/Fieries/Quiz|Fiery-meteor astronomy quiz]] * [[Gamma-ray astronomy/Quiz]] * [[Green astronomy/Quiz]] * [[Radiation astronomy/Hydrometeors/Quiz|Hydrometeor astronomy quiz]] * [[Infrared astronomy/Quiz]] * [[Radiation astronomy/Lightnings/Quiz|Lightning astronomy quiz]] * [[Radiation astronomy/Lithometeors/Quiz|Lithometeor astronomy quiz]] * [[Meteor astronomy/Quiz]] * [[Meteorites/Quiz]] * [[Radiation astronomy/Meteoroids/Quiz|Meteoroid astronomy quiz]] * [[Radiation/Meteors/Quiz|Meteor radiation/Quiz]] * [[Radiation astronomy/Showers/Quiz|Meteor-shower astronomy quiz]] * [[Optical astronomy/Quiz]] * [[Radiation astronomy/Plasmas/Quiz|Plasma-meteors quiz]] * [[Positron astronomy/Quiz]] * [[Proton astronomy/Quiz]] * [[Radar astronomy/Quiz]] * [[Radiation/Quiz]] * [[Radiation/Meteors/Quiz|Radiation meteors/Quiz]] * [[Radiation satellites/Quiz]] * [[Radiation telescopes/Quiz]] * [[Radio astronomy/Quiz]] * [[Red astronomy/Quiz]] * [[Source astronomy/Quiz]] * [[Star fission/Quiz]] * [[Star-forming region/Quiz]] * [[Stellar active region/Quiz]] * [[Radiation astronomy/Stars/Quiz|Stellar astronomy quiz]] * [[Stellar surface fusion/Quiz]] * [[Radiation astronomy/Subatomics/Quiz|Subatomic astronomy quiz]] * [[Submillimeter astronomy/Quiz]] * [[Radiation astronomy/Synchrotrons/Quiz|Synchrotron astronomy quiz]] * [[Theoretical radiation astronomy/Quiz]] * [[Ultraviolet astronomy/Quiz]] * [[Violet astronomy/Quiz]] * [[Visual astronomy/Quiz]] * [[X-ray astronomy/Quiz]] {{Div col end}} ==External links== <!-- footer templates --> {{Principles of radiation astronomy}}{{tlx|Radiation astronomy resources}}{{Sisterlinks|Lightning radiation astronomy}} <!-- categories --> [[Category:Atmospheric sciences quizzes]] [[Category:Charges quizzes]] [[Category:Lightning quizzes]] [[Category:Physics quizzes]] [[Category:Radiation astronomy quizzes]] 6qpm48lgvrpme0wzkx3wy65sluwctok 0 250750 2416451 2027605 2022-08-19T07:01:37Z Marshallsumter 311529 wikitext text/x-wiki [[Image:Orionid meteor.jpg|thumb|right|250px|This image is of a green and red Orionid meteor. Credit: [https://sites.google.com/site/thebrockeninglory/ Brocken Inaglory].{{tlx|free media}}]] '''[[Radiation astronomy/Meteoroids|Meteoroid radiation astronomy]]''' is a lecture for the [[Portal:Radiation astronomy|radiation astronomy department]] course on the [[principles of radiation astronomy]]. You are free to take this quiz based on [[Radiation astronomy/Meteoroids|meteoroid radiation astronomy]] at any time. To improve your score, read and study the lecture, the links contained within, listed under [[Radiation astronomy/Meteoroids/Quiz#See also|'''See also''']], [[Radiation astronomy/Meteoroids/Quiz#External links|'''External links''']], and in the {{tlx|principles of radiation astronomy}} template. This should give you adequate background to get 100 %. As a "learning by doing" resource, this quiz helps you to assess your knowledge and understanding of the information, and it is a quiz you may take over and over as a learning resource to improve your knowledge, understanding, test-taking skills, and your score. '''Suggestion:''' Have the lecture available in a separate window. To master the information and use only your memory while taking the quiz, try rewriting the information from more familiar points of view, or be creative with association. Enjoy learning by doing! {{clear}} ==Quiz== <quiz> {What may be the first astronomical source of the rocky objects in the solar system? |type="[]"} + Jupiter - the solar wind - the diffuse X-ray background + Saturn - Mount Redoubt in Alaska + the asteroid belt + the former protoplanetary disc around the Sun + the Oort cloud + the Kuiper belt {Which of the following is not a radiation phenomenon associated with a comet? |type="()"} - airglow - elongated dust particles - high albedo - olivine - Rayleighs + coronal mass ejection {With respect to protoplanetary disks what green mineral has been found? |type="{}"} { olivine (i) } {Which of the following are astronomical phenomena associated with the Sun as a likely source? |type="[]"} + coronal clouds + cosmic rays + neutrinos - Oh My God type particles + meteors - comets + blue rays {Complete the text: |type="{}"} Match up the item letter with each of the cosmogonic possibilities below: interior models of the giant planets - A high interest for cosmogony, geophysics and nuclear physics - B hierarchical accumulation - C clouds and globular clusters - D cosmic helium abundance - E deuterium fusion - F a large deficiency of light elements - G after galactic sized systems had collapsed - H the motions of hydrogen { D (i) } formation of luminous quasars { H (i) }. stars with an initial mass less than the solar mass { G (i) }. rotating liquid drops { B|F (i) }. primordial is less than 26 per cent { E (i) }. a solar mixture of elements dominated by hydrogen and helium gas { A (i) }. around 13 Jupiter masses { F (i) }. smaller rocky objects { C (i) }. {The charge on a planetary dust particle may change with? |type="{}"} { latitude (i) } {Which weather phenomena are most likely to interfere with observing stars? |type="[]"} + a late-summer rainstorm - a clear sky + an approaching dust storm - below normal temperatures + a typhoon + a snow fall + fog {A thin-section of a meteorite that may be analyzed has which of the following? |type="[]"} + a great many light gray and dark gray chondrules present + large grain cross sections making sizing easy + an available sizing or magnification marker + independent verification as a meteorite + correctable diameters and relative abundances + optical conditions {Various gaps and density minima have been observed in the Saturnian |type="{}"} { ring|rings (i) } system. {Which of the following are characteristic of a binary formed via gravitational fragmentation? |type="[]"} + the local Jeans length + the local speed of sound + the mean molecular weight - the electron neutrino + the mean particle density - neutrons {Complete the text: |type="{}"} The moon was formed independently of the earth and later { captured (i) }, presumably by a { three-body| three body (i) } interaction. {True or False, A hydrometeor is a precipitation product. |type="()"} + TRUE - FALSE {Usually associated with clouds filling the sky, thunder and lightning, wind and what water based meteorites |type="{}"} { rain (i) } {True or False, A lithometeor is a rocky-object meteor. |type="()"} + TRUE - FALSE {Meteorites found on Earth may be from which of the following? |type="[]"} - Saturn - Mercury + the Moon + the asteroid belt - Jupiter + Mars + Earth {True or False, A cryometeor is a very large chunk of amber. |type="()"} - TRUE + FALSE {Complete the text: |type="{}"} An aerometeor is a { discrete (i) } unit of air traveling or { falling (i) } through an atmosphere. {True or False, A metallic or stony object that is the remains of a meteor is called a meteoroid. |type="()"} - TRUE + FALSE {On 19 May 1910, the Earth actually passed through the tail of what comet. |type="{}"} { Halley's (i) } {True or False, The observations of planetary motion agree with computed orbits to the accuracy of the observations. |type="()"} - TRUE + FALSE {A cloud at nearly a quarter of the distance to Proxima Centauri is called the |type="{}"} { Oort cloud (i) } {True or False, Clouds are a form of radiation because they have a temperature gradient. |type="()"} - TRUE + FALSE {Which phenomena are associated with the heliosphere? |type="[]"} + a region of space where the interstellar medium is blown away by the solar wind + a bubble in space + virtually all the material emanates from the Sun itself - Voyager 2 + Voyager 1 + the termination shock {True or False, The prismoidal method provides a good approximation of the dust emission peak for cold sources. |type="()"} + TRUE - FALSE {A natural rocky source of chemicals from the sky to the ground may originate from what astronomical source? |type="()"} - Jupiter - the solar wind - the diffuse X-ray background - Mount Redoubt in Alaska + the asteroid belt - the International Space Station {True or False, A new cosmogony predicts that all particles in one stream have the same mineral composition. |type="()"} + TRUE - FALSE {Complete the text: |type="{}"} The majority of known asteroids orbit the Sun between the orbits of { Mars|Jupiter (i) } and { Jupiter|Mars (i) }. {True or False, Mars may have suffered asteroid impacts. |type="()"} + TRUE - FALSE {Which of the following are radiation astronomy phenomena associated with the gaseous-object Neptune? |type="[]"} - Voyager 2 + blue rays + clouds - neutron emission - polar coronal holes + meteor emission - rotation {True or False, The visible path of a meteoroid that has entered the Earth's atmosphere is called a meteorite. |type="()"} - TRUE + FALSE {Complete the text: |type="{}"} The 0.6 m Tortugas Mountain Observatory is used to monitor the temporal changes in the { Jovian|Jupiter (i) } cloud deck and equatorial activity on { Saturn (i) }. {True or False, Mercury may have once been a satellite of Jupiter, or possibly of Saturn. |type="()"} + TRUE - FALSE {Which of the following is not a characteristic of showers? |type="()"} + throwing a beam - meteors - rain - snow - hail {True or False, Micrometeoroids have less stable orbits than meteoroids. |type="()"} + TRUE - FALSE {Which of the following are radiation astronomy phenomena associated with the rocky-object Io? |type="[]"} + surface regions reflecting or emitting violet or purple - an excess brightness at or near the edge + red regions that may be phosphorus - neutron emission - polar coronal holes + meteor emission - rotation {True or False, To date, all of the reported hypervelocity stars (HVSs), which are believed to be ejected from the Galactic center, are blue. |type="()"} + TRUE - FALSE {Any natural object radiating through a portion or all of a natural object's atmosphere may be called a what? |type="{}"} { meteor (i) } {True or False, Olivine is a silicate mineral that may be detected in cometary coma dust with green astronomy. |type="()"} + TRUE - FALSE {Observations of comets have benefited greatly from what phenomenon of cyan astronomy? |type="()"} - Cherenkov radiation - the electric blue glow of lightning - gas-expansion velocity decreases with increasing heliocentric distance - methane possesses prominent absorption bands in the visible - adaptive optics + the light of the neutral CN-radical {Yes or No, A cometary orbit about the Sun is a radiation astronomy phenomenon. |type="()"} + Yes - No {Which of the following is not a radiation phenomenon associated with a planet? |type="()"} - airglow - elongated dust particles - high albedo - olivine - Rayleighs + coronal mass ejection </quiz> ==Hypotheses== {{main|Hypotheses}} # Meteors range in size from galaxy clusters to dust grains and molecular clusters. ==See also== {{div col|colwidth=20em}} * [[Radiation astronomy/Aerometeors/Quiz|Aerometeor astronomy quiz]] * [[Radiation astronomy/Asteroids/Quiz|Asteroid astronomy quiz]] * [[Astronomical observatories/Quiz]] * [[Radiation astronomy/Beta particles/Quiz|Beta-particles astronomy/Quiz]] * [[Blue astronomy/Quiz]] * [[Radiation astronomy/Clouds/Quiz|Cloud astronomy quiz]] * [[Radiation astronomy/Comets/Quiz|Cometary astronomy quiz]] * [[Cosmic-ray astronomy/Quiz]] * [[Crater astronomy/Quiz]] * [[Radiation astronomy/Cryometeors/Quiz|Cryometeor astronomy quiz]] * [[Cyan astronomy/Quiz]] * [[Radiation astronomy/Electromagnetics/Quiz|Electromagnetic astronomy/Quiz]] * [[Radiation/Electromagnetics/Quiz|Electromagnetic radiation/Quiz]] * [[Electron astronomy/Quiz]] * [[Radiation astronomy/Fieries/Quiz|Fiery-meteor astronomy quiz]] * [[First astronomical source/Quiz]] * [[First astronomical X-ray source/Quiz]] * [[Gamma-ray astronomy/Quiz]] * [[Green astronomy/Quiz]] * [[Radiation astronomy/Hydrometeors/Quiz|Hydrometeor astronomy quiz]] * [[Infrared astronomy/Quiz]] * [[Intergalactic medium/Quiz]] * [[Interplanetary medium/Quiz]] * [[Interstellar medium/Quiz]] * [[Radiation astronomy/Kuiper belts/Quiz|Kuiper belt astronomy quiz]] * [[Radiation astronomy/Lightnings/Quiz|Lightning astronomy quiz]] * [[Radiation astronomy/Lithometeors/Quiz|Lithometeor astronomy quiz]] * [[Meteor astronomy/Quiz]] * [[Meteorites/Quiz]] * [[Radiation astronomy/Meteoroids/Quiz|Meteoroid astronomy quiz]] * [[Radiation/Meteors/Quiz|Meteor radiation/Quiz]] * [[Radiation astronomy/Showers/Quiz|Meteor-shower astronomy quiz]] * [[Microwave astronomy/Quiz]] * [[Radiation astronomy/Nebulas/Quiz|Nebula quiz]] * [[Radiation astronomy/Neutrals/Quiz|Neutrals astronomy quiz]] * [[Radiation astronomy/Oort clouds/Quiz|Oort cloud astronomy quiz]] * [[Optical astronomy/Quiz]] * [[Orange astronomy/Quiz]] * [[Radiation astronomy/Particles/Quiz|Particle astronomy quiz]] * [[Radiation astronomy/Planets/Quiz|Planetary astronomy quiz]] * [[Radiation astronomy/Plasmas/Quiz|Plasma-meteors quiz]] * [[Radar astronomy/Quiz]] * [[Radiation/Quiz]] * [[Radiation/Meteors/Quiz|Radiation meteors/Quiz]] * [[Red astronomy/Quiz]] * [[Radiation astronomy/Rocks/Quiz|Rock astronomy quiz]] * [[Rocky-object astronomy/Quiz]] * [[Radiation astronomy/Scattered disks/Quiz|Scattered disk astronomy quiz]] * [[Solar binary/Quiz]] * [[Source astronomy/Quiz]] * [[Submillimeter astronomy/Quiz]] * [[Ultraviolet astronomy/Quiz]] * [[Violet astronomy/Quiz]] * [[Visual astronomy/Quiz]] * [[X-ray astronomy/Quiz]] * [[Yellow astronomy/Quiz]] {{Div col end}} ==External links== * [http://www.iau.org/ International Astronomical Union] * [http://nedwww.ipac.caltech.edu/ NASA/IPAC Extragalactic Database - NED] * [http://nssdc.gsfc.nasa.gov/ NASA's National Space Science Data Center] * [http://www.adsabs.harvard.edu/ The SAO/NASA Astrophysics Data System] * [http://cas.sdss.org/astrodr6/en/tools/quicklook/quickobj.asp SDSS Quick Look tool: SkyServer] * [http://simbad.u-strasbg.fr/simbad/ SIMBAD Astronomical Database] * [http://simbad.harvard.edu/simbad/ SIMBAD Web interface, Harvard alternate] * [http://nssdc.gsfc.nasa.gov/nmc/SpacecraftQuery.jsp Spacecraft Query at NASA] * [http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/convcoord/convcoord.pl Universal coordinate converter] <!-- footer templates --> {{tlx|Radiation astronomy resources}}{{Principles of radiation astronomy}}{{Sisterlinks|Meteoroid radiation astronomy}} <!-- categories --> [[Category:Astrophysics quizzes]] [[Category:Meteorology quizzes]] [[Category:Meteors quizzes]] [[Category:Radiation astronomy quizzes]] [[Category:Resources last modified in June 2019]] n7ila555azm8nx3wv4mwemgmwyaybpx 2416452 2416451 2022-08-19T07:01:51Z Marshallsumter 311529 /* External links */ wikitext text/x-wiki [[Image:Orionid meteor.jpg|thumb|right|250px|This image is of a green and red Orionid meteor. Credit: [https://sites.google.com/site/thebrockeninglory/ Brocken Inaglory].{{tlx|free media}}]] '''[[Radiation astronomy/Meteoroids|Meteoroid radiation astronomy]]''' is a lecture for the [[Portal:Radiation astronomy|radiation astronomy department]] course on the [[principles of radiation astronomy]]. You are free to take this quiz based on [[Radiation astronomy/Meteoroids|meteoroid radiation astronomy]] at any time. To improve your score, read and study the lecture, the links contained within, listed under [[Radiation astronomy/Meteoroids/Quiz#See also|'''See also''']], [[Radiation astronomy/Meteoroids/Quiz#External links|'''External links''']], and in the {{tlx|principles of radiation astronomy}} template. This should give you adequate background to get 100 %. As a "learning by doing" resource, this quiz helps you to assess your knowledge and understanding of the information, and it is a quiz you may take over and over as a learning resource to improve your knowledge, understanding, test-taking skills, and your score. '''Suggestion:''' Have the lecture available in a separate window. To master the information and use only your memory while taking the quiz, try rewriting the information from more familiar points of view, or be creative with association. Enjoy learning by doing! {{clear}} ==Quiz== <quiz> {What may be the first astronomical source of the rocky objects in the solar system? |type="[]"} + Jupiter - the solar wind - the diffuse X-ray background + Saturn - Mount Redoubt in Alaska + the asteroid belt + the former protoplanetary disc around the Sun + the Oort cloud + the Kuiper belt {Which of the following is not a radiation phenomenon associated with a comet? |type="()"} - airglow - elongated dust particles - high albedo - olivine - Rayleighs + coronal mass ejection {With respect to protoplanetary disks what green mineral has been found? |type="{}"} { olivine (i) } {Which of the following are astronomical phenomena associated with the Sun as a likely source? |type="[]"} + coronal clouds + cosmic rays + neutrinos - Oh My God type particles + meteors - comets + blue rays {Complete the text: |type="{}"} Match up the item letter with each of the cosmogonic possibilities below: interior models of the giant planets - A high interest for cosmogony, geophysics and nuclear physics - B hierarchical accumulation - C clouds and globular clusters - D cosmic helium abundance - E deuterium fusion - F a large deficiency of light elements - G after galactic sized systems had collapsed - H the motions of hydrogen { D (i) } formation of luminous quasars { H (i) }. stars with an initial mass less than the solar mass { G (i) }. rotating liquid drops { B|F (i) }. primordial is less than 26 per cent { E (i) }. a solar mixture of elements dominated by hydrogen and helium gas { A (i) }. around 13 Jupiter masses { F (i) }. smaller rocky objects { C (i) }. {The charge on a planetary dust particle may change with? |type="{}"} { latitude (i) } {Which weather phenomena are most likely to interfere with observing stars? |type="[]"} + a late-summer rainstorm - a clear sky + an approaching dust storm - below normal temperatures + a typhoon + a snow fall + fog {A thin-section of a meteorite that may be analyzed has which of the following? |type="[]"} + a great many light gray and dark gray chondrules present + large grain cross sections making sizing easy + an available sizing or magnification marker + independent verification as a meteorite + correctable diameters and relative abundances + optical conditions {Various gaps and density minima have been observed in the Saturnian |type="{}"} { ring|rings (i) } system. {Which of the following are characteristic of a binary formed via gravitational fragmentation? |type="[]"} + the local Jeans length + the local speed of sound + the mean molecular weight - the electron neutrino + the mean particle density - neutrons {Complete the text: |type="{}"} The moon was formed independently of the earth and later { captured (i) }, presumably by a { three-body| three body (i) } interaction. {True or False, A hydrometeor is a precipitation product. |type="()"} + TRUE - FALSE {Usually associated with clouds filling the sky, thunder and lightning, wind and what water based meteorites |type="{}"} { rain (i) } {True or False, A lithometeor is a rocky-object meteor. |type="()"} + TRUE - FALSE {Meteorites found on Earth may be from which of the following? |type="[]"} - Saturn - Mercury + the Moon + the asteroid belt - Jupiter + Mars + Earth {True or False, A cryometeor is a very large chunk of amber. |type="()"} - TRUE + FALSE {Complete the text: |type="{}"} An aerometeor is a { discrete (i) } unit of air traveling or { falling (i) } through an atmosphere. {True or False, A metallic or stony object that is the remains of a meteor is called a meteoroid. |type="()"} - TRUE + FALSE {On 19 May 1910, the Earth actually passed through the tail of what comet. |type="{}"} { Halley's (i) } {True or False, The observations of planetary motion agree with computed orbits to the accuracy of the observations. |type="()"} - TRUE + FALSE {A cloud at nearly a quarter of the distance to Proxima Centauri is called the |type="{}"} { Oort cloud (i) } {True or False, Clouds are a form of radiation because they have a temperature gradient. |type="()"} - TRUE + FALSE {Which phenomena are associated with the heliosphere? |type="[]"} + a region of space where the interstellar medium is blown away by the solar wind + a bubble in space + virtually all the material emanates from the Sun itself - Voyager 2 + Voyager 1 + the termination shock {True or False, The prismoidal method provides a good approximation of the dust emission peak for cold sources. |type="()"} + TRUE - FALSE {A natural rocky source of chemicals from the sky to the ground may originate from what astronomical source? |type="()"} - Jupiter - the solar wind - the diffuse X-ray background - Mount Redoubt in Alaska + the asteroid belt - the International Space Station {True or False, A new cosmogony predicts that all particles in one stream have the same mineral composition. |type="()"} + TRUE - FALSE {Complete the text: |type="{}"} The majority of known asteroids orbit the Sun between the orbits of { Mars|Jupiter (i) } and { Jupiter|Mars (i) }. {True or False, Mars may have suffered asteroid impacts. |type="()"} + TRUE - FALSE {Which of the following are radiation astronomy phenomena associated with the gaseous-object Neptune? |type="[]"} - Voyager 2 + blue rays + clouds - neutron emission - polar coronal holes + meteor emission - rotation {True or False, The visible path of a meteoroid that has entered the Earth's atmosphere is called a meteorite. |type="()"} - TRUE + FALSE {Complete the text: |type="{}"} The 0.6 m Tortugas Mountain Observatory is used to monitor the temporal changes in the { Jovian|Jupiter (i) } cloud deck and equatorial activity on { Saturn (i) }. {True or False, Mercury may have once been a satellite of Jupiter, or possibly of Saturn. |type="()"} + TRUE - FALSE {Which of the following is not a characteristic of showers? |type="()"} + throwing a beam - meteors - rain - snow - hail {True or False, Micrometeoroids have less stable orbits than meteoroids. |type="()"} + TRUE - FALSE {Which of the following are radiation astronomy phenomena associated with the rocky-object Io? |type="[]"} + surface regions reflecting or emitting violet or purple - an excess brightness at or near the edge + red regions that may be phosphorus - neutron emission - polar coronal holes + meteor emission - rotation {True or False, To date, all of the reported hypervelocity stars (HVSs), which are believed to be ejected from the Galactic center, are blue. |type="()"} + TRUE - FALSE {Any natural object radiating through a portion or all of a natural object's atmosphere may be called a what? |type="{}"} { meteor (i) } {True or False, Olivine is a silicate mineral that may be detected in cometary coma dust with green astronomy. |type="()"} + TRUE - FALSE {Observations of comets have benefited greatly from what phenomenon of cyan astronomy? |type="()"} - Cherenkov radiation - the electric blue glow of lightning - gas-expansion velocity decreases with increasing heliocentric distance - methane possesses prominent absorption bands in the visible - adaptive optics + the light of the neutral CN-radical {Yes or No, A cometary orbit about the Sun is a radiation astronomy phenomenon. |type="()"} + Yes - No {Which of the following is not a radiation phenomenon associated with a planet? |type="()"} - airglow - elongated dust particles - high albedo - olivine - Rayleighs + coronal mass ejection </quiz> ==Hypotheses== {{main|Hypotheses}} # Meteors range in size from galaxy clusters to dust grains and molecular clusters. ==See also== {{div col|colwidth=20em}} * [[Radiation astronomy/Aerometeors/Quiz|Aerometeor astronomy quiz]] * [[Radiation astronomy/Asteroids/Quiz|Asteroid astronomy quiz]] * [[Astronomical observatories/Quiz]] * [[Radiation astronomy/Beta particles/Quiz|Beta-particles astronomy/Quiz]] * [[Blue astronomy/Quiz]] * [[Radiation astronomy/Clouds/Quiz|Cloud astronomy quiz]] * [[Radiation astronomy/Comets/Quiz|Cometary astronomy quiz]] * [[Cosmic-ray astronomy/Quiz]] * [[Crater astronomy/Quiz]] * [[Radiation astronomy/Cryometeors/Quiz|Cryometeor astronomy quiz]] * [[Cyan astronomy/Quiz]] * [[Radiation astronomy/Electromagnetics/Quiz|Electromagnetic astronomy/Quiz]] * [[Radiation/Electromagnetics/Quiz|Electromagnetic radiation/Quiz]] * [[Electron astronomy/Quiz]] * [[Radiation astronomy/Fieries/Quiz|Fiery-meteor astronomy quiz]] * [[First astronomical source/Quiz]] * [[First astronomical X-ray source/Quiz]] * [[Gamma-ray astronomy/Quiz]] * [[Green astronomy/Quiz]] * [[Radiation astronomy/Hydrometeors/Quiz|Hydrometeor astronomy quiz]] * [[Infrared astronomy/Quiz]] * [[Intergalactic medium/Quiz]] * [[Interplanetary medium/Quiz]] * [[Interstellar medium/Quiz]] * [[Radiation astronomy/Kuiper belts/Quiz|Kuiper belt astronomy quiz]] * [[Radiation astronomy/Lightnings/Quiz|Lightning astronomy quiz]] * [[Radiation astronomy/Lithometeors/Quiz|Lithometeor astronomy quiz]] * [[Meteor astronomy/Quiz]] * [[Meteorites/Quiz]] * [[Radiation astronomy/Meteoroids/Quiz|Meteoroid astronomy quiz]] * [[Radiation/Meteors/Quiz|Meteor radiation/Quiz]] * [[Radiation astronomy/Showers/Quiz|Meteor-shower astronomy quiz]] * [[Microwave astronomy/Quiz]] * [[Radiation astronomy/Nebulas/Quiz|Nebula quiz]] * [[Radiation astronomy/Neutrals/Quiz|Neutrals astronomy quiz]] * [[Radiation astronomy/Oort clouds/Quiz|Oort cloud astronomy quiz]] * [[Optical astronomy/Quiz]] * [[Orange astronomy/Quiz]] * [[Radiation astronomy/Particles/Quiz|Particle astronomy quiz]] * [[Radiation astronomy/Planets/Quiz|Planetary astronomy quiz]] * [[Radiation astronomy/Plasmas/Quiz|Plasma-meteors quiz]] * [[Radar astronomy/Quiz]] * [[Radiation/Quiz]] * [[Radiation/Meteors/Quiz|Radiation meteors/Quiz]] * [[Red astronomy/Quiz]] * [[Radiation astronomy/Rocks/Quiz|Rock astronomy quiz]] * [[Rocky-object astronomy/Quiz]] * [[Radiation astronomy/Scattered disks/Quiz|Scattered disk astronomy quiz]] * [[Solar binary/Quiz]] * [[Source astronomy/Quiz]] * [[Submillimeter astronomy/Quiz]] * [[Ultraviolet astronomy/Quiz]] * [[Violet astronomy/Quiz]] * [[Visual astronomy/Quiz]] * [[X-ray astronomy/Quiz]] * [[Yellow astronomy/Quiz]] {{Div col end}} ==External links== * [http://www.iau.org/ International Astronomical Union] * [http://nedwww.ipac.caltech.edu/ NASA/IPAC Extragalactic Database - NED] * [http://nssdc.gsfc.nasa.gov/ NASA's National Space Science Data Center] * [http://www.adsabs.harvard.edu/ The SAO/NASA Astrophysics Data System] * [http://cas.sdss.org/astrodr6/en/tools/quicklook/quickobj.asp SDSS Quick Look tool: SkyServer] * [http://simbad.u-strasbg.fr/simbad/ SIMBAD Astronomical Database] * [http://simbad.harvard.edu/simbad/ SIMBAD Web interface, Harvard alternate] * [http://nssdc.gsfc.nasa.gov/nmc/SpacecraftQuery.jsp Spacecraft Query at NASA] * [http://heasarc.gsfc.nasa.gov/cgi-bin/Tools/convcoord/convcoord.pl Universal coordinate converter] <!-- footer templates --> {{tlx|Radiation astronomy resources}}{{Principles of radiation astronomy}}{{Sisterlinks|Meteoroid radiation astronomy}} <!-- categories --> [[Category:Astrophysics quizzes]] [[Category:Meteorology quizzes]] [[Category:Meteors quizzes]] [[Category:Radiation astronomy quizzes]] 4rxmp2fgz01apjh164opmy3r9ve36hz 2 257428 2416391 2416076 2022-08-19T01:29:47Z 73.25.23.186 /* 0 */ wikitext text/x-wiki =🎱= [https://www.overleaf.com/learn/latex/TikZ_package TikZ¿¡⚽] [https://tikz.dev/ ฿ ₪ ₰] {| +BBwLTX Beaver Bank w LaTeX |- || 🎱 || main.pdf || destination || best distribution of purposeful print worthy content |- || ⛐ || readme.txt || 1st place to look for exceptions and exceptionaltions |- || 🈁 || main.aux || unknown |- || ɸ || Campus.jpeg || dot by dot picture w/o transparency |- || d͡ʒ || panther.jpeg || ligature |- || θ || mascot.jpg || the t a |- || ¶ || main.nav || unknown, but exciting place for an anchor icon |- || ‡.out || main.out || unknown |- ||『』.png || photo.png || dot by dot picture w alpha/transparency |- ||「」.snm || main.snm || 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style="color:pink; font-size: 11px; "> <center>⟁ ◁ pink ○ gray △ ⟁</span> |- || 人 | style="background:#000;" | <span style="color:#fff; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#fff ;" | <span style="color:#000; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:gray;" | <span style="color:pink; font-size: 11px;"><center> ⟁ </center></span> || [http://www.amp-what.com/unicode/search/triangle |- || 1 | style="background:#111;" | <span style="color:#ff0; font-size: 11px; "> <center>◁ #ff0 ○ #111△</center></span> || 人 || 人 | style="background:#111;" | <span style="color:#F0f; font-size: 11px; "> <center>◁ #f0f ○ #111△</center></span> || 人 || 人 | style="background:#111;" | <span style="color:#0ff; font-size: 11px; "> <center>◁ #0ff ○ #111 △</center></span> |- || 人 | style="background:#111;" | <span style="color:#ff0; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#111;" | <span style="color:#f0f; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#111;" | <span style="color:#0ff; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#333;" | <span style="color:#f00; font-size: 11px; "> <center>◁ #ff0 ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#0f0; font-size: 11px; "> <center>◁ #f0f ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#00f; font-size: 11px; "> <center>◁ #0ff ○ #333 △</center></span> |- || 人 | style="background:#333;" | <span style="color:#f00; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#333;" | <span style="color:#0f0; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#333;" | <span style="color:#00f; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#555;" | <span style="color:black; font-size: 11px; "> ◁ black ≡ #000 △ </span> || 人 || 人 | style="background:#555;" | <span style="color:indigo; font-size: 9px; "> ◁ indigo ≡ #4B0082△</span> || 人 || 人 | style="background:#555;" | <span style="color:midnightblue; font-size: 11px; "> ◁ midnightblue ≡ #191970△</span> |- || 人 | style="background:#555;" | <span style="color:black; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#555;" | <span style="color:indigo; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#555;" | <span style="color:midnightblue; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#777;" | <span style="color:#fff9c4; font-size: 11px; "> ◁ #fff9c4 ○ #777 △ </span> || 人 || 人 | style="background:#777;" | <span style="color:#FFCDD2; font-size: 11px; "> ◁ #FFCDD2 ○ #777 △ </span> || 人 || 人 | style="background:#777;" | <span style="color:#BBDEFB; font-size: 11px; "> ◁ #BBDEFB ○ #777 △ </span> |- || 人 | style="background:#777;" | <span style="color:#fff9c4; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#777;" | <span style="color:#FFCDD2; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#777;" | <span style="color:#BBDEFB; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#999;" | <span style="color:#ffe0b2; font-size: 11px; "> ◁ #ffe0b2 ○ #999 △ </span> || 人 || 人 | style="background:#999;" | <span style="color:#e1bee7; font-size: 11px; "> ◁ #e1bee7 ○ #999 △ </span> || 人 || 人 | style="background:#999;" | <span style="color:#C8E6C9; font-size: 11px; "> ◁ #C8E6C9 ○ #999 △ </span> |- || 人 | style="background:#999;" | <span style="color:#ffe0b2; font-size: 11px;"><center> ⌂ </center></span> || || | style="background:#999;" | <span style="color:#e1bee7; font-size: 11px;"><center> ⌂ </center></span> || || | style="background:#999;" | <span style="color:#C8E6C9; font-size: 11px;"><center> ⌂ </center></span> || |- || ⟁ | style="background:#aaa;" | <span style="color:#efefef; font-size: 11px; "> ◁ #efefef ○ #aaa △ </span> || 人 || 人 | style="background:#aaa;" | <span style="color:#222; font-size: 11px; "> ◁ #e1bee7 ○ #aaa △ </span> || 人 || 人 | style="background:#aaa;" | <span style="color:#777; font-size: 11px; "> ◁ #ababab ○ #aaa △ </span> |- || 人 | style="background:#aaa;" | <span style="color:#efefef ; font-size: 33px;"><center> ⚞人⚟ </center></span> || || | style="background:#aaa;" | <span style="color:#222; font-size: 33px;"><center> ⚞人⚟ </center></span> || || | style="background:#aaa;" | <span style="color:#777; font-size: 33px;"><center> ⚞人⚟ </center></span> || |} {| + font-size: 25px; |- || 3 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 25px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 4 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} ==WashingtonCounty.OR.US.case.22CN03334== [http://themetawiki.clu/w/index.php/WashingtonCounty.OR.US.case.22CN03334 http://themetawiki.clu/w/index.php/WashingtonCounty.OR.US.case.22CN03334] This is the case holding up plans for a physical wikiversity site in 97305-2644 Next court date is 9/8/2022 == Salem Accord == ===/əˈkôrd/ {phoenetic w _ } === ===アコード {phoenetic w kana} === ===符合 {kanji} === ===与える { hiragana | "one's own"? "tribal"? } === ==◯== <center style="background-color: indigo; color: yellow">⬛◯⬤⚪ ◁ yellow ○ indigo △ ⟁</center> <center style="background-color: indigo; color: magenta">⬛◯⬤⚪ ◁ magenta ○ indigo △ ⟁</center> <center style="background-color: indigo; color: cyan">⬛◯⬤⚪ ◁ cyan ○ indigo △ ⟁</center> <center style="background-color: indigo; color: lime">⬛◯⬤⚪ ◁ lime ○ indigo △ ⟁</center> <center style="background-color: midnightblue; color: indigo">⬛◯⬤⚪ ◁ indigo ○ midnightblue △ ⟁</center> <center style="background-color: midnightblue; color: black">⬛◯⬤⚪ ◁ black ○ midnightblue△ ⟁</center> <center style="background-color: indigo; color: midnightblue">⬛◯⬤⚪ ◁ midnightblue ○ indigo △ ⟁</center> <center style="background-color: indigo; color: black">⬛◯⬤⚪ ◁ black ○ indigo △ ⟁</center> <center style="background-color: black; color: indigo">⬛◯⬤⚪ ◁ indigo○ black △ ⟁</center> <center style="background-color: black; color: midnightblue">⬛◯⬤⚪ ◁ midnightblue ○ black△ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> ===人=== {| + font-size: 222px; primary |- || ◯ | style="background:#000 ;" | <span style="color:#fff "> <center>◁ #fff ○ #000 △ ⟁ </span> || 人 || 人 | style="background:#fff ;" | <span style="color:#000 "> <center>◁ #000 ○ #fff △ ⧋</span> || 人 || 人 | style="background:gray ;" | <span style="color:pink "> <center>◁ pink ○ gray △ 📐</span> |- || 人 | style="background:#000;" | <span style="color:#fff; font-size: 77px;"><center> ⟁ </center></span> || || | style="background:#fff ;" | <span style="color:#000; font-size: 77px;"><center> ⧋ </center></span> || || | style="background:gray;" | <span style="color:pink; font-size: 77px;"><center> 📐 </center></span> || [http://www.amp-what.com/unicode/search/triangle |- || 1 | style="background:#333;" | <span style="color:#ff0"> <center>◁ #ff0 ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#F0f"> <center>◁ #f0f ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#0ff"> <center>◁ #0ff ○ #333 △</center></span> |- || 人 | style="background:#333;" | <span style="color:#ff0; font-size: 277px;"><center> ⌂ </center></span> || || | style="background:#333;" | <span style="color:#f0f; font-size: 277px;"><center> ⌂ </center></span> || || | style="background:#333;" | <span style="color:#0ff; font-size: 277px;"><center> ⌂ </center></span> || |- || 2 | style="background:#808080;" | <span style="color:pink"> ◁ pink ○ #808080 △</span> || 人 || 人 | style="background:#808080;" | <span style="color:midnightblue"> ◁ midnightblue ○ #808080 △</span> || 人 || 人 | style="background:#808080;" | <span style="color:indigo"> ◁ indigo ○ #808080 △</span> |- || 人 | style="background:#808080;" | <span style="color:pink; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || || | style="background:#808080;" | <span style="color:midnightblue; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || || | style="background:#808080;" | <span style="color:indigo; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || |} or ∨↯∧ & + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + {| + font-size: 100px; |- || ◯ | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 46px;"><center> ∨↯∧ </center></span> || |} {| + font-size: 222px; primary |- || 1 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ #fff9c4 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ #FFCDD2 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ #BBDEFB ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 222px;"><center> ⌂ </center></span> || |} {| + font-size: 111px; secondary |- || 2 | style="background:#808080;" | <span style="color:#ffe0b2"> ◁ #ffe0b2 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#e1bee7"> ◁ #e1bee7 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#C8E6C9"> ◁ #C8E6C9 ○</span> |- || 人 | style="background:#808080;" | <span style="color:#ffe0b2; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#e1bee7; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#C8E6C9; font-size: 222px;"><center> ⌂ </center></span> || |} {| + font-size: 25px; |- || 3 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 25px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 4 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} {| + font-size: 100px; |- || 5 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 6 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} == 0 == {| |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家] || 小 |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] |} meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} {| + font-size: 100px; |- || 人 | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 100px;"><center> ∨↯∧ </center></span> || || | style="background:#808080;" | <span style="color:#fff9c4; font-size: 100px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 100px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 100px;"><center> ⌂ </center></span> || |- || |- || |} == or ∨↯∧ & + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + .us == ==⌂== {| + font-size: 100px; |- || 人 | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 100px;"><center> ⌂ </center></span> || |- || |- || |} [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ==✪⚪⬤🔥◍⚫⨁ 👀 ≡ odd → +1 [ { ( East ⚫🔴⚪○💮⭕ West ) } ] iff even ⇒ ÷2== ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] || ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪ || || ○人∘🧿⚪⬤◍⚫ || || ⨁⚉⨂❂✧✷✸✡✵ |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |} == ya == ◯ ○ ∘ ⬤ 大 【 ヤ / や 】 (ya) 人 【 ジン、 ニン / ひと 】(jean、he toe) 大家 (ya ya) Ya (hiragana: や, katakana: ヤ) ヤフー 屋 - Wiktionaryhttps://en.wiktionary.org › wiki › 屋 Semantic compound of 尸+至. 尸 does not represent the radical for death, but is a pictogram depicting a cloth draped. 至 means "dead end". Home (家) A home, or domicile, is a space used as a permanent or semi-permanent residence for one or many humans. It is a fully or semi sheltered space and can have both interior and exterior aspects to it. Ya (や) Kana Ya is one of the Japanese kana, each of which represents one mora. The hiragana is written in three strokes, while the katakana is written in two. Both represent. Their shapes have origins in the character 也. Wikipedia hiragana origin: 也 spelling kana: 大和のヤ Yamato no "ya" transliteration: ya unicode: U+3084, U+30E4 What is the pronunciation of Ya line? In historical kana orthography, it is written as "yau", "say", and "eu", and read as "yo", "you", and "yo", respectively. Even in modern times, "saying" and "going" are sometimes pronounced as "yu" and "yuku . " From the above, it can be said that Ya line is the yoon of that line . [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] == g == {| + |- || g || ⌂ || 時 |- || nono || ⌂ || ◁ || 前 || の || ノ || 名 || "Salmon" || #fa8072 || || || || || || rgb(250, 128, 114) || :: || hsl(6, 93%, 71%) |- || "のノ" || ⌂ || <span style="color:salmon;"><center> ∅ </center></span> | style="background:#000;"|<span style="color:salmon;"><center> #000 </center></span> | style="background:#808080;" | <span style="color:salmon;"><center> #808080 </center></span> | style="background:#fff;" | <span style="color:salmon;"><center> #fff </center></span> | style="background:#333;" | <span style="color:salmon;"><center> #333 </center></span> | style="background:#333;" | <span style="color:#fa8072;"><center> #fa8072 </center></span> | style="background:#333;" | <span style="color:#fa8072ff;"><center> #fa8072ff </center></span> | style="background:#333;" | <span style="color:#fa8072aa;"><center> #fa8072aa </center></span> | style="background:#333;" | <span style="color:#fa807280;"><center> #fa807280 </center></span> | style="background:#333;" | <span style="color:#fa807233;"><center> #fa807233 </center></span> | style="background:#333;" | <span style="color:#fa807200;"><center> #fa807200 </center></span> |- || know no ノノ || ⌂ || <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#000;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#808080;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#fff;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072ff; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072aa; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807280; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807233; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807200; font-size: 100px;"> ☂ </span> |} {| + |- || hsl(0-33, 93%, 71%) || <span style="color:hsl(0, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(1, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(2, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(3, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(4, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(5, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(7, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(8, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(9, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(10, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(11, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(12, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(13, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(14, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(15, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(16, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(17, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(18, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(19, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(20, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(21, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(22, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(23, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(24, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(25, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(26, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(27, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(28, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(29, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(30, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(31, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(32, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(33, 93%, 71%);"><center>⌂ </center></span> |} {| + |- || hsl(6-606, 93%, 71%) || <span style="color:hsl(6, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(36, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(66, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(96, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(126, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(156, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(186, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(216, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(246, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(276, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(306, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(336, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(366, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(396, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(426, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(456, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(486, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(516, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(546, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(576, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(606, 93%, 71%);"><center>⌂ </center></span> |} {| + |- || hsl(6, 93%, 0-100%) || <span style="color:hsl(6, 93%, 0%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 10%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 20%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 30%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 40%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 50%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 60%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 70%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 80%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 90%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 100%);"><center>⌂ </center></span> |} {| + |- || hsl(6, 0-100%, 71%) || <span style="color:hsl(6, 0%, 71%);"><center> ⌂ </center></span> || <span style="color:hsl(6, 10%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 20%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 30%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 40%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 50%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 60%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 70%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 80%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 90%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 100%, 71%);"><center>⌂ </center></span> |} meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |- ||ternary ||3 |- ||quaternary ||4 |- ||quinary ||5 |- ||senary ||6 |- ||septenary ||7 |- ||octal ||8 |- ||nonary |- ||decimal |- ||undenary |- ||duodecimal |- ||hexadecimal ||16 |- ||vigesimal ||20 |- ||sexagesimal ||60 |} How do you want your water served when you get here? https://pubs.usgs.gov/sir/2005/5168/pdf/sir2005-5168.pdf Robert Lee Stinson %VOX "tautology club says hi" w 11am "Naturalist Society for the Humane Treatment of Monsters" from dnd game on twitter [https://www.youtube.com/watch?v=n7uNA5fO1iI rice ex in CA] https://www.oregonwild.org/about/blog/oregon-grizzly-country https://therevelator.org/yellowstone-grizzlies-unbearable-divides/ https://developer.mozilla.org/en-US/docs/Web/CSS/color_value/hsl https://www.researchgate.net/about Amare, Nicole & Manning, A.. (2012). Seeing typeface personality: Emotional responses to form as tone. IEEE International Professional Communication Conference. 1-9. 10.1109/IPCC.2012.6408605. Various studies have correlated specific visual characteristics of typefaces with specific overall emotional effects: curvilinear forms and open letter shapes generally feel “friendly” but also “formal” or “informal,” depending on other factors; large contrasts in stroke widths, cap height, and aspect ratio generally feel “interesting,” but also “attractive” or “aggressive,” depending on other factors; low-variety and low-contrast forms generally feel “professional” but also “reliable” or “boring.” Although the current findings on typeface personality are useful, they have not indicated a systematic explanation for why specific physical typeface forms have the specific emotion effects that they do. This paper will report results of an empirical study in which 102 participants indicated their immediate emotional responses to each of 36 distinct typeface designs. Results support correlation between specific typeface features (variety vs. contrast vs. pattern) and specific emotional parameters (amusement vs. agitation vs. focus), explaining findings of previous studies, suggesting various classroom approaches to purpose-driven typeface selection. {{User alternative account|VeronicaJeanAnderson}} {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( A B E ) } ] </span> | style="background:black;" | <span style="color:white"> [ { ( [https://www.twitch.tv/archie97305 👀] ) } ]</span> |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> index.html</span> || notepad/atom (atom is deprecated) |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> vue </span> || [https://www.vim.org/ vim] [https://github.com/vim/vim-win32-installer/releases installer] |- || Tertiary | style="background:white;" | <span style="color:black"> css </span> || global css @ || gg css @ || NPC css @ |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> pug </span> |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> |- || Senary || b |} https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/sandbox trying to create a 1 -> 2 -> 3 -> 4 -> 5 -> 6 system in the apartment here that can be copied from site to site using artistic threads to help a Nice And Proper NAP-er navigate between properties with ease while maintaining adequate supportive care that we all require to enable us to focus on whatever catches our fancy. sun; natural light; breathe; BGs carbs; hygiene; laundry away bedroom; needles; blood; garbage out kitchen/nutritional/study social/outreach/linking worlds back porch 0 -- Computer Science, information and general works {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( T O P ) } ] </span> | style="background:black;" | <span style="color:white"> [ ℳ ] </span> | style="background:white;" | <span style="color:black"> { ¢ } </span> | style="background:#F9F9F9;" | <span style="color:pink"> ( ৳ ) </span> | style="background:black;" | <span style="color:white"> [ { ( I.n C.ase of E.mergency ) } ] </span> | style="background:teal;" | <span style="color:lime"> ᐪ l i p s c h i t z </span> || [https://www.youtube.com/watch?v=qrrz54UtkCc ᐪ] |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> physical</span> | style="background:#FFE6E6;" | <span style="color:black"> emotional</span> | style="background:#E6EAFF;" | <span style="color:black"> social</span> || This reflects health enough to communicate with people intimately enough to address real immediate issues | style="background:#FFFFE6;" | <span style="color:teal"> ^ torikomu </span> ||[https://www.youtube.com/watch?v=YxvBPH4sArQ ^] |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> occupational</span> | style="background:#F2E6FF;" | <span style="color:black"> intellectual</span> | style="background:#E6FFEA;" | <span style="color:black"> environmental</span> || This reflects living somewhere promoting healthy reasoning | style="background:#FFE6E6;" | <span style="color: teal"> | kaizen | </span> || | |- || Tertiary | style="background:white;" | <span style="color:black"> spiritual</span> | style="background:#BFBFBF;" | <span style="color:white"> factual </span> | style="background:#F2F2F2;" | <span style="color:black"> nutritional</span> || This reflects healthy mindful every habits | style="background:#E6EAFF;" | <span style="color:teal"> . genkiness . .</span> || . |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> generational</span> | style="background:#E6FFFF;" | <span style="color:black"> miscellaneal</span> | style="background:#F2E0CE;" | <span style="color:black"> punctuational</span> || This reflects having it all together enough to enjoy the holidays | style="background:#FFF2E6;" | <span style="color:lime"> # goblin </span> || # |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> | style="background:white;" | <span style="color:black"> {verb} </span> | style="background:black;" | <span style="color:white"> [noun] </span> || This reflects deliberate professional progress | style="background:#F2E6FF;" | <span style="color:lime"> / tsugu /</span> || / |- || Senary || b || 〇 || x || This reflects influencing others | style="background:#E6FFEA;" | <span style="color:lime"> @ g @ g @ </span> || [https://www.youtube.com/watch?v=SYnVYJDxu2Q @] |} == 100 -- Philosophy and psychology == How can I use color to manipulate behavior and improve communication? ===named=== ==== Re⋮Beccaδ#639 ==== ===== rebeccapurple :: #663399 ===== https://meyerweb.com/eric/thoughts/2014/06/19/rebeccapurple/ ====black==== ====white==== ====græy==== ====pink==== ====indigo==== ====midnightblue==== ===hex=== ====#fff==== ====#fff==== {| class="wikitable" style="text-align: center;" |+ |- | style="background:black;" | <span style="color:white"> [ white { on black ⚞🧿⚟ #fff on #000 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:pink"> [ pink { on 50% grey ⚞🧿⚟ #ffc0cb on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#191970"> [ midnightblue { on 50% grey ⚞🧿⚟ #191970 on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#4b0082"> [ indigo { on 50% grey ⚞🧿⚟ #4b0082 on #808080 ⚞🧿⚟ } ] </span> |} ===cmyk=== https://colordesigner.io/convert/cmyktohex ====gg on cmyk(0,0,0,33.3) w|materializecss.com==== https://materializecss.com/color.html {| class="wikitable" style="text-align: center;" |+ |- | style="background:#ababab" | <span style="color:#fff9c4"> [ gg_yellow { on cmyk(0,0,0,33.3) ⚞🧿⚟ #fff9c4 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffe0b2"> [ gg_orange { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffe0b2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffcdd2"> [ gg_red { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffcdd2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#e1bee7"> [ gg_purple { on cmyk(0,0,0,33.3) ⚞🧿⚟ #e1bee7 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#bbdefb"> [ gg_blue { on cmyk(0,0,0,33.3) ⚞🧿⚟ #bbdefb on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#c8e6c9"> [ gg_green { on cmyk(0,0,0,33.3) ⚞🧿⚟ #c8e6c9 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#efefef"> [ gg_white { on cmyk(0,0,0,33.3) ⚞🧿⚟ #efefef on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#111"> [ gg_black { on cmyk(0,0,0,33.3) ⚞🧿⚟ #111 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#808080"> [ gg_grey { on cmyk(0,0,0,33.3) ⚞🧿⚟ #808080 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#f8bbd0"> [ gg_pink { on cmyk(0,0,0,33.3) ⚞🧿⚟ #f8bbd0 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#b2ebf2"> [ gg_cyan { on cmyk(0,0,0,33.3) ⚞🧿⚟ #b2ebf2 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#d7ccc8"> [ gg_brown { on cmyk(0,0,0,33.3) ⚞🧿⚟ #d7ccc8 on #ababab )⚞🧿⚟} ] </span> |} ===rgba=== === TrumPutin-ism === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. == 200 -- Religion == Royal We "spiritual warfare" 1000 things I did 1992-2022 other than lie my way onto the supreme court to overturn Roe v Wade {| |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |} == 300 -- Social sciences == https://wattention.com/traditional-rice-harvesting-in-japan/ https://www.wwoofjapan.com/home/index.php?lang=en == 400 -- Language == === Programming === ==== .png ==== ==== Esperanto ==== ==== HTML ==== ==== PUG ==== == 500 -- Pure Science == == 600 -- Technology == === local hosts=== [http://localhost:8080/ 8080] file:///D:/index.html === Roland SP 404MKII === https://www.roland.com/global/products/sp-404mk2/ https://www.roland.com/global/support/by_product/sp-404mk2/owners_manuals/ @https://static.roland.com/manuals/sp-404mk2_app/eng/19610757.html === VIM === https://vim-adventures.com/ == 700 -- Arts and recreation == == 800 -- Literature == == 900 -- History and geography == https://geology.com/stories/13/bear-areas/ === pre-2030 === 2022 "booked" by Hillsboro Police for sending email addressing "Christian Hate" and "Spiritual War" along with "exorcisms" and "Halloween Hysteria" in Marion County, OR where Salem Police Department abdicated from protecting some children in Salem from 2016-2021. 2021 Kaiser Permanente promised cash settlement to mitigate their abdication in Marion County. KP lawyer with intimate details about my vagina: terrence .j . loeber@kp.org 2012 "unliked" on FB by some Nazarene peers after openly questioning Alex Jones' allegation that Sandy Hook didn't happen and asking for compassion for parents who were called actors while they grieved publicly through no choice of their own. 2011 Lupron given by KP for menorrhagia as alternative to b/c pills first rxd in 86. How many women who have "mostly" been on b/c pills from 87-11 are obese? Why no menorrhagia while immersed in Japan? How close to a traditional Japanese diet can I get in the Willamette Valley and how close to no meat will my body allow? === TrumPutish War Against Humanity === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. === Ring of Fire === === Post "Roe v Wade" === Who did Roe v Wade protect? Why would a Nazarene raised pro-life support an "underground" network post Roe v Wade? === Our Contemporary "Underground Railroad" needs a submarine? === Why did Portland, OR close the Shanghai Tunnels recently? Human Trafficking through Astoria, OR has been going on "forever". How do we align an "underground railroad" with contemporary supports? == 10 -- A & + == == 11 -- B * x == == 12 -- C f(◯) == == 13 -- D Δ δ ƍ ≜ 𝜟 𝝳 == == 14 -- E 🐘 𓃰 == == 15 -- F == == 16 -- G == == 17 -- H == == 18 -- I == == 19 -- J == == 20 -- K == == 21 -- L == == 22 -- M == == 23 -- N == == 24 -- O == == 25 -- P == == 26 -- Q == == 27 -- R == == 28 -- S == == 29 -- T == == 30 -- U == == 31 -- V == == 32 -- W == == 33 -- X == == 34 -- Y == == 35 -- Z == j8umanpvoys6j3v48it0yzsa99dgx0p 2416393 2416391 2022-08-19T01:30:00Z 73.25.23.186 wikitext text/x-wiki =0= == 00 == === 000 === ==== 0000 ==== ===== 00000 ===== ====== 000000 ====== =🎱= [https://www.overleaf.com/learn/latex/TikZ_package TikZ¿¡⚽] [https://tikz.dev/ ฿ ₪ ₰] {| +BBwLTX Beaver Bank w LaTeX |- || 🎱 || main.pdf || destination || best distribution of purposeful print worthy content |- || ⛐ || readme.txt || 1st place to look for exceptions and exceptionaltions |- || 🈁 || main.aux || unknown |- || ɸ || Campus.jpeg || dot by dot picture w/o transparency |- || d͡ʒ || panther.jpeg || ligature |- || θ || mascot.jpg || the t a |- || ¶ || main.nav || unknown, but exciting place for an anchor icon |- || ‡.out || main.out || unknown |- ||『』.png || photo.png || dot by dot picture w alpha/transparency |- ||「」.snm || main.snm || wft? |- ||《》.tex || main.tex || this is the mistress J wants me to love |- || ⟨⟩ .log || main.log || sounds like somewhere stuff gets auto-logged |- || «» .toc || main.toc || table of contents: creative content starts here |} [https://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes ⛐] | git bash 🈁 | | pdflatex <your d🎱cument> | | [https://en.wikibooks.org/wiki/LaTeX/Basics#Compilation (ℝ\ℚ)] | | 1 = R\Q / (ℝ\ℚ) = ⬤ [https://www.latex-project.org/ 🎱 git a ⚽ clu & <3 🎮 LaTeX 🈁 👣 🏊 ] > https://ggf333ttu.blogspot.com/ https://www.youtube.com/watch?v=H3QmruFdfgo ==new habits== https://architecturenotes.co/redis/ 𝑓(◯) allostatis https://vim.rtorr.com/ https://github.com/rtorr/vim-cheat-sheet {| + 𝑓(◯) |- || ◯ | style="background:#000 ;" | <span style="color:#fff; font-size: 11px; "> <center>⟁ ◁ #fff ○ #000 △ ⟁ </span> || 人 || 人 | style="background:#fff ;" | <span style="color:#000; font-size: 11px; "> <center>⟁ ◁ #000 ○ #fff △ ⟁</span> || 人 || 人 | style="background:gray ;" | <span style="color:pink; font-size: 11px; "> <center>⟁ ◁ pink ○ gray △ ⟁</span> |- || 人 | style="background:#000;" | <span style="color:#fff; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#fff ;" | <span style="color:#000; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:gray;" | <span style="color:pink; font-size: 11px;"><center> ⟁ </center></span> || [http://www.amp-what.com/unicode/search/triangle |- || 1 | style="background:#111;" | <span style="color:#ff0; font-size: 11px; "> <center>◁ #ff0 ○ #111△</center></span> || 人 || 人 | style="background:#111;" | <span style="color:#F0f; font-size: 11px; "> <center>◁ #f0f ○ #111△</center></span> || 人 || 人 | style="background:#111;" | <span style="color:#0ff; font-size: 11px; "> <center>◁ #0ff ○ #111 △</center></span> |- || 人 | style="background:#111;" | <span style="color:#ff0; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#111;" | <span style="color:#f0f; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#111;" | <span style="color:#0ff; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#333;" | <span style="color:#f00; font-size: 11px; "> <center>◁ #ff0 ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#0f0; font-size: 11px; "> <center>◁ #f0f ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#00f; font-size: 11px; "> <center>◁ #0ff ○ #333 △</center></span> |- || 人 | style="background:#333;" | <span style="color:#f00; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#333;" | <span style="color:#0f0; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#333;" | <span style="color:#00f; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#555;" | <span style="color:black; font-size: 11px; "> ◁ black ≡ #000 △ </span> || 人 || 人 | style="background:#555;" | <span style="color:indigo; font-size: 9px; "> ◁ indigo ≡ #4B0082△</span> || 人 || 人 | style="background:#555;" | <span style="color:midnightblue; font-size: 11px; "> ◁ midnightblue ≡ #191970△</span> |- || 人 | style="background:#555;" | <span style="color:black; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#555;" | <span style="color:indigo; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#555;" | <span style="color:midnightblue; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#777;" | <span style="color:#fff9c4; font-size: 11px; "> ◁ #fff9c4 ○ #777 △ </span> || 人 || 人 | style="background:#777;" | <span style="color:#FFCDD2; font-size: 11px; "> ◁ #FFCDD2 ○ #777 △ </span> || 人 || 人 | style="background:#777;" | <span style="color:#BBDEFB; font-size: 11px; "> ◁ #BBDEFB ○ #777 △ </span> |- || 人 | style="background:#777;" | <span style="color:#fff9c4; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#777;" | <span style="color:#FFCDD2; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#777;" | <span style="color:#BBDEFB; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#999;" | <span style="color:#ffe0b2; font-size: 11px; "> ◁ #ffe0b2 ○ #999 △ </span> || 人 || 人 | style="background:#999;" | <span style="color:#e1bee7; font-size: 11px; "> ◁ #e1bee7 ○ #999 △ </span> || 人 || 人 | style="background:#999;" | <span style="color:#C8E6C9; font-size: 11px; "> ◁ #C8E6C9 ○ #999 △ </span> |- || 人 | style="background:#999;" | <span style="color:#ffe0b2; font-size: 11px;"><center> ⌂ </center></span> || || | style="background:#999;" | <span style="color:#e1bee7; font-size: 11px;"><center> ⌂ </center></span> || || | style="background:#999;" | <span style="color:#C8E6C9; font-size: 11px;"><center> ⌂ </center></span> || |- || ⟁ | style="background:#aaa;" | <span style="color:#efefef; font-size: 11px; "> ◁ #efefef ○ #aaa △ </span> || 人 || 人 | style="background:#aaa;" | <span style="color:#222; font-size: 11px; "> ◁ #e1bee7 ○ #aaa △ </span> || 人 || 人 | style="background:#aaa;" | <span style="color:#777; font-size: 11px; "> ◁ #ababab ○ #aaa △ </span> |- || 人 | style="background:#aaa;" | <span style="color:#efefef ; font-size: 33px;"><center> ⚞人⚟ </center></span> || || | style="background:#aaa;" | <span style="color:#222; font-size: 33px;"><center> ⚞人⚟ </center></span> || || | style="background:#aaa;" | <span style="color:#777; font-size: 33px;"><center> ⚞人⚟ </center></span> || |} {| + font-size: 25px; |- || 3 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 25px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 4 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} ==WashingtonCounty.OR.US.case.22CN03334== [http://themetawiki.clu/w/index.php/WashingtonCounty.OR.US.case.22CN03334 http://themetawiki.clu/w/index.php/WashingtonCounty.OR.US.case.22CN03334] This is the case holding up plans for a physical wikiversity site in 97305-2644 Next court date is 9/8/2022 == Salem Accord == ===/əˈkôrd/ {phoenetic w _ } === ===アコード {phoenetic w kana} === ===符合 {kanji} === ===与える { hiragana | "one's own"? "tribal"? } === ==◯== <center style="background-color: indigo; color: yellow">⬛◯⬤⚪ ◁ yellow ○ indigo △ ⟁</center> <center style="background-color: indigo; color: magenta">⬛◯⬤⚪ ◁ magenta ○ indigo △ ⟁</center> <center style="background-color: indigo; color: cyan">⬛◯⬤⚪ ◁ cyan ○ indigo △ ⟁</center> <center style="background-color: indigo; color: lime">⬛◯⬤⚪ ◁ lime ○ indigo △ ⟁</center> <center style="background-color: midnightblue; color: indigo">⬛◯⬤⚪ ◁ indigo ○ midnightblue △ ⟁</center> <center style="background-color: midnightblue; color: black">⬛◯⬤⚪ ◁ black ○ midnightblue△ ⟁</center> <center style="background-color: indigo; color: midnightblue">⬛◯⬤⚪ ◁ midnightblue ○ indigo △ ⟁</center> <center style="background-color: indigo; color: black">⬛◯⬤⚪ ◁ black ○ indigo △ ⟁</center> <center style="background-color: black; color: indigo">⬛◯⬤⚪ ◁ indigo○ black △ ⟁</center> <center style="background-color: black; color: midnightblue">⬛◯⬤⚪ ◁ midnightblue ○ black△ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> ===人=== {| + font-size: 222px; primary |- || ◯ | style="background:#000 ;" | <span style="color:#fff "> <center>◁ #fff ○ #000 △ ⟁ </span> || 人 || 人 | style="background:#fff ;" | <span style="color:#000 "> <center>◁ #000 ○ #fff △ ⧋</span> || 人 || 人 | style="background:gray ;" | <span style="color:pink "> <center>◁ pink ○ gray △ 📐</span> |- || 人 | style="background:#000;" | <span style="color:#fff; font-size: 77px;"><center> ⟁ </center></span> || || | style="background:#fff ;" | <span style="color:#000; font-size: 77px;"><center> ⧋ </center></span> || || | style="background:gray;" | <span style="color:pink; font-size: 77px;"><center> 📐 </center></span> || [http://www.amp-what.com/unicode/search/triangle |- || 1 | style="background:#333;" | <span style="color:#ff0"> <center>◁ #ff0 ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#F0f"> <center>◁ #f0f ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#0ff"> <center>◁ #0ff ○ #333 △</center></span> |- || 人 | style="background:#333;" | <span style="color:#ff0; font-size: 277px;"><center> ⌂ </center></span> || || | style="background:#333;" | <span style="color:#f0f; font-size: 277px;"><center> ⌂ </center></span> || || | style="background:#333;" | <span style="color:#0ff; font-size: 277px;"><center> ⌂ </center></span> || |- || 2 | style="background:#808080;" | <span style="color:pink"> ◁ pink ○ #808080 △</span> || 人 || 人 | style="background:#808080;" | <span style="color:midnightblue"> ◁ midnightblue ○ #808080 △</span> || 人 || 人 | style="background:#808080;" | <span style="color:indigo"> ◁ indigo ○ #808080 △</span> |- || 人 | style="background:#808080;" | <span style="color:pink; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || || | style="background:#808080;" | <span style="color:midnightblue; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || || | style="background:#808080;" | <span style="color:indigo; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || |} or ∨↯∧ & + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + {| + font-size: 100px; |- || ◯ | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 46px;"><center> ∨↯∧ </center></span> || |} {| + font-size: 222px; primary |- || 1 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ #fff9c4 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ #FFCDD2 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ #BBDEFB ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 222px;"><center> ⌂ </center></span> || |} {| + font-size: 111px; secondary |- || 2 | style="background:#808080;" | <span style="color:#ffe0b2"> ◁ #ffe0b2 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#e1bee7"> ◁ #e1bee7 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#C8E6C9"> ◁ #C8E6C9 ○</span> |- || 人 | style="background:#808080;" | <span style="color:#ffe0b2; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#e1bee7; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#C8E6C9; font-size: 222px;"><center> ⌂ </center></span> || |} {| + font-size: 25px; |- || 3 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 25px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 4 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} {| + font-size: 100px; |- || 5 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 6 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} == 0 == {| |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家] || 小 |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] |} meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} {| + font-size: 100px; |- || 人 | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 100px;"><center> ∨↯∧ </center></span> || || | style="background:#808080;" | <span style="color:#fff9c4; font-size: 100px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 100px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 100px;"><center> ⌂ </center></span> || |- || |- || |} == or ∨↯∧ & + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + .us == ==⌂== {| + font-size: 100px; |- || 人 | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 100px;"><center> ⌂ </center></span> || |- || |- || |} [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ==✪⚪⬤🔥◍⚫⨁ 👀 ≡ odd → +1 [ { ( East ⚫🔴⚪○💮⭕ West ) } ] iff even ⇒ ÷2== ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] || ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪ || || ○人∘🧿⚪⬤◍⚫ || || ⨁⚉⨂❂✧✷✸✡✵ |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |} == ya == ◯ ○ ∘ ⬤ 大 【 ヤ / や 】 (ya) 人 【 ジン、 ニン / ひと 】(jean、he toe) 大家 (ya ya) Ya (hiragana: や, katakana: ヤ) ヤフー 屋 - Wiktionaryhttps://en.wiktionary.org › wiki › 屋 Semantic compound of 尸+至. 尸 does not represent the radical for death, but is a pictogram depicting a cloth draped. 至 means "dead end". Home (家) A home, or domicile, is a space used as a permanent or semi-permanent residence for one or many humans. It is a fully or semi sheltered space and can have both interior and exterior aspects to it. Ya (や) Kana Ya is one of the Japanese kana, each of which represents one mora. The hiragana is written in three strokes, while the katakana is written in two. Both represent. Their shapes have origins in the character 也. Wikipedia hiragana origin: 也 spelling kana: 大和のヤ Yamato no "ya" transliteration: ya unicode: U+3084, U+30E4 What is the pronunciation of Ya line? In historical kana orthography, it is written as "yau", "say", and "eu", and read as "yo", "you", and "yo", respectively. Even in modern times, "saying" and "going" are sometimes pronounced as "yu" and "yuku . " From the above, it can be said that Ya line is the yoon of that line . [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] == g == {| + |- || g || ⌂ || 時 |- || nono || ⌂ || ◁ || 前 || の || ノ || 名 || "Salmon" || #fa8072 || || || || || || rgb(250, 128, 114) || :: || hsl(6, 93%, 71%) |- || "のノ" || ⌂ || <span style="color:salmon;"><center> ∅ </center></span> | style="background:#000;"|<span style="color:salmon;"><center> #000 </center></span> | style="background:#808080;" | <span style="color:salmon;"><center> #808080 </center></span> | style="background:#fff;" | <span style="color:salmon;"><center> #fff </center></span> | style="background:#333;" | <span style="color:salmon;"><center> #333 </center></span> | style="background:#333;" | <span style="color:#fa8072;"><center> #fa8072 </center></span> | style="background:#333;" | <span style="color:#fa8072ff;"><center> #fa8072ff </center></span> | style="background:#333;" | <span style="color:#fa8072aa;"><center> #fa8072aa </center></span> | style="background:#333;" | <span style="color:#fa807280;"><center> #fa807280 </center></span> | style="background:#333;" | <span style="color:#fa807233;"><center> #fa807233 </center></span> | style="background:#333;" | <span style="color:#fa807200;"><center> #fa807200 </center></span> |- || know no ノノ || ⌂ || <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#000;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#808080;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#fff;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072ff; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072aa; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807280; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807233; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807200; font-size: 100px;"> ☂ </span> |} {| + |- || hsl(0-33, 93%, 71%) || <span style="color:hsl(0, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(1, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(2, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(3, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(4, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(5, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(7, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(8, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(9, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(10, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(11, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(12, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(13, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(14, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(15, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(16, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(17, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(18, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(19, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(20, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(21, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(22, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(23, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(24, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(25, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(26, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(27, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(28, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(29, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(30, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(31, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(32, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(33, 93%, 71%);"><center>⌂ </center></span> |} {| + |- || hsl(6-606, 93%, 71%) || <span style="color:hsl(6, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(36, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(66, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(96, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(126, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(156, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(186, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(216, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(246, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(276, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(306, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(336, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(366, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(396, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(426, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(456, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(486, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(516, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(546, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(576, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(606, 93%, 71%);"><center>⌂ </center></span> |} {| + |- || hsl(6, 93%, 0-100%) || <span style="color:hsl(6, 93%, 0%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 10%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 20%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 30%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 40%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 50%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 60%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 70%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 80%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 90%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 100%);"><center>⌂ </center></span> |} {| + |- || hsl(6, 0-100%, 71%) || <span style="color:hsl(6, 0%, 71%);"><center> ⌂ </center></span> || <span style="color:hsl(6, 10%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 20%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 30%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 40%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 50%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 60%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 70%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 80%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 90%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 100%, 71%);"><center>⌂ </center></span> |} meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |- ||ternary ||3 |- ||quaternary ||4 |- ||quinary ||5 |- ||senary ||6 |- ||septenary ||7 |- ||octal ||8 |- ||nonary |- ||decimal |- ||undenary |- ||duodecimal |- ||hexadecimal ||16 |- ||vigesimal ||20 |- ||sexagesimal ||60 |} How do you want your water served when you get here? https://pubs.usgs.gov/sir/2005/5168/pdf/sir2005-5168.pdf Robert Lee Stinson %VOX "tautology club says hi" w 11am "Naturalist Society for the Humane Treatment of Monsters" from dnd game on twitter [https://www.youtube.com/watch?v=n7uNA5fO1iI rice ex in CA] https://www.oregonwild.org/about/blog/oregon-grizzly-country https://therevelator.org/yellowstone-grizzlies-unbearable-divides/ https://developer.mozilla.org/en-US/docs/Web/CSS/color_value/hsl https://www.researchgate.net/about Amare, Nicole & Manning, A.. (2012). Seeing typeface personality: Emotional responses to form as tone. IEEE International Professional Communication Conference. 1-9. 10.1109/IPCC.2012.6408605. Various studies have correlated specific visual characteristics of typefaces with specific overall emotional effects: curvilinear forms and open letter shapes generally feel “friendly” but also “formal” or “informal,” depending on other factors; large contrasts in stroke widths, cap height, and aspect ratio generally feel “interesting,” but also “attractive” or “aggressive,” depending on other factors; low-variety and low-contrast forms generally feel “professional” but also “reliable” or “boring.” Although the current findings on typeface personality are useful, they have not indicated a systematic explanation for why specific physical typeface forms have the specific emotion effects that they do. This paper will report results of an empirical study in which 102 participants indicated their immediate emotional responses to each of 36 distinct typeface designs. Results support correlation between specific typeface features (variety vs. contrast vs. pattern) and specific emotional parameters (amusement vs. agitation vs. focus), explaining findings of previous studies, suggesting various classroom approaches to purpose-driven typeface selection. {{User alternative account|VeronicaJeanAnderson}} {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( A B E ) } ] </span> | style="background:black;" | <span style="color:white"> [ { ( [https://www.twitch.tv/archie97305 👀] ) } ]</span> |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> index.html</span> || notepad/atom (atom is deprecated) |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> vue </span> || [https://www.vim.org/ vim] [https://github.com/vim/vim-win32-installer/releases installer] |- || Tertiary | style="background:white;" | <span style="color:black"> css </span> || global css @ || gg css @ || NPC css @ |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> pug </span> |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> |- || Senary || b |} https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/sandbox trying to create a 1 -> 2 -> 3 -> 4 -> 5 -> 6 system in the apartment here that can be copied from site to site using artistic threads to help a Nice And Proper NAP-er navigate between properties with ease while maintaining adequate supportive care that we all require to enable us to focus on whatever catches our fancy. sun; natural light; breathe; BGs carbs; hygiene; laundry away bedroom; needles; blood; garbage out kitchen/nutritional/study social/outreach/linking worlds back porch 0 -- Computer Science, information and general works {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( T O P ) } ] </span> | style="background:black;" | <span style="color:white"> [ ℳ ] </span> | style="background:white;" | <span style="color:black"> { ¢ } </span> | style="background:#F9F9F9;" | <span style="color:pink"> ( ৳ ) </span> | style="background:black;" | <span style="color:white"> [ { ( I.n C.ase of E.mergency ) } ] </span> | style="background:teal;" | <span style="color:lime"> ᐪ l i p s c h i t z </span> || [https://www.youtube.com/watch?v=qrrz54UtkCc ᐪ] |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> physical</span> | style="background:#FFE6E6;" | <span style="color:black"> emotional</span> | style="background:#E6EAFF;" | <span style="color:black"> social</span> || This reflects health enough to communicate with people intimately enough to address real immediate issues | style="background:#FFFFE6;" | <span style="color:teal"> ^ torikomu </span> ||[https://www.youtube.com/watch?v=YxvBPH4sArQ ^] |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> occupational</span> | style="background:#F2E6FF;" | <span style="color:black"> intellectual</span> | style="background:#E6FFEA;" | <span style="color:black"> environmental</span> || This reflects living somewhere promoting healthy reasoning | style="background:#FFE6E6;" | <span style="color: teal"> | kaizen | </span> || | |- || Tertiary | style="background:white;" | <span style="color:black"> spiritual</span> | style="background:#BFBFBF;" | <span style="color:white"> factual </span> | style="background:#F2F2F2;" | <span style="color:black"> nutritional</span> || This reflects healthy mindful every habits | style="background:#E6EAFF;" | <span style="color:teal"> . genkiness . .</span> || . |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> generational</span> | style="background:#E6FFFF;" | <span style="color:black"> miscellaneal</span> | style="background:#F2E0CE;" | <span style="color:black"> punctuational</span> || This reflects having it all together enough to enjoy the holidays | style="background:#FFF2E6;" | <span style="color:lime"> # goblin </span> || # |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> | style="background:white;" | <span style="color:black"> {verb} </span> | style="background:black;" | <span style="color:white"> [noun] </span> || This reflects deliberate professional progress | style="background:#F2E6FF;" | <span style="color:lime"> / tsugu /</span> || / |- || Senary || b || 〇 || x || This reflects influencing others | style="background:#E6FFEA;" | <span style="color:lime"> @ g @ g @ </span> || [https://www.youtube.com/watch?v=SYnVYJDxu2Q @] |} == 100 -- Philosophy and psychology == How can I use color to manipulate behavior and improve communication? ===named=== ==== Re⋮Beccaδ#639 ==== ===== rebeccapurple :: #663399 ===== https://meyerweb.com/eric/thoughts/2014/06/19/rebeccapurple/ ====black==== ====white==== ====græy==== ====pink==== ====indigo==== ====midnightblue==== ===hex=== ====#fff==== ====#fff==== {| class="wikitable" style="text-align: center;" |+ |- | style="background:black;" | <span style="color:white"> [ white { on black ⚞🧿⚟ #fff on #000 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:pink"> [ pink { on 50% grey ⚞🧿⚟ #ffc0cb on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#191970"> [ midnightblue { on 50% grey ⚞🧿⚟ #191970 on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#4b0082"> [ indigo { on 50% grey ⚞🧿⚟ #4b0082 on #808080 ⚞🧿⚟ } ] </span> |} ===cmyk=== https://colordesigner.io/convert/cmyktohex ====gg on cmyk(0,0,0,33.3) w|materializecss.com==== https://materializecss.com/color.html {| class="wikitable" style="text-align: center;" |+ |- | style="background:#ababab" | <span style="color:#fff9c4"> [ gg_yellow { on cmyk(0,0,0,33.3) ⚞🧿⚟ #fff9c4 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffe0b2"> [ gg_orange { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffe0b2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffcdd2"> [ gg_red { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffcdd2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#e1bee7"> [ gg_purple { on cmyk(0,0,0,33.3) ⚞🧿⚟ #e1bee7 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#bbdefb"> [ gg_blue { on cmyk(0,0,0,33.3) ⚞🧿⚟ #bbdefb on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#c8e6c9"> [ gg_green { on cmyk(0,0,0,33.3) ⚞🧿⚟ #c8e6c9 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#efefef"> [ gg_white { on cmyk(0,0,0,33.3) ⚞🧿⚟ #efefef on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#111"> [ gg_black { on cmyk(0,0,0,33.3) ⚞🧿⚟ #111 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#808080"> [ gg_grey { on cmyk(0,0,0,33.3) ⚞🧿⚟ #808080 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#f8bbd0"> [ gg_pink { on cmyk(0,0,0,33.3) ⚞🧿⚟ #f8bbd0 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#b2ebf2"> [ gg_cyan { on cmyk(0,0,0,33.3) ⚞🧿⚟ #b2ebf2 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#d7ccc8"> [ gg_brown { on cmyk(0,0,0,33.3) ⚞🧿⚟ #d7ccc8 on #ababab )⚞🧿⚟} ] </span> |} ===rgba=== === TrumPutin-ism === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. == 200 -- Religion == Royal We "spiritual warfare" 1000 things I did 1992-2022 other than lie my way onto the supreme court to overturn Roe v Wade {| |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |} == 300 -- Social sciences == https://wattention.com/traditional-rice-harvesting-in-japan/ https://www.wwoofjapan.com/home/index.php?lang=en == 400 -- Language == === Programming === ==== .png ==== ==== Esperanto ==== ==== HTML ==== ==== PUG ==== == 500 -- Pure Science == == 600 -- Technology == === local hosts=== [http://localhost:8080/ 8080] file:///D:/index.html === Roland SP 404MKII === https://www.roland.com/global/products/sp-404mk2/ https://www.roland.com/global/support/by_product/sp-404mk2/owners_manuals/ @https://static.roland.com/manuals/sp-404mk2_app/eng/19610757.html === VIM === https://vim-adventures.com/ == 700 -- Arts and recreation == == 800 -- Literature == == 900 -- History and geography == https://geology.com/stories/13/bear-areas/ === pre-2030 === 2022 "booked" by Hillsboro Police for sending email addressing "Christian Hate" and "Spiritual War" along with "exorcisms" and "Halloween Hysteria" in Marion County, OR where Salem Police Department abdicated from protecting some children in Salem from 2016-2021. 2021 Kaiser Permanente promised cash settlement to mitigate their abdication in Marion County. KP lawyer with intimate details about my vagina: terrence .j . loeber@kp.org 2012 "unliked" on FB by some Nazarene peers after openly questioning Alex Jones' allegation that Sandy Hook didn't happen and asking for compassion for parents who were called actors while they grieved publicly through no choice of their own. 2011 Lupron given by KP for menorrhagia as alternative to b/c pills first rxd in 86. How many women who have "mostly" been on b/c pills from 87-11 are obese? Why no menorrhagia while immersed in Japan? How close to a traditional Japanese diet can I get in the Willamette Valley and how close to no meat will my body allow? === TrumPutish War Against Humanity === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. === Ring of Fire === === Post "Roe v Wade" === Who did Roe v Wade protect? Why would a Nazarene raised pro-life support an "underground" network post Roe v Wade? === Our Contemporary "Underground Railroad" needs a submarine? === Why did Portland, OR close the Shanghai Tunnels recently? Human Trafficking through Astoria, OR has been going on "forever". How do we align an "underground railroad" with contemporary supports? == 10 -- A & + == == 11 -- B * x == == 12 -- C f(◯) == == 13 -- D Δ δ ƍ ≜ 𝜟 𝝳 == == 14 -- E 🐘 𓃰 == == 15 -- F == == 16 -- G == == 17 -- H == == 18 -- I == == 19 -- J == == 20 -- K == == 21 -- L == == 22 -- M == == 23 -- N == == 24 -- O == == 25 -- P == == 26 -- Q == == 27 -- R == == 28 -- S == == 29 -- T == == 30 -- U == == 31 -- V == == 32 -- W == == 33 -- X == == 34 -- Y == == 35 -- Z == ab2jfe4tmmzgwg0e3t72olxkb75ye94 2416394 2416393 2022-08-19T01:30:37Z 73.25.23.186 /* 000000 */ wikitext text/x-wiki =0= == 00 == === 000 === ==== 0000 ==== ===== 00000 ===== ====== 000000 hex # ====== =🎱= [https://www.overleaf.com/learn/latex/TikZ_package TikZ¿¡⚽] [https://tikz.dev/ ฿ ₪ ₰] {| +BBwLTX Beaver Bank w LaTeX |- || 🎱 || main.pdf || destination || best distribution of purposeful print worthy content |- || ⛐ || readme.txt || 1st place to look for exceptions and exceptionaltions |- || 🈁 || main.aux || unknown |- || ɸ || Campus.jpeg || dot by dot picture w/o transparency |- || d͡ʒ || panther.jpeg || ligature |- || θ || mascot.jpg || the t a |- || ¶ || main.nav || unknown, but exciting place for an anchor icon |- || ‡.out || main.out || unknown |- ||『』.png || photo.png || dot by dot picture w alpha/transparency |- ||「」.snm || main.snm || wft? |- ||《》.tex || main.tex || this is the mistress J wants me to love |- || ⟨⟩ .log || main.log || sounds like somewhere stuff gets auto-logged |- || «» .toc || main.toc || table of contents: creative content starts here |} [https://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes ⛐] | git bash 🈁 | | pdflatex <your d🎱cument> | | [https://en.wikibooks.org/wiki/LaTeX/Basics#Compilation (ℝ\ℚ)] | | 1 = R\Q / (ℝ\ℚ) = ⬤ [https://www.latex-project.org/ 🎱 git a ⚽ clu & <3 🎮 LaTeX 🈁 👣 🏊 ] > https://ggf333ttu.blogspot.com/ https://www.youtube.com/watch?v=H3QmruFdfgo ==new habits== https://architecturenotes.co/redis/ 𝑓(◯) allostatis https://vim.rtorr.com/ https://github.com/rtorr/vim-cheat-sheet {| + 𝑓(◯) |- || ◯ | style="background:#000 ;" | <span style="color:#fff; font-size: 11px; "> <center>⟁ ◁ #fff ○ #000 △ ⟁ </span> || 人 || 人 | style="background:#fff ;" | <span style="color:#000; font-size: 11px; "> <center>⟁ ◁ #000 ○ #fff △ ⟁</span> || 人 || 人 | style="background:gray ;" | <span style="color:pink; font-size: 11px; "> <center>⟁ ◁ pink ○ gray △ ⟁</span> |- || 人 | style="background:#000;" | <span style="color:#fff; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#fff ;" | <span style="color:#000; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:gray;" | <span style="color:pink; font-size: 11px;"><center> ⟁ </center></span> || [http://www.amp-what.com/unicode/search/triangle |- || 1 | style="background:#111;" | <span style="color:#ff0; font-size: 11px; "> <center>◁ #ff0 ○ #111△</center></span> || 人 || 人 | style="background:#111;" | <span style="color:#F0f; font-size: 11px; "> <center>◁ #f0f ○ #111△</center></span> || 人 || 人 | style="background:#111;" | <span style="color:#0ff; font-size: 11px; "> <center>◁ #0ff ○ #111 △</center></span> |- || 人 | style="background:#111;" | <span style="color:#ff0; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#111;" | <span style="color:#f0f; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#111;" | <span style="color:#0ff; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#333;" | <span style="color:#f00; font-size: 11px; "> <center>◁ #ff0 ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#0f0; font-size: 11px; "> <center>◁ #f0f ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#00f; font-size: 11px; "> <center>◁ #0ff ○ #333 △</center></span> |- || 人 | style="background:#333;" | <span style="color:#f00; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#333;" | <span style="color:#0f0; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#333;" | <span style="color:#00f; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#555;" | <span style="color:black; font-size: 11px; "> ◁ black ≡ #000 △ </span> || 人 || 人 | style="background:#555;" | <span style="color:indigo; font-size: 9px; "> ◁ indigo ≡ #4B0082△</span> || 人 || 人 | style="background:#555;" | <span style="color:midnightblue; font-size: 11px; "> ◁ midnightblue ≡ #191970△</span> |- || 人 | style="background:#555;" | <span style="color:black; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#555;" | <span style="color:indigo; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#555;" | <span style="color:midnightblue; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#777;" | <span style="color:#fff9c4; font-size: 11px; "> ◁ #fff9c4 ○ #777 △ </span> || 人 || 人 | style="background:#777;" | <span style="color:#FFCDD2; font-size: 11px; "> ◁ #FFCDD2 ○ #777 △ </span> || 人 || 人 | style="background:#777;" | <span style="color:#BBDEFB; font-size: 11px; "> ◁ #BBDEFB ○ #777 △ </span> |- || 人 | style="background:#777;" | <span style="color:#fff9c4; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#777;" | <span style="color:#FFCDD2; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#777;" | <span style="color:#BBDEFB; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#999;" | <span style="color:#ffe0b2; font-size: 11px; "> ◁ #ffe0b2 ○ #999 △ </span> || 人 || 人 | style="background:#999;" | <span style="color:#e1bee7; font-size: 11px; "> ◁ #e1bee7 ○ #999 △ </span> || 人 || 人 | style="background:#999;" | <span style="color:#C8E6C9; font-size: 11px; "> ◁ #C8E6C9 ○ #999 △ </span> |- || 人 | style="background:#999;" | <span style="color:#ffe0b2; font-size: 11px;"><center> ⌂ </center></span> || || | style="background:#999;" | <span style="color:#e1bee7; font-size: 11px;"><center> ⌂ </center></span> || || | style="background:#999;" | <span style="color:#C8E6C9; font-size: 11px;"><center> ⌂ </center></span> || |- || ⟁ | style="background:#aaa;" | <span style="color:#efefef; font-size: 11px; "> ◁ #efefef ○ #aaa △ </span> || 人 || 人 | style="background:#aaa;" | <span style="color:#222; font-size: 11px; "> ◁ #e1bee7 ○ #aaa △ </span> || 人 || 人 | style="background:#aaa;" | <span style="color:#777; font-size: 11px; "> ◁ #ababab ○ #aaa △ </span> |- || 人 | style="background:#aaa;" | <span style="color:#efefef ; font-size: 33px;"><center> ⚞人⚟ </center></span> || || | style="background:#aaa;" | <span style="color:#222; font-size: 33px;"><center> ⚞人⚟ </center></span> || || | style="background:#aaa;" | <span style="color:#777; font-size: 33px;"><center> ⚞人⚟ </center></span> || |} {| + font-size: 25px; |- || 3 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 25px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 4 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} ==WashingtonCounty.OR.US.case.22CN03334== [http://themetawiki.clu/w/index.php/WashingtonCounty.OR.US.case.22CN03334 http://themetawiki.clu/w/index.php/WashingtonCounty.OR.US.case.22CN03334] This is the case holding up plans for a physical wikiversity site in 97305-2644 Next court date is 9/8/2022 == Salem Accord == ===/əˈkôrd/ {phoenetic w _ } === ===アコード {phoenetic w kana} === ===符合 {kanji} === ===与える { hiragana | "one's own"? "tribal"? } === ==◯== <center style="background-color: indigo; color: yellow">⬛◯⬤⚪ ◁ yellow ○ indigo △ ⟁</center> <center style="background-color: indigo; color: magenta">⬛◯⬤⚪ ◁ magenta ○ indigo △ ⟁</center> <center style="background-color: indigo; color: cyan">⬛◯⬤⚪ ◁ cyan ○ indigo △ ⟁</center> <center style="background-color: indigo; color: lime">⬛◯⬤⚪ ◁ lime ○ indigo △ ⟁</center> <center style="background-color: midnightblue; color: indigo">⬛◯⬤⚪ ◁ indigo ○ midnightblue △ ⟁</center> <center style="background-color: midnightblue; color: black">⬛◯⬤⚪ ◁ black ○ midnightblue△ ⟁</center> <center style="background-color: indigo; color: midnightblue">⬛◯⬤⚪ ◁ midnightblue ○ indigo △ ⟁</center> <center style="background-color: indigo; color: black">⬛◯⬤⚪ ◁ black ○ indigo △ ⟁</center> <center style="background-color: black; color: indigo">⬛◯⬤⚪ ◁ indigo○ black △ ⟁</center> <center style="background-color: black; color: midnightblue">⬛◯⬤⚪ ◁ midnightblue ○ black△ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> ===人=== {| + font-size: 222px; primary |- || ◯ | style="background:#000 ;" | <span style="color:#fff "> <center>◁ #fff ○ #000 △ ⟁ </span> || 人 || 人 | style="background:#fff ;" | <span style="color:#000 "> <center>◁ #000 ○ #fff △ ⧋</span> || 人 || 人 | style="background:gray ;" | <span style="color:pink "> <center>◁ pink ○ gray △ 📐</span> |- || 人 | style="background:#000;" | <span style="color:#fff; font-size: 77px;"><center> ⟁ </center></span> || || | style="background:#fff ;" | <span style="color:#000; font-size: 77px;"><center> ⧋ </center></span> || || | style="background:gray;" | <span style="color:pink; font-size: 77px;"><center> 📐 </center></span> || [http://www.amp-what.com/unicode/search/triangle |- || 1 | style="background:#333;" | <span style="color:#ff0"> <center>◁ #ff0 ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#F0f"> <center>◁ #f0f ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#0ff"> <center>◁ #0ff ○ #333 △</center></span> |- || 人 | style="background:#333;" | <span style="color:#ff0; font-size: 277px;"><center> ⌂ </center></span> || || | style="background:#333;" | <span style="color:#f0f; font-size: 277px;"><center> ⌂ </center></span> || || | style="background:#333;" | <span style="color:#0ff; font-size: 277px;"><center> ⌂ </center></span> || |- || 2 | style="background:#808080;" | <span style="color:pink"> ◁ pink ○ #808080 △</span> || 人 || 人 | style="background:#808080;" | <span style="color:midnightblue"> ◁ midnightblue ○ #808080 △</span> || 人 || 人 | style="background:#808080;" | <span style="color:indigo"> ◁ indigo ○ #808080 △</span> |- || 人 | style="background:#808080;" | <span style="color:pink; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || || | style="background:#808080;" | <span style="color:midnightblue; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || || | style="background:#808080;" | <span style="color:indigo; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || |} or ∨↯∧ & + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + {| + font-size: 100px; |- || ◯ | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 46px;"><center> ∨↯∧ </center></span> || |} {| + font-size: 222px; primary |- || 1 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ #fff9c4 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ #FFCDD2 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ #BBDEFB ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 222px;"><center> ⌂ </center></span> || |} {| + font-size: 111px; secondary |- || 2 | style="background:#808080;" | <span style="color:#ffe0b2"> ◁ #ffe0b2 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#e1bee7"> ◁ #e1bee7 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#C8E6C9"> ◁ #C8E6C9 ○</span> |- || 人 | style="background:#808080;" | <span style="color:#ffe0b2; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#e1bee7; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#C8E6C9; font-size: 222px;"><center> ⌂ </center></span> || |} {| + font-size: 25px; |- || 3 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 25px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 4 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} {| + font-size: 100px; |- || 5 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 6 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} == 0 == {| |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家] || 小 |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] |} meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} {| + font-size: 100px; |- || 人 | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 100px;"><center> ∨↯∧ </center></span> || || | style="background:#808080;" | <span style="color:#fff9c4; font-size: 100px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 100px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 100px;"><center> ⌂ </center></span> || |- || |- || |} == or ∨↯∧ & + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + .us == ==⌂== {| + font-size: 100px; |- || 人 | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 100px;"><center> ⌂ </center></span> || |- || |- || |} [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ==✪⚪⬤🔥◍⚫⨁ 👀 ≡ odd → +1 [ { ( East ⚫🔴⚪○💮⭕ West ) } ] iff even ⇒ ÷2== ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] || ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪ || || ○人∘🧿⚪⬤◍⚫ || || ⨁⚉⨂❂✧✷✸✡✵ |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |} == ya == ◯ ○ ∘ ⬤ 大 【 ヤ / や 】 (ya) 人 【 ジン、 ニン / ひと 】(jean、he toe) 大家 (ya ya) Ya (hiragana: や, katakana: ヤ) ヤフー 屋 - Wiktionaryhttps://en.wiktionary.org › wiki › 屋 Semantic compound of 尸+至. 尸 does not represent the radical for death, but is a pictogram depicting a cloth draped. 至 means "dead end". Home (家) A home, or domicile, is a space used as a permanent or semi-permanent residence for one or many humans. It is a fully or semi sheltered space and can have both interior and exterior aspects to it. Ya (や) Kana Ya is one of the Japanese kana, each of which represents one mora. The hiragana is written in three strokes, while the katakana is written in two. Both represent. Their shapes have origins in the character 也. Wikipedia hiragana origin: 也 spelling kana: 大和のヤ Yamato no "ya" transliteration: ya unicode: U+3084, U+30E4 What is the pronunciation of Ya line? In historical kana orthography, it is written as "yau", "say", and "eu", and read as "yo", "you", and "yo", respectively. Even in modern times, "saying" and "going" are sometimes pronounced as "yu" and "yuku . " From the above, it can be said that Ya line is the yoon of that line . [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] == g == {| + |- || g || ⌂ || 時 |- || nono || ⌂ || ◁ || 前 || の || ノ || 名 || "Salmon" || #fa8072 || || || || || || rgb(250, 128, 114) || :: || hsl(6, 93%, 71%) |- || "のノ" || ⌂ || <span style="color:salmon;"><center> ∅ </center></span> | style="background:#000;"|<span style="color:salmon;"><center> #000 </center></span> | style="background:#808080;" | <span style="color:salmon;"><center> #808080 </center></span> | style="background:#fff;" | <span style="color:salmon;"><center> #fff </center></span> | style="background:#333;" | <span style="color:salmon;"><center> #333 </center></span> | style="background:#333;" | <span style="color:#fa8072;"><center> #fa8072 </center></span> | style="background:#333;" | <span style="color:#fa8072ff;"><center> #fa8072ff </center></span> | style="background:#333;" | <span style="color:#fa8072aa;"><center> #fa8072aa </center></span> | style="background:#333;" | <span style="color:#fa807280;"><center> #fa807280 </center></span> | style="background:#333;" | <span style="color:#fa807233;"><center> #fa807233 </center></span> | style="background:#333;" | <span style="color:#fa807200;"><center> #fa807200 </center></span> |- || know no ノノ || ⌂ || <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#000;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#808080;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#fff;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072ff; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072aa; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807280; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807233; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807200; font-size: 100px;"> ☂ </span> |} {| + |- || hsl(0-33, 93%, 71%) || <span style="color:hsl(0, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(1, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(2, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(3, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(4, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(5, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(7, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(8, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(9, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(10, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(11, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(12, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(13, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(14, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(15, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(16, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(17, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(18, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(19, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(20, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(21, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(22, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(23, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(24, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(25, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(26, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(27, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(28, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(29, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(30, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(31, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(32, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(33, 93%, 71%);"><center>⌂ </center></span> |} {| + |- || hsl(6-606, 93%, 71%) || <span style="color:hsl(6, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(36, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(66, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(96, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(126, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(156, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(186, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(216, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(246, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(276, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(306, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(336, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(366, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(396, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(426, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(456, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(486, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(516, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(546, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(576, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(606, 93%, 71%);"><center>⌂ </center></span> |} {| + |- || hsl(6, 93%, 0-100%) || <span style="color:hsl(6, 93%, 0%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 10%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 20%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 30%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 40%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 50%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 60%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 70%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 80%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 90%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 100%);"><center>⌂ </center></span> |} {| + |- || hsl(6, 0-100%, 71%) || <span style="color:hsl(6, 0%, 71%);"><center> ⌂ </center></span> || <span style="color:hsl(6, 10%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 20%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 30%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 40%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 50%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 60%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 70%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 80%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 90%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 100%, 71%);"><center>⌂ </center></span> |} meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |- ||ternary ||3 |- ||quaternary ||4 |- ||quinary ||5 |- ||senary ||6 |- ||septenary ||7 |- ||octal ||8 |- ||nonary |- ||decimal |- ||undenary |- ||duodecimal |- ||hexadecimal ||16 |- ||vigesimal ||20 |- ||sexagesimal ||60 |} How do you want your water served when you get here? https://pubs.usgs.gov/sir/2005/5168/pdf/sir2005-5168.pdf Robert Lee Stinson %VOX "tautology club says hi" w 11am "Naturalist Society for the Humane Treatment of Monsters" from dnd game on twitter [https://www.youtube.com/watch?v=n7uNA5fO1iI rice ex in CA] https://www.oregonwild.org/about/blog/oregon-grizzly-country https://therevelator.org/yellowstone-grizzlies-unbearable-divides/ https://developer.mozilla.org/en-US/docs/Web/CSS/color_value/hsl https://www.researchgate.net/about Amare, Nicole & Manning, A.. (2012). Seeing typeface personality: Emotional responses to form as tone. IEEE International Professional Communication Conference. 1-9. 10.1109/IPCC.2012.6408605. Various studies have correlated specific visual characteristics of typefaces with specific overall emotional effects: curvilinear forms and open letter shapes generally feel “friendly” but also “formal” or “informal,” depending on other factors; large contrasts in stroke widths, cap height, and aspect ratio generally feel “interesting,” but also “attractive” or “aggressive,” depending on other factors; low-variety and low-contrast forms generally feel “professional” but also “reliable” or “boring.” Although the current findings on typeface personality are useful, they have not indicated a systematic explanation for why specific physical typeface forms have the specific emotion effects that they do. This paper will report results of an empirical study in which 102 participants indicated their immediate emotional responses to each of 36 distinct typeface designs. Results support correlation between specific typeface features (variety vs. contrast vs. pattern) and specific emotional parameters (amusement vs. agitation vs. focus), explaining findings of previous studies, suggesting various classroom approaches to purpose-driven typeface selection. {{User alternative account|VeronicaJeanAnderson}} {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( A B E ) } ] </span> | style="background:black;" | <span style="color:white"> [ { ( [https://www.twitch.tv/archie97305 👀] ) } ]</span> |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> index.html</span> || notepad/atom (atom is deprecated) |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> vue </span> || [https://www.vim.org/ vim] [https://github.com/vim/vim-win32-installer/releases installer] |- || Tertiary | style="background:white;" | <span style="color:black"> css </span> || global css @ || gg css @ || NPC css @ |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> pug </span> |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> |- || Senary || b |} https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/sandbox trying to create a 1 -> 2 -> 3 -> 4 -> 5 -> 6 system in the apartment here that can be copied from site to site using artistic threads to help a Nice And Proper NAP-er navigate between properties with ease while maintaining adequate supportive care that we all require to enable us to focus on whatever catches our fancy. sun; natural light; breathe; BGs carbs; hygiene; laundry away bedroom; needles; blood; garbage out kitchen/nutritional/study social/outreach/linking worlds back porch 0 -- Computer Science, information and general works {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( T O P ) } ] </span> | style="background:black;" | <span style="color:white"> [ ℳ ] </span> | style="background:white;" | <span style="color:black"> { ¢ } </span> | style="background:#F9F9F9;" | <span style="color:pink"> ( ৳ ) </span> | style="background:black;" | <span style="color:white"> [ { ( I.n C.ase of E.mergency ) } ] </span> | style="background:teal;" | <span style="color:lime"> ᐪ l i p s c h i t z </span> || [https://www.youtube.com/watch?v=qrrz54UtkCc ᐪ] |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> physical</span> | style="background:#FFE6E6;" | <span style="color:black"> emotional</span> | style="background:#E6EAFF;" | <span style="color:black"> social</span> || This reflects health enough to communicate with people intimately enough to address real immediate issues | style="background:#FFFFE6;" | <span style="color:teal"> ^ torikomu </span> ||[https://www.youtube.com/watch?v=YxvBPH4sArQ ^] |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> occupational</span> | style="background:#F2E6FF;" | <span style="color:black"> intellectual</span> | style="background:#E6FFEA;" | <span style="color:black"> environmental</span> || This reflects living somewhere promoting healthy reasoning | style="background:#FFE6E6;" | <span style="color: teal"> | kaizen | </span> || | |- || Tertiary | style="background:white;" | <span style="color:black"> spiritual</span> | style="background:#BFBFBF;" | <span style="color:white"> factual </span> | style="background:#F2F2F2;" | <span style="color:black"> nutritional</span> || This reflects healthy mindful every habits | style="background:#E6EAFF;" | <span style="color:teal"> . genkiness . .</span> || . |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> generational</span> | style="background:#E6FFFF;" | <span style="color:black"> miscellaneal</span> | style="background:#F2E0CE;" | <span style="color:black"> punctuational</span> || This reflects having it all together enough to enjoy the holidays | style="background:#FFF2E6;" | <span style="color:lime"> # goblin </span> || # |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> | style="background:white;" | <span style="color:black"> {verb} </span> | style="background:black;" | <span style="color:white"> [noun] </span> || This reflects deliberate professional progress | style="background:#F2E6FF;" | <span style="color:lime"> / tsugu /</span> || / |- || Senary || b || 〇 || x || This reflects influencing others | style="background:#E6FFEA;" | <span style="color:lime"> @ g @ g @ </span> || [https://www.youtube.com/watch?v=SYnVYJDxu2Q @] |} == 100 -- Philosophy and psychology == How can I use color to manipulate behavior and improve communication? ===named=== ==== Re⋮Beccaδ#639 ==== ===== rebeccapurple :: #663399 ===== https://meyerweb.com/eric/thoughts/2014/06/19/rebeccapurple/ ====black==== ====white==== ====græy==== ====pink==== ====indigo==== ====midnightblue==== ===hex=== ====#fff==== ====#fff==== {| class="wikitable" style="text-align: center;" |+ |- | style="background:black;" | <span style="color:white"> [ white { on black ⚞🧿⚟ #fff on #000 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:pink"> [ pink { on 50% grey ⚞🧿⚟ #ffc0cb on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#191970"> [ midnightblue { on 50% grey ⚞🧿⚟ #191970 on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#4b0082"> [ indigo { on 50% grey ⚞🧿⚟ #4b0082 on #808080 ⚞🧿⚟ } ] </span> |} ===cmyk=== https://colordesigner.io/convert/cmyktohex ====gg on cmyk(0,0,0,33.3) w|materializecss.com==== https://materializecss.com/color.html {| class="wikitable" style="text-align: center;" |+ |- | style="background:#ababab" | <span style="color:#fff9c4"> [ gg_yellow { on cmyk(0,0,0,33.3) ⚞🧿⚟ #fff9c4 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffe0b2"> [ gg_orange { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffe0b2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffcdd2"> [ gg_red { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffcdd2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#e1bee7"> [ gg_purple { on cmyk(0,0,0,33.3) ⚞🧿⚟ #e1bee7 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#bbdefb"> [ gg_blue { on cmyk(0,0,0,33.3) ⚞🧿⚟ #bbdefb on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#c8e6c9"> [ gg_green { on cmyk(0,0,0,33.3) ⚞🧿⚟ #c8e6c9 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#efefef"> [ gg_white { on cmyk(0,0,0,33.3) ⚞🧿⚟ #efefef on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#111"> [ gg_black { on cmyk(0,0,0,33.3) ⚞🧿⚟ #111 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#808080"> [ gg_grey { on cmyk(0,0,0,33.3) ⚞🧿⚟ #808080 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#f8bbd0"> [ gg_pink { on cmyk(0,0,0,33.3) ⚞🧿⚟ #f8bbd0 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#b2ebf2"> [ gg_cyan { on cmyk(0,0,0,33.3) ⚞🧿⚟ #b2ebf2 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#d7ccc8"> [ gg_brown { on cmyk(0,0,0,33.3) ⚞🧿⚟ #d7ccc8 on #ababab )⚞🧿⚟} ] </span> |} ===rgba=== === TrumPutin-ism === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. == 200 -- Religion == Royal We "spiritual warfare" 1000 things I did 1992-2022 other than lie my way onto the supreme court to overturn Roe v Wade {| |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |} == 300 -- Social sciences == https://wattention.com/traditional-rice-harvesting-in-japan/ https://www.wwoofjapan.com/home/index.php?lang=en == 400 -- Language == === Programming === ==== .png ==== ==== Esperanto ==== ==== HTML ==== ==== PUG ==== == 500 -- Pure Science == == 600 -- Technology == === local hosts=== [http://localhost:8080/ 8080] file:///D:/index.html === Roland SP 404MKII === https://www.roland.com/global/products/sp-404mk2/ https://www.roland.com/global/support/by_product/sp-404mk2/owners_manuals/ @https://static.roland.com/manuals/sp-404mk2_app/eng/19610757.html === VIM === https://vim-adventures.com/ == 700 -- Arts and recreation == == 800 -- Literature == == 900 -- History and geography == https://geology.com/stories/13/bear-areas/ === pre-2030 === 2022 "booked" by Hillsboro Police for sending email addressing "Christian Hate" and "Spiritual War" along with "exorcisms" and "Halloween Hysteria" in Marion County, OR where Salem Police Department abdicated from protecting some children in Salem from 2016-2021. 2021 Kaiser Permanente promised cash settlement to mitigate their abdication in Marion County. KP lawyer with intimate details about my vagina: terrence .j . loeber@kp.org 2012 "unliked" on FB by some Nazarene peers after openly questioning Alex Jones' allegation that Sandy Hook didn't happen and asking for compassion for parents who were called actors while they grieved publicly through no choice of their own. 2011 Lupron given by KP for menorrhagia as alternative to b/c pills first rxd in 86. How many women who have "mostly" been on b/c pills from 87-11 are obese? Why no menorrhagia while immersed in Japan? How close to a traditional Japanese diet can I get in the Willamette Valley and how close to no meat will my body allow? === TrumPutish War Against Humanity === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. === Ring of Fire === === Post "Roe v Wade" === Who did Roe v Wade protect? Why would a Nazarene raised pro-life support an "underground" network post Roe v Wade? === Our Contemporary "Underground Railroad" needs a submarine? === Why did Portland, OR close the Shanghai Tunnels recently? Human Trafficking through Astoria, OR has been going on "forever". How do we align an "underground railroad" with contemporary supports? == 10 -- A & + == == 11 -- B * x == == 12 -- C f(◯) == == 13 -- D Δ δ ƍ ≜ 𝜟 𝝳 == == 14 -- E 🐘 𓃰 == == 15 -- F == == 16 -- G == == 17 -- H == == 18 -- I == == 19 -- J == == 20 -- K == == 21 -- L == == 22 -- M == == 23 -- N == == 24 -- O == == 25 -- P == == 26 -- Q == == 27 -- R == == 28 -- S == == 29 -- T == == 30 -- U == == 31 -- V == == 32 -- W == == 33 -- X == == 34 -- Y == == 35 -- Z == 9sroicrj4i1z6u4zsz7g7i932k28obc 2416395 2416394 2022-08-19T01:32:02Z 73.25.23.186 /* 100 -- Philosophy and psychology */ wikitext text/x-wiki =0= == 00 == === 000 === ==== 0000 ==== ===== 00000 ===== ====== 000000 hex # ====== =🎱= [https://www.overleaf.com/learn/latex/TikZ_package TikZ¿¡⚽] [https://tikz.dev/ ฿ ₪ ₰] {| +BBwLTX Beaver Bank w LaTeX |- || 🎱 || main.pdf || destination || best distribution of purposeful print worthy content |- || ⛐ || readme.txt || 1st place to look for exceptions and exceptionaltions |- || 🈁 || main.aux || unknown |- || ɸ || Campus.jpeg || dot by dot picture w/o transparency |- || d͡ʒ || panther.jpeg || ligature |- || θ || mascot.jpg || the t a |- || ¶ || main.nav || unknown, but exciting place for an anchor icon |- || ‡.out || main.out || unknown |- ||『』.png || photo.png || dot by dot picture w alpha/transparency |- ||「」.snm || main.snm || wft? |- ||《》.tex || main.tex || this is the mistress J wants me to love |- || ⟨⟩ .log || main.log || sounds like somewhere stuff gets auto-logged |- || «» .toc || main.toc || table of contents: creative content starts here |} [https://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes ⛐] | git bash 🈁 | | pdflatex <your d🎱cument> | | [https://en.wikibooks.org/wiki/LaTeX/Basics#Compilation (ℝ\ℚ)] | | 1 = R\Q / (ℝ\ℚ) = ⬤ [https://www.latex-project.org/ 🎱 git a ⚽ clu & <3 🎮 LaTeX 🈁 👣 🏊 ] > https://ggf333ttu.blogspot.com/ https://www.youtube.com/watch?v=H3QmruFdfgo ==new habits== https://architecturenotes.co/redis/ 𝑓(◯) allostatis https://vim.rtorr.com/ https://github.com/rtorr/vim-cheat-sheet {| + 𝑓(◯) |- || ◯ | style="background:#000 ;" | <span style="color:#fff; font-size: 11px; "> <center>⟁ ◁ #fff ○ #000 △ ⟁ </span> || 人 || 人 | style="background:#fff ;" | <span style="color:#000; font-size: 11px; "> <center>⟁ ◁ #000 ○ #fff △ ⟁</span> || 人 || 人 | style="background:gray ;" | <span style="color:pink; font-size: 11px; "> <center>⟁ ◁ pink ○ gray △ ⟁</span> |- || 人 | style="background:#000;" | <span style="color:#fff; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#fff ;" | <span style="color:#000; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:gray;" | <span style="color:pink; font-size: 11px;"><center> ⟁ </center></span> || [http://www.amp-what.com/unicode/search/triangle |- || 1 | style="background:#111;" | <span style="color:#ff0; font-size: 11px; "> <center>◁ #ff0 ○ #111△</center></span> || 人 || 人 | style="background:#111;" | <span style="color:#F0f; font-size: 11px; "> <center>◁ #f0f ○ #111△</center></span> || 人 || 人 | style="background:#111;" | <span style="color:#0ff; font-size: 11px; "> <center>◁ #0ff ○ #111 △</center></span> |- || 人 | style="background:#111;" | <span style="color:#ff0; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#111;" | <span style="color:#f0f; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#111;" | <span style="color:#0ff; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#333;" | <span style="color:#f00; font-size: 11px; "> <center>◁ #ff0 ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#0f0; font-size: 11px; "> <center>◁ #f0f ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#00f; font-size: 11px; "> <center>◁ #0ff ○ #333 △</center></span> |- || 人 | style="background:#333;" | <span style="color:#f00; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#333;" | <span style="color:#0f0; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#333;" | <span style="color:#00f; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#555;" | <span style="color:black; font-size: 11px; "> ◁ black ≡ #000 △ </span> || 人 || 人 | style="background:#555;" | <span style="color:indigo; font-size: 9px; "> ◁ indigo ≡ #4B0082△</span> || 人 || 人 | style="background:#555;" | <span style="color:midnightblue; font-size: 11px; "> ◁ midnightblue ≡ #191970△</span> |- || 人 | style="background:#555;" | <span style="color:black; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#555;" | <span style="color:indigo; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#555;" | <span style="color:midnightblue; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#777;" | <span style="color:#fff9c4; font-size: 11px; "> ◁ #fff9c4 ○ #777 △ </span> || 人 || 人 | style="background:#777;" | <span style="color:#FFCDD2; font-size: 11px; "> ◁ #FFCDD2 ○ #777 △ </span> || 人 || 人 | style="background:#777;" | <span style="color:#BBDEFB; font-size: 11px; "> ◁ #BBDEFB ○ #777 △ </span> |- || 人 | style="background:#777;" | <span style="color:#fff9c4; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#777;" | <span style="color:#FFCDD2; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#777;" | <span style="color:#BBDEFB; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#999;" | <span style="color:#ffe0b2; font-size: 11px; "> ◁ #ffe0b2 ○ #999 △ </span> || 人 || 人 | style="background:#999;" | <span style="color:#e1bee7; font-size: 11px; "> ◁ #e1bee7 ○ #999 △ </span> || 人 || 人 | style="background:#999;" | <span style="color:#C8E6C9; font-size: 11px; "> ◁ #C8E6C9 ○ #999 △ </span> |- || 人 | style="background:#999;" | <span style="color:#ffe0b2; font-size: 11px;"><center> ⌂ </center></span> || || | style="background:#999;" | <span style="color:#e1bee7; font-size: 11px;"><center> ⌂ </center></span> || || | style="background:#999;" | <span style="color:#C8E6C9; font-size: 11px;"><center> ⌂ </center></span> || |- || ⟁ | style="background:#aaa;" | <span style="color:#efefef; font-size: 11px; "> ◁ #efefef ○ #aaa △ </span> || 人 || 人 | style="background:#aaa;" | <span style="color:#222; font-size: 11px; "> ◁ #e1bee7 ○ #aaa △ </span> || 人 || 人 | style="background:#aaa;" | <span style="color:#777; font-size: 11px; "> ◁ #ababab ○ #aaa △ </span> |- || 人 | style="background:#aaa;" | <span style="color:#efefef ; font-size: 33px;"><center> ⚞人⚟ </center></span> || || | style="background:#aaa;" | <span style="color:#222; font-size: 33px;"><center> ⚞人⚟ </center></span> || || | style="background:#aaa;" | <span style="color:#777; font-size: 33px;"><center> ⚞人⚟ </center></span> || |} {| + font-size: 25px; |- || 3 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 25px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 4 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} ==WashingtonCounty.OR.US.case.22CN03334== [http://themetawiki.clu/w/index.php/WashingtonCounty.OR.US.case.22CN03334 http://themetawiki.clu/w/index.php/WashingtonCounty.OR.US.case.22CN03334] This is the case holding up plans for a physical wikiversity site in 97305-2644 Next court date is 9/8/2022 == Salem Accord == ===/əˈkôrd/ {phoenetic w _ } === ===アコード {phoenetic w kana} === ===符合 {kanji} === ===与える { hiragana | "one's own"? "tribal"? } === ==◯== <center style="background-color: indigo; color: yellow">⬛◯⬤⚪ ◁ yellow ○ indigo △ ⟁</center> <center style="background-color: indigo; color: magenta">⬛◯⬤⚪ ◁ magenta ○ indigo △ ⟁</center> <center style="background-color: indigo; color: cyan">⬛◯⬤⚪ ◁ cyan ○ indigo △ ⟁</center> <center style="background-color: indigo; color: lime">⬛◯⬤⚪ ◁ lime ○ indigo △ ⟁</center> <center style="background-color: midnightblue; color: indigo">⬛◯⬤⚪ ◁ indigo ○ midnightblue △ ⟁</center> <center style="background-color: midnightblue; color: black">⬛◯⬤⚪ ◁ black ○ midnightblue△ ⟁</center> <center style="background-color: indigo; color: midnightblue">⬛◯⬤⚪ ◁ midnightblue ○ indigo △ ⟁</center> <center style="background-color: indigo; color: black">⬛◯⬤⚪ ◁ black ○ indigo △ ⟁</center> <center style="background-color: black; color: indigo">⬛◯⬤⚪ ◁ indigo○ black △ ⟁</center> <center style="background-color: black; color: midnightblue">⬛◯⬤⚪ ◁ midnightblue ○ black△ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> ===人=== {| + font-size: 222px; primary |- || ◯ | style="background:#000 ;" | <span style="color:#fff "> <center>◁ #fff ○ #000 △ ⟁ </span> || 人 || 人 | style="background:#fff ;" | <span style="color:#000 "> <center>◁ #000 ○ #fff △ ⧋</span> || 人 || 人 | style="background:gray ;" | <span style="color:pink "> <center>◁ pink ○ gray △ 📐</span> |- || 人 | style="background:#000;" | <span style="color:#fff; font-size: 77px;"><center> ⟁ </center></span> || || | style="background:#fff ;" | <span style="color:#000; font-size: 77px;"><center> ⧋ </center></span> || || | style="background:gray;" | <span style="color:pink; font-size: 77px;"><center> 📐 </center></span> || [http://www.amp-what.com/unicode/search/triangle |- || 1 | style="background:#333;" | <span style="color:#ff0"> <center>◁ #ff0 ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#F0f"> <center>◁ #f0f ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#0ff"> <center>◁ #0ff ○ #333 △</center></span> |- || 人 | style="background:#333;" | <span style="color:#ff0; font-size: 277px;"><center> ⌂ </center></span> || || | style="background:#333;" | <span style="color:#f0f; font-size: 277px;"><center> ⌂ </center></span> || || | style="background:#333;" | <span style="color:#0ff; font-size: 277px;"><center> ⌂ </center></span> || |- || 2 | style="background:#808080;" | <span style="color:pink"> ◁ pink ○ #808080 △</span> || 人 || 人 | style="background:#808080;" | <span style="color:midnightblue"> ◁ midnightblue ○ #808080 △</span> || 人 || 人 | style="background:#808080;" | <span style="color:indigo"> ◁ indigo ○ #808080 △</span> |- || 人 | style="background:#808080;" | <span style="color:pink; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || || | style="background:#808080;" | <span style="color:midnightblue; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || || | style="background:#808080;" | <span style="color:indigo; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || |} or ∨↯∧ & + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + {| + font-size: 100px; |- || ◯ | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 46px;"><center> ∨↯∧ </center></span> || |} {| + font-size: 222px; primary |- || 1 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ #fff9c4 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ #FFCDD2 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ #BBDEFB ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 222px;"><center> ⌂ </center></span> || |} {| + font-size: 111px; secondary |- || 2 | style="background:#808080;" | <span style="color:#ffe0b2"> ◁ #ffe0b2 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#e1bee7"> ◁ #e1bee7 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#C8E6C9"> ◁ #C8E6C9 ○</span> |- || 人 | style="background:#808080;" | <span style="color:#ffe0b2; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#e1bee7; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#C8E6C9; font-size: 222px;"><center> ⌂ </center></span> || |} {| + font-size: 25px; |- || 3 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 25px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 4 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} {| + font-size: 100px; |- || 5 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 6 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} == 0 == {| |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家] || 小 |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] |} meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} {| + font-size: 100px; |- || 人 | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 100px;"><center> ∨↯∧ </center></span> || || | style="background:#808080;" | <span style="color:#fff9c4; font-size: 100px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 100px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 100px;"><center> ⌂ </center></span> || |- || |- || |} == or ∨↯∧ & + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + .us == ==⌂== {| + font-size: 100px; |- || 人 | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 100px;"><center> ⌂ </center></span> || |- || |- || |} [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ==✪⚪⬤🔥◍⚫⨁ 👀 ≡ odd → +1 [ { ( East ⚫🔴⚪○💮⭕ West ) } ] iff even ⇒ ÷2== ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] || ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪ || || ○人∘🧿⚪⬤◍⚫ || || ⨁⚉⨂❂✧✷✸✡✵ |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |} == ya == ◯ ○ ∘ ⬤ 大 【 ヤ / や 】 (ya) 人 【 ジン、 ニン / ひと 】(jean、he toe) 大家 (ya ya) Ya (hiragana: や, katakana: ヤ) ヤフー 屋 - Wiktionaryhttps://en.wiktionary.org › wiki › 屋 Semantic compound of 尸+至. 尸 does not represent the radical for death, but is a pictogram depicting a cloth draped. 至 means "dead end". Home (家) A home, or domicile, is a space used as a permanent or semi-permanent residence for one or many humans. It is a fully or semi sheltered space and can have both interior and exterior aspects to it. Ya (や) Kana Ya is one of the Japanese kana, each of which represents one mora. The hiragana is written in three strokes, while the katakana is written in two. Both represent. Their shapes have origins in the character 也. Wikipedia hiragana origin: 也 spelling kana: 大和のヤ Yamato no "ya" transliteration: ya unicode: U+3084, U+30E4 What is the pronunciation of Ya line? In historical kana orthography, it is written as "yau", "say", and "eu", and read as "yo", "you", and "yo", respectively. Even in modern times, "saying" and "going" are sometimes pronounced as "yu" and "yuku . " From the above, it can be said that Ya line is the yoon of that line . [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] == g == {| + |- || g || ⌂ || 時 |- || nono || ⌂ || ◁ || 前 || の || ノ || 名 || "Salmon" || #fa8072 || || || || || || rgb(250, 128, 114) || :: || hsl(6, 93%, 71%) |- || "のノ" || ⌂ || <span style="color:salmon;"><center> ∅ </center></span> | style="background:#000;"|<span style="color:salmon;"><center> #000 </center></span> | style="background:#808080;" | <span style="color:salmon;"><center> #808080 </center></span> | style="background:#fff;" | <span style="color:salmon;"><center> #fff </center></span> | style="background:#333;" | <span style="color:salmon;"><center> #333 </center></span> | style="background:#333;" | <span style="color:#fa8072;"><center> #fa8072 </center></span> | style="background:#333;" | <span style="color:#fa8072ff;"><center> #fa8072ff </center></span> | style="background:#333;" | <span style="color:#fa8072aa;"><center> #fa8072aa </center></span> | style="background:#333;" | <span style="color:#fa807280;"><center> #fa807280 </center></span> | style="background:#333;" | <span style="color:#fa807233;"><center> #fa807233 </center></span> | style="background:#333;" | <span style="color:#fa807200;"><center> #fa807200 </center></span> |- || know no ノノ || ⌂ || <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#000;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#808080;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#fff;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072ff; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072aa; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807280; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807233; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807200; font-size: 100px;"> ☂ </span> |} {| + |- || hsl(0-33, 93%, 71%) || <span style="color:hsl(0, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(1, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(2, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(3, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(4, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(5, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(7, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(8, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(9, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(10, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(11, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(12, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(13, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(14, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(15, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(16, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(17, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(18, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(19, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(20, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(21, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(22, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(23, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(24, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(25, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(26, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(27, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(28, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(29, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(30, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(31, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(32, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(33, 93%, 71%);"><center>⌂ </center></span> |} {| + |- || hsl(6-606, 93%, 71%) || <span style="color:hsl(6, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(36, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(66, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(96, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(126, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(156, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(186, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(216, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(246, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(276, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(306, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(336, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(366, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(396, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(426, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(456, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(486, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(516, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(546, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(576, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(606, 93%, 71%);"><center>⌂ </center></span> |} {| + |- || hsl(6, 93%, 0-100%) || <span style="color:hsl(6, 93%, 0%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 10%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 20%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 30%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 40%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 50%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 60%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 70%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 80%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 90%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 100%);"><center>⌂ </center></span> |} {| + |- || hsl(6, 0-100%, 71%) || <span style="color:hsl(6, 0%, 71%);"><center> ⌂ </center></span> || <span style="color:hsl(6, 10%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 20%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 30%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 40%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 50%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 60%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 70%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 80%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 90%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 100%, 71%);"><center>⌂ </center></span> |} meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |- ||ternary ||3 |- ||quaternary ||4 |- ||quinary ||5 |- ||senary ||6 |- ||septenary ||7 |- ||octal ||8 |- ||nonary |- ||decimal |- ||undenary |- ||duodecimal |- ||hexadecimal ||16 |- ||vigesimal ||20 |- ||sexagesimal ||60 |} How do you want your water served when you get here? https://pubs.usgs.gov/sir/2005/5168/pdf/sir2005-5168.pdf Robert Lee Stinson %VOX "tautology club says hi" w 11am "Naturalist Society for the Humane Treatment of Monsters" from dnd game on twitter [https://www.youtube.com/watch?v=n7uNA5fO1iI rice ex in CA] https://www.oregonwild.org/about/blog/oregon-grizzly-country https://therevelator.org/yellowstone-grizzlies-unbearable-divides/ https://developer.mozilla.org/en-US/docs/Web/CSS/color_value/hsl https://www.researchgate.net/about Amare, Nicole & Manning, A.. (2012). Seeing typeface personality: Emotional responses to form as tone. IEEE International Professional Communication Conference. 1-9. 10.1109/IPCC.2012.6408605. Various studies have correlated specific visual characteristics of typefaces with specific overall emotional effects: curvilinear forms and open letter shapes generally feel “friendly” but also “formal” or “informal,” depending on other factors; large contrasts in stroke widths, cap height, and aspect ratio generally feel “interesting,” but also “attractive” or “aggressive,” depending on other factors; low-variety and low-contrast forms generally feel “professional” but also “reliable” or “boring.” Although the current findings on typeface personality are useful, they have not indicated a systematic explanation for why specific physical typeface forms have the specific emotion effects that they do. This paper will report results of an empirical study in which 102 participants indicated their immediate emotional responses to each of 36 distinct typeface designs. Results support correlation between specific typeface features (variety vs. contrast vs. pattern) and specific emotional parameters (amusement vs. agitation vs. focus), explaining findings of previous studies, suggesting various classroom approaches to purpose-driven typeface selection. {{User alternative account|VeronicaJeanAnderson}} {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( A B E ) } ] </span> | style="background:black;" | <span style="color:white"> [ { ( [https://www.twitch.tv/archie97305 👀] ) } ]</span> |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> index.html</span> || notepad/atom (atom is deprecated) |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> vue </span> || [https://www.vim.org/ vim] [https://github.com/vim/vim-win32-installer/releases installer] |- || Tertiary | style="background:white;" | <span style="color:black"> css </span> || global css @ || gg css @ || NPC css @ |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> pug </span> |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> |- || Senary || b |} https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/sandbox trying to create a 1 -> 2 -> 3 -> 4 -> 5 -> 6 system in the apartment here that can be copied from site to site using artistic threads to help a Nice And Proper NAP-er navigate between properties with ease while maintaining adequate supportive care that we all require to enable us to focus on whatever catches our fancy. sun; natural light; breathe; BGs carbs; hygiene; laundry away bedroom; needles; blood; garbage out kitchen/nutritional/study social/outreach/linking worlds back porch 0 -- Computer Science, information and general works {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( T O P ) } ] </span> | style="background:black;" | <span style="color:white"> [ ℳ ] </span> | style="background:white;" | <span style="color:black"> { ¢ } </span> | style="background:#F9F9F9;" | <span style="color:pink"> ( ৳ ) </span> | style="background:black;" | <span style="color:white"> [ { ( I.n C.ase of E.mergency ) } ] </span> | style="background:teal;" | <span style="color:lime"> ᐪ l i p s c h i t z </span> || [https://www.youtube.com/watch?v=qrrz54UtkCc ᐪ] |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> physical</span> | style="background:#FFE6E6;" | <span style="color:black"> emotional</span> | style="background:#E6EAFF;" | <span style="color:black"> social</span> || This reflects health enough to communicate with people intimately enough to address real immediate issues | style="background:#FFFFE6;" | <span style="color:teal"> ^ torikomu </span> ||[https://www.youtube.com/watch?v=YxvBPH4sArQ ^] |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> occupational</span> | style="background:#F2E6FF;" | <span style="color:black"> intellectual</span> | style="background:#E6FFEA;" | <span style="color:black"> environmental</span> || This reflects living somewhere promoting healthy reasoning | style="background:#FFE6E6;" | <span style="color: teal"> | kaizen | </span> || | |- || Tertiary | style="background:white;" | <span style="color:black"> spiritual</span> | style="background:#BFBFBF;" | <span style="color:white"> factual </span> | style="background:#F2F2F2;" | <span style="color:black"> nutritional</span> || This reflects healthy mindful every habits | style="background:#E6EAFF;" | <span style="color:teal"> . genkiness . .</span> || . |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> generational</span> | style="background:#E6FFFF;" | <span style="color:black"> miscellaneal</span> | style="background:#F2E0CE;" | <span style="color:black"> punctuational</span> || This reflects having it all together enough to enjoy the holidays | style="background:#FFF2E6;" | <span style="color:lime"> # goblin </span> || # |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> | style="background:white;" | <span style="color:black"> {verb} </span> | style="background:black;" | <span style="color:white"> [noun] </span> || This reflects deliberate professional progress | style="background:#F2E6FF;" | <span style="color:lime"> / tsugu /</span> || / |- || Senary || b || 〇 || x || This reflects influencing others | style="background:#E6FFEA;" | <span style="color:lime"> @ g @ g @ </span> || [https://www.youtube.com/watch?v=SYnVYJDxu2Q @] |} == 000 *itches== == 100 -- Philosophy and psychology == How can I use color to manipulate behavior and improve communication? ===named=== ==== Re⋮Beccaδ#639 ==== ===== rebeccapurple :: #663399 ===== https://meyerweb.com/eric/thoughts/2014/06/19/rebeccapurple/ ====black==== ====white==== ====græy==== ====pink==== ====indigo==== ====midnightblue==== ===hex=== ====#fff==== ====#fff==== {| class="wikitable" style="text-align: center;" |+ |- | style="background:black;" | <span style="color:white"> [ white { on black ⚞🧿⚟ #fff on #000 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:pink"> [ pink { on 50% grey ⚞🧿⚟ #ffc0cb on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#191970"> [ midnightblue { on 50% grey ⚞🧿⚟ #191970 on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#4b0082"> [ indigo { on 50% grey ⚞🧿⚟ #4b0082 on #808080 ⚞🧿⚟ } ] </span> |} ===cmyk=== https://colordesigner.io/convert/cmyktohex ====gg on cmyk(0,0,0,33.3) w|materializecss.com==== https://materializecss.com/color.html {| class="wikitable" style="text-align: center;" |+ |- | style="background:#ababab" | <span style="color:#fff9c4"> [ gg_yellow { on cmyk(0,0,0,33.3) ⚞🧿⚟ #fff9c4 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffe0b2"> [ gg_orange { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffe0b2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffcdd2"> [ gg_red { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffcdd2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#e1bee7"> [ gg_purple { on cmyk(0,0,0,33.3) ⚞🧿⚟ #e1bee7 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#bbdefb"> [ gg_blue { on cmyk(0,0,0,33.3) ⚞🧿⚟ #bbdefb on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#c8e6c9"> [ gg_green { on cmyk(0,0,0,33.3) ⚞🧿⚟ #c8e6c9 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#efefef"> [ gg_white { on cmyk(0,0,0,33.3) ⚞🧿⚟ #efefef on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#111"> [ gg_black { on cmyk(0,0,0,33.3) ⚞🧿⚟ #111 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#808080"> [ gg_grey { on cmyk(0,0,0,33.3) ⚞🧿⚟ #808080 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#f8bbd0"> [ gg_pink { on cmyk(0,0,0,33.3) ⚞🧿⚟ #f8bbd0 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#b2ebf2"> [ gg_cyan { on cmyk(0,0,0,33.3) ⚞🧿⚟ #b2ebf2 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#d7ccc8"> [ gg_brown { on cmyk(0,0,0,33.3) ⚞🧿⚟ #d7ccc8 on #ababab )⚞🧿⚟} ] </span> |} ===rgba=== === TrumPutin-ism === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. == 200 -- Religion == Royal We "spiritual warfare" 1000 things I did 1992-2022 other than lie my way onto the supreme court to overturn Roe v Wade {| |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |} == 300 -- Social sciences == https://wattention.com/traditional-rice-harvesting-in-japan/ https://www.wwoofjapan.com/home/index.php?lang=en == 400 -- Language == === Programming === ==== .png ==== ==== Esperanto ==== ==== HTML ==== ==== PUG ==== == 500 -- Pure Science == == 600 -- Technology == === local hosts=== [http://localhost:8080/ 8080] file:///D:/index.html === Roland SP 404MKII === https://www.roland.com/global/products/sp-404mk2/ https://www.roland.com/global/support/by_product/sp-404mk2/owners_manuals/ @https://static.roland.com/manuals/sp-404mk2_app/eng/19610757.html === VIM === https://vim-adventures.com/ == 700 -- Arts and recreation == == 800 -- Literature == == 900 -- History and geography == https://geology.com/stories/13/bear-areas/ === pre-2030 === 2022 "booked" by Hillsboro Police for sending email addressing "Christian Hate" and "Spiritual War" along with "exorcisms" and "Halloween Hysteria" in Marion County, OR where Salem Police Department abdicated from protecting some children in Salem from 2016-2021. 2021 Kaiser Permanente promised cash settlement to mitigate their abdication in Marion County. KP lawyer with intimate details about my vagina: terrence .j . loeber@kp.org 2012 "unliked" on FB by some Nazarene peers after openly questioning Alex Jones' allegation that Sandy Hook didn't happen and asking for compassion for parents who were called actors while they grieved publicly through no choice of their own. 2011 Lupron given by KP for menorrhagia as alternative to b/c pills first rxd in 86. How many women who have "mostly" been on b/c pills from 87-11 are obese? Why no menorrhagia while immersed in Japan? How close to a traditional Japanese diet can I get in the Willamette Valley and how close to no meat will my body allow? === TrumPutish War Against Humanity === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. === Ring of Fire === === Post "Roe v Wade" === Who did Roe v Wade protect? Why would a Nazarene raised pro-life support an "underground" network post Roe v Wade? === Our Contemporary "Underground Railroad" needs a submarine? === Why did Portland, OR close the Shanghai Tunnels recently? Human Trafficking through Astoria, OR has been going on "forever". How do we align an "underground railroad" with contemporary supports? == 10 -- A & + == == 11 -- B * x == == 12 -- C f(◯) == == 13 -- D Δ δ ƍ ≜ 𝜟 𝝳 == == 14 -- E 🐘 𓃰 == == 15 -- F == == 16 -- G == == 17 -- H == == 18 -- I == == 19 -- J == == 20 -- K == == 21 -- L == == 22 -- M == == 23 -- N == == 24 -- O == == 25 -- P == == 26 -- Q == == 27 -- R == == 28 -- S == == 29 -- T == == 30 -- U == == 31 -- V == == 32 -- W == == 33 -- X == == 34 -- Y == == 35 -- Z == s0oan0npu92r1h0c48434cklbov76v9 2416396 2416395 2022-08-19T01:34:11Z 73.25.23.186 /* 000 *itches */ wikitext text/x-wiki =0= == 00 == === 000 === ==== 0000 ==== ===== 00000 ===== ====== 000000 hex # ====== =🎱= [https://www.overleaf.com/learn/latex/TikZ_package TikZ¿¡⚽] [https://tikz.dev/ ฿ ₪ ₰] {| +BBwLTX Beaver Bank w LaTeX |- || 🎱 || main.pdf || destination || best distribution of purposeful print worthy content |- || ⛐ || readme.txt || 1st place to look for exceptions and exceptionaltions |- || 🈁 || main.aux || unknown |- || ɸ || Campus.jpeg || dot by dot picture w/o transparency |- || d͡ʒ || panther.jpeg || ligature |- || θ || mascot.jpg || the t a |- || ¶ || main.nav || unknown, but exciting place for an anchor icon |- || ‡.out || main.out || unknown |- ||『』.png || photo.png || dot by dot picture w alpha/transparency |- ||「」.snm || main.snm || wft? |- ||《》.tex || main.tex || this is the mistress J wants me to love |- || ⟨⟩ .log || main.log || sounds like somewhere stuff gets auto-logged |- || «» .toc || main.toc || table of contents: creative content starts here |} [https://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes ⛐] | git bash 🈁 | | pdflatex <your d🎱cument> | | [https://en.wikibooks.org/wiki/LaTeX/Basics#Compilation (ℝ\ℚ)] | | 1 = R\Q / (ℝ\ℚ) = ⬤ [https://www.latex-project.org/ 🎱 git a ⚽ clu & <3 🎮 LaTeX 🈁 👣 🏊 ] > https://ggf333ttu.blogspot.com/ https://www.youtube.com/watch?v=H3QmruFdfgo ==new habits== https://architecturenotes.co/redis/ 𝑓(◯) allostatis https://vim.rtorr.com/ https://github.com/rtorr/vim-cheat-sheet {| + 𝑓(◯) |- || ◯ | style="background:#000 ;" | <span style="color:#fff; font-size: 11px; "> <center>⟁ ◁ #fff ○ #000 △ ⟁ </span> || 人 || 人 | style="background:#fff ;" | <span style="color:#000; font-size: 11px; "> <center>⟁ ◁ #000 ○ #fff △ ⟁</span> || 人 || 人 | style="background:gray ;" | <span style="color:pink; font-size: 11px; "> <center>⟁ ◁ pink ○ gray △ ⟁</span> |- || 人 | style="background:#000;" | <span style="color:#fff; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#fff ;" | <span style="color:#000; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:gray;" | <span style="color:pink; font-size: 11px;"><center> ⟁ </center></span> || [http://www.amp-what.com/unicode/search/triangle |- || 1 | style="background:#111;" | <span style="color:#ff0; font-size: 11px; "> <center>◁ #ff0 ○ #111△</center></span> || 人 || 人 | style="background:#111;" | <span style="color:#F0f; font-size: 11px; "> <center>◁ #f0f ○ #111△</center></span> || 人 || 人 | style="background:#111;" | <span style="color:#0ff; font-size: 11px; "> <center>◁ #0ff ○ #111 △</center></span> |- || 人 | style="background:#111;" | <span style="color:#ff0; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#111;" | <span style="color:#f0f; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#111;" | <span style="color:#0ff; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#333;" | <span style="color:#f00; font-size: 11px; "> <center>◁ #ff0 ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#0f0; font-size: 11px; "> <center>◁ #f0f ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#00f; font-size: 11px; "> <center>◁ #0ff ○ #333 △</center></span> |- || 人 | style="background:#333;" | <span style="color:#f00; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#333;" | <span style="color:#0f0; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#333;" | <span style="color:#00f; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#555;" | <span style="color:black; font-size: 11px; "> ◁ black ≡ #000 △ </span> || 人 || 人 | style="background:#555;" | <span style="color:indigo; font-size: 9px; "> ◁ indigo ≡ #4B0082△</span> || 人 || 人 | style="background:#555;" | <span style="color:midnightblue; font-size: 11px; "> ◁ midnightblue ≡ #191970△</span> |- || 人 | style="background:#555;" | <span style="color:black; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#555;" | <span style="color:indigo; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#555;" | <span style="color:midnightblue; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#777;" | <span style="color:#fff9c4; font-size: 11px; "> ◁ #fff9c4 ○ #777 △ </span> || 人 || 人 | style="background:#777;" | <span style="color:#FFCDD2; font-size: 11px; "> ◁ #FFCDD2 ○ #777 △ </span> || 人 || 人 | style="background:#777;" | <span style="color:#BBDEFB; font-size: 11px; "> ◁ #BBDEFB ○ #777 △ </span> |- || 人 | style="background:#777;" | <span style="color:#fff9c4; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#777;" | <span style="color:#FFCDD2; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#777;" | <span style="color:#BBDEFB; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#999;" | <span style="color:#ffe0b2; font-size: 11px; "> ◁ #ffe0b2 ○ #999 △ </span> || 人 || 人 | style="background:#999;" | <span style="color:#e1bee7; font-size: 11px; "> ◁ #e1bee7 ○ #999 △ </span> || 人 || 人 | style="background:#999;" | <span style="color:#C8E6C9; font-size: 11px; "> ◁ #C8E6C9 ○ #999 △ </span> |- || 人 | style="background:#999;" | <span style="color:#ffe0b2; font-size: 11px;"><center> ⌂ </center></span> || || | style="background:#999;" | <span style="color:#e1bee7; font-size: 11px;"><center> ⌂ </center></span> || || | style="background:#999;" | <span style="color:#C8E6C9; font-size: 11px;"><center> ⌂ </center></span> || |- || ⟁ | style="background:#aaa;" | <span style="color:#efefef; font-size: 11px; "> ◁ #efefef ○ #aaa △ </span> || 人 || 人 | style="background:#aaa;" | <span style="color:#222; font-size: 11px; "> ◁ #e1bee7 ○ #aaa △ </span> || 人 || 人 | style="background:#aaa;" | <span style="color:#777; font-size: 11px; "> ◁ #ababab ○ #aaa △ </span> |- || 人 | style="background:#aaa;" | <span style="color:#efefef ; font-size: 33px;"><center> ⚞人⚟ </center></span> || || | style="background:#aaa;" | <span style="color:#222; font-size: 33px;"><center> ⚞人⚟ </center></span> || || | style="background:#aaa;" | <span style="color:#777; font-size: 33px;"><center> ⚞人⚟ </center></span> || |} {| + font-size: 25px; |- || 3 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 25px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 4 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} ==WashingtonCounty.OR.US.case.22CN03334== [http://themetawiki.clu/w/index.php/WashingtonCounty.OR.US.case.22CN03334 http://themetawiki.clu/w/index.php/WashingtonCounty.OR.US.case.22CN03334] This is the case holding up plans for a physical wikiversity site in 97305-2644 Next court date is 9/8/2022 == Salem Accord == ===/əˈkôrd/ {phoenetic w _ } === ===アコード {phoenetic w kana} === ===符合 {kanji} === ===与える { hiragana | "one's own"? "tribal"? } === ==◯== <center style="background-color: indigo; color: yellow">⬛◯⬤⚪ ◁ yellow ○ indigo △ ⟁</center> <center style="background-color: indigo; color: magenta">⬛◯⬤⚪ ◁ magenta ○ indigo △ ⟁</center> <center style="background-color: indigo; color: cyan">⬛◯⬤⚪ ◁ cyan ○ indigo △ ⟁</center> <center style="background-color: indigo; color: lime">⬛◯⬤⚪ ◁ lime ○ indigo △ ⟁</center> <center style="background-color: midnightblue; color: indigo">⬛◯⬤⚪ ◁ indigo ○ midnightblue △ ⟁</center> <center style="background-color: midnightblue; color: black">⬛◯⬤⚪ ◁ black ○ midnightblue△ ⟁</center> <center style="background-color: indigo; color: midnightblue">⬛◯⬤⚪ ◁ midnightblue ○ indigo △ ⟁</center> <center style="background-color: indigo; color: black">⬛◯⬤⚪ ◁ black ○ indigo △ ⟁</center> <center style="background-color: black; color: indigo">⬛◯⬤⚪ ◁ indigo○ black △ ⟁</center> <center style="background-color: black; color: midnightblue">⬛◯⬤⚪ ◁ midnightblue ○ black△ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> ===人=== {| + font-size: 222px; primary |- || ◯ | style="background:#000 ;" | <span style="color:#fff "> <center>◁ #fff ○ #000 △ ⟁ </span> || 人 || 人 | style="background:#fff ;" | <span style="color:#000 "> <center>◁ #000 ○ #fff △ ⧋</span> || 人 || 人 | style="background:gray ;" | <span style="color:pink "> <center>◁ pink ○ gray △ 📐</span> |- || 人 | style="background:#000;" | <span style="color:#fff; font-size: 77px;"><center> ⟁ </center></span> || || | style="background:#fff ;" | <span style="color:#000; font-size: 77px;"><center> ⧋ </center></span> || || | style="background:gray;" | <span style="color:pink; font-size: 77px;"><center> 📐 </center></span> || [http://www.amp-what.com/unicode/search/triangle |- || 1 | style="background:#333;" | <span style="color:#ff0"> <center>◁ #ff0 ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#F0f"> <center>◁ #f0f ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#0ff"> <center>◁ #0ff ○ #333 △</center></span> |- || 人 | style="background:#333;" | <span style="color:#ff0; font-size: 277px;"><center> ⌂ </center></span> || || | style="background:#333;" | <span style="color:#f0f; font-size: 277px;"><center> ⌂ </center></span> || || | style="background:#333;" | <span style="color:#0ff; font-size: 277px;"><center> ⌂ </center></span> || |- || 2 | style="background:#808080;" | <span style="color:pink"> ◁ pink ○ #808080 △</span> || 人 || 人 | style="background:#808080;" | <span style="color:midnightblue"> ◁ midnightblue ○ #808080 △</span> || 人 || 人 | style="background:#808080;" | <span style="color:indigo"> ◁ indigo ○ #808080 △</span> |- || 人 | style="background:#808080;" | <span style="color:pink; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || || | style="background:#808080;" | <span style="color:midnightblue; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || || | style="background:#808080;" | <span style="color:indigo; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || |} or ∨↯∧ & + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + {| + font-size: 100px; |- || ◯ | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 46px;"><center> ∨↯∧ </center></span> || |} {| + font-size: 222px; primary |- || 1 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ #fff9c4 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ #FFCDD2 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ #BBDEFB ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 222px;"><center> ⌂ </center></span> || |} {| + font-size: 111px; secondary |- || 2 | style="background:#808080;" | <span style="color:#ffe0b2"> ◁ #ffe0b2 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#e1bee7"> ◁ #e1bee7 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#C8E6C9"> ◁ #C8E6C9 ○</span> |- || 人 | style="background:#808080;" | <span style="color:#ffe0b2; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#e1bee7; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#C8E6C9; font-size: 222px;"><center> ⌂ </center></span> || |} {| + font-size: 25px; |- || 3 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 25px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 4 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} {| + font-size: 100px; |- || 5 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 6 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} == 0 == {| |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家] || 小 |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] |} meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} {| + font-size: 100px; |- || 人 | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 100px;"><center> ∨↯∧ </center></span> || || | style="background:#808080;" | <span style="color:#fff9c4; font-size: 100px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 100px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 100px;"><center> ⌂ </center></span> || |- || |- || |} == or ∨↯∧ & + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + .us == ==⌂== {| + font-size: 100px; |- || 人 | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 100px;"><center> ⌂ </center></span> || |- || |- || |} [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ==✪⚪⬤🔥◍⚫⨁ 👀 ≡ odd → +1 [ { ( East ⚫🔴⚪○💮⭕ West ) } ] iff even ⇒ ÷2== ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] || ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪ || || ○人∘🧿⚪⬤◍⚫ || || ⨁⚉⨂❂✧✷✸✡✵ |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |} == ya == ◯ ○ ∘ ⬤ 大 【 ヤ / や 】 (ya) 人 【 ジン、 ニン / ひと 】(jean、he toe) 大家 (ya ya) Ya (hiragana: や, katakana: ヤ) ヤフー 屋 - Wiktionaryhttps://en.wiktionary.org › wiki › 屋 Semantic compound of 尸+至. 尸 does not represent the radical for death, but is a pictogram depicting a cloth draped. 至 means "dead end". Home (家) A home, or domicile, is a space used as a permanent or semi-permanent residence for one or many humans. It is a fully or semi sheltered space and can have both interior and exterior aspects to it. Ya (や) Kana Ya is one of the Japanese kana, each of which represents one mora. The hiragana is written in three strokes, while the katakana is written in two. Both represent. Their shapes have origins in the character 也. Wikipedia hiragana origin: 也 spelling kana: 大和のヤ Yamato no "ya" transliteration: ya unicode: U+3084, U+30E4 What is the pronunciation of Ya line? In historical kana orthography, it is written as "yau", "say", and "eu", and read as "yo", "you", and "yo", respectively. Even in modern times, "saying" and "going" are sometimes pronounced as "yu" and "yuku . " From the above, it can be said that Ya line is the yoon of that line . [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] == g == {| + |- || g || ⌂ || 時 |- || nono || ⌂ || ◁ || 前 || の || ノ || 名 || "Salmon" || #fa8072 || || || || || || rgb(250, 128, 114) || :: || hsl(6, 93%, 71%) |- || "のノ" || ⌂ || <span style="color:salmon;"><center> ∅ </center></span> | style="background:#000;"|<span style="color:salmon;"><center> #000 </center></span> | style="background:#808080;" | <span style="color:salmon;"><center> #808080 </center></span> | style="background:#fff;" | <span style="color:salmon;"><center> #fff </center></span> | style="background:#333;" | <span style="color:salmon;"><center> #333 </center></span> | style="background:#333;" | <span style="color:#fa8072;"><center> #fa8072 </center></span> | style="background:#333;" | <span style="color:#fa8072ff;"><center> #fa8072ff </center></span> | style="background:#333;" | <span style="color:#fa8072aa;"><center> #fa8072aa </center></span> | style="background:#333;" | <span style="color:#fa807280;"><center> #fa807280 </center></span> | style="background:#333;" | <span style="color:#fa807233;"><center> #fa807233 </center></span> | style="background:#333;" | <span style="color:#fa807200;"><center> #fa807200 </center></span> |- || know no ノノ || ⌂ || <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#000;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#808080;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#fff;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072ff; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072aa; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807280; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807233; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807200; font-size: 100px;"> ☂ </span> |} {| + |- || hsl(0-33, 93%, 71%) || <span style="color:hsl(0, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(1, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(2, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(3, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(4, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(5, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(7, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(8, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(9, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(10, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(11, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(12, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(13, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(14, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(15, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(16, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(17, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(18, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(19, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(20, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(21, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(22, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(23, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(24, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(25, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(26, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(27, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(28, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(29, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(30, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(31, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(32, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(33, 93%, 71%);"><center>⌂ </center></span> |} {| + |- || hsl(6-606, 93%, 71%) || <span style="color:hsl(6, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(36, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(66, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(96, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(126, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(156, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(186, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(216, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(246, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(276, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(306, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(336, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(366, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(396, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(426, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(456, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(486, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(516, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(546, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(576, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(606, 93%, 71%);"><center>⌂ </center></span> |} {| + |- || hsl(6, 93%, 0-100%) || <span style="color:hsl(6, 93%, 0%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 10%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 20%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 30%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 40%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 50%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 60%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 70%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 80%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 90%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 100%);"><center>⌂ </center></span> |} {| + |- || hsl(6, 0-100%, 71%) || <span style="color:hsl(6, 0%, 71%);"><center> ⌂ </center></span> || <span style="color:hsl(6, 10%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 20%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 30%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 40%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 50%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 60%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 70%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 80%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 90%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 100%, 71%);"><center>⌂ </center></span> |} meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |- ||ternary ||3 |- ||quaternary ||4 |- ||quinary ||5 |- ||senary ||6 |- ||septenary ||7 |- ||octal ||8 |- ||nonary |- ||decimal |- ||undenary |- ||duodecimal |- ||hexadecimal ||16 |- ||vigesimal ||20 |- ||sexagesimal ||60 |} How do you want your water served when you get here? https://pubs.usgs.gov/sir/2005/5168/pdf/sir2005-5168.pdf Robert Lee Stinson %VOX "tautology club says hi" w 11am "Naturalist Society for the Humane Treatment of Monsters" from dnd game on twitter [https://www.youtube.com/watch?v=n7uNA5fO1iI rice ex in CA] https://www.oregonwild.org/about/blog/oregon-grizzly-country https://therevelator.org/yellowstone-grizzlies-unbearable-divides/ https://developer.mozilla.org/en-US/docs/Web/CSS/color_value/hsl https://www.researchgate.net/about Amare, Nicole & Manning, A.. (2012). Seeing typeface personality: Emotional responses to form as tone. IEEE International Professional Communication Conference. 1-9. 10.1109/IPCC.2012.6408605. Various studies have correlated specific visual characteristics of typefaces with specific overall emotional effects: curvilinear forms and open letter shapes generally feel “friendly” but also “formal” or “informal,” depending on other factors; large contrasts in stroke widths, cap height, and aspect ratio generally feel “interesting,” but also “attractive” or “aggressive,” depending on other factors; low-variety and low-contrast forms generally feel “professional” but also “reliable” or “boring.” Although the current findings on typeface personality are useful, they have not indicated a systematic explanation for why specific physical typeface forms have the specific emotion effects that they do. This paper will report results of an empirical study in which 102 participants indicated their immediate emotional responses to each of 36 distinct typeface designs. Results support correlation between specific typeface features (variety vs. contrast vs. pattern) and specific emotional parameters (amusement vs. agitation vs. focus), explaining findings of previous studies, suggesting various classroom approaches to purpose-driven typeface selection. {{User alternative account|VeronicaJeanAnderson}} {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( A B E ) } ] </span> | style="background:black;" | <span style="color:white"> [ { ( [https://www.twitch.tv/archie97305 👀] ) } ]</span> |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> index.html</span> || notepad/atom (atom is deprecated) |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> vue </span> || [https://www.vim.org/ vim] [https://github.com/vim/vim-win32-installer/releases installer] |- || Tertiary | style="background:white;" | <span style="color:black"> css </span> || global css @ || gg css @ || NPC css @ |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> pug </span> |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> |- || Senary || b |} https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/sandbox trying to create a 1 -> 2 -> 3 -> 4 -> 5 -> 6 system in the apartment here that can be copied from site to site using artistic threads to help a Nice And Proper NAP-er navigate between properties with ease while maintaining adequate supportive care that we all require to enable us to focus on whatever catches our fancy. sun; natural light; breathe; BGs carbs; hygiene; laundry away bedroom; needles; blood; garbage out kitchen/nutritional/study social/outreach/linking worlds back porch 0 -- Computer Science, information and general works {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( T O P ) } ] </span> | style="background:black;" | <span style="color:white"> [ ℳ ] </span> | style="background:white;" | <span style="color:black"> { ¢ } </span> | style="background:#F9F9F9;" | <span style="color:pink"> ( ৳ ) </span> | style="background:black;" | <span style="color:white"> [ { ( I.n C.ase of E.mergency ) } ] </span> | style="background:teal;" | <span style="color:lime"> ᐪ l i p s c h i t z </span> || [https://www.youtube.com/watch?v=qrrz54UtkCc ᐪ] |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> physical</span> | style="background:#FFE6E6;" | <span style="color:black"> emotional</span> | style="background:#E6EAFF;" | <span style="color:black"> social</span> || This reflects health enough to communicate with people intimately enough to address real immediate issues | style="background:#FFFFE6;" | <span style="color:teal"> ^ torikomu </span> ||[https://www.youtube.com/watch?v=YxvBPH4sArQ ^] |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> occupational</span> | style="background:#F2E6FF;" | <span style="color:black"> intellectual</span> | style="background:#E6FFEA;" | <span style="color:black"> environmental</span> || This reflects living somewhere promoting healthy reasoning | style="background:#FFE6E6;" | <span style="color: teal"> | kaizen | </span> || | |- || Tertiary | style="background:white;" | <span style="color:black"> spiritual</span> | style="background:#BFBFBF;" | <span style="color:white"> factual </span> | style="background:#F2F2F2;" | <span style="color:black"> nutritional</span> || This reflects healthy mindful every habits | style="background:#E6EAFF;" | <span style="color:teal"> . genkiness . .</span> || . |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> generational</span> | style="background:#E6FFFF;" | <span style="color:black"> miscellaneal</span> | style="background:#F2E0CE;" | <span style="color:black"> punctuational</span> || This reflects having it all together enough to enjoy the holidays | style="background:#FFF2E6;" | <span style="color:lime"> # goblin </span> || # |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> | style="background:white;" | <span style="color:black"> {verb} </span> | style="background:black;" | <span style="color:white"> [noun] </span> || This reflects deliberate professional progress | style="background:#F2E6FF;" | <span style="color:lime"> / tsugu /</span> || / |- || Senary || b || 〇 || x || This reflects influencing others | style="background:#E6FFEA;" | <span style="color:lime"> @ g @ g @ </span> || [https://www.youtube.com/watch?v=SYnVYJDxu2Q @] |} === 000 *itches=== == 100 -- Philosophy and psychology == How can I use color to manipulate behavior and improve communication? ===named=== ==== Re⋮Beccaδ#639 ==== ===== rebeccapurple :: #663399 ===== https://meyerweb.com/eric/thoughts/2014/06/19/rebeccapurple/ ====black==== ====white==== ====græy==== ====pink==== ====indigo==== ====midnightblue==== ===hex=== ====#fff==== ====#fff==== {| class="wikitable" style="text-align: center;" |+ |- | style="background:black;" | <span style="color:white"> [ white { on black ⚞🧿⚟ #fff on #000 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:pink"> [ pink { on 50% grey ⚞🧿⚟ #ffc0cb on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#191970"> [ midnightblue { on 50% grey ⚞🧿⚟ #191970 on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#4b0082"> [ indigo { on 50% grey ⚞🧿⚟ #4b0082 on #808080 ⚞🧿⚟ } ] </span> |} ===cmyk=== https://colordesigner.io/convert/cmyktohex ====gg on cmyk(0,0,0,33.3) w|materializecss.com==== https://materializecss.com/color.html {| class="wikitable" style="text-align: center;" |+ |- | style="background:#ababab" | <span style="color:#fff9c4"> [ gg_yellow { on cmyk(0,0,0,33.3) ⚞🧿⚟ #fff9c4 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffe0b2"> [ gg_orange { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffe0b2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffcdd2"> [ gg_red { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffcdd2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#e1bee7"> [ gg_purple { on cmyk(0,0,0,33.3) ⚞🧿⚟ #e1bee7 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#bbdefb"> [ gg_blue { on cmyk(0,0,0,33.3) ⚞🧿⚟ #bbdefb on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#c8e6c9"> [ gg_green { on cmyk(0,0,0,33.3) ⚞🧿⚟ #c8e6c9 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#efefef"> [ gg_white { on cmyk(0,0,0,33.3) ⚞🧿⚟ #efefef on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#111"> [ gg_black { on cmyk(0,0,0,33.3) ⚞🧿⚟ #111 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#808080"> [ gg_grey { on cmyk(0,0,0,33.3) ⚞🧿⚟ #808080 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#f8bbd0"> [ gg_pink { on cmyk(0,0,0,33.3) ⚞🧿⚟ #f8bbd0 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#b2ebf2"> [ gg_cyan { on cmyk(0,0,0,33.3) ⚞🧿⚟ #b2ebf2 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#d7ccc8"> [ gg_brown { on cmyk(0,0,0,33.3) ⚞🧿⚟ #d7ccc8 on #ababab )⚞🧿⚟} ] </span> |} ===rgba=== === TrumPutin-ism === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. == 200 -- Religion == Royal We "spiritual warfare" 1000 things I did 1992-2022 other than lie my way onto the supreme court to overturn Roe v Wade {| |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |} == 300 -- Social sciences == https://wattention.com/traditional-rice-harvesting-in-japan/ https://www.wwoofjapan.com/home/index.php?lang=en == 400 -- Language == === Programming === ==== .png ==== ==== Esperanto ==== ==== HTML ==== ==== PUG ==== == 500 -- Pure Science == == 600 -- Technology == === local hosts=== [http://localhost:8080/ 8080] file:///D:/index.html === Roland SP 404MKII === https://www.roland.com/global/products/sp-404mk2/ https://www.roland.com/global/support/by_product/sp-404mk2/owners_manuals/ @https://static.roland.com/manuals/sp-404mk2_app/eng/19610757.html === VIM === https://vim-adventures.com/ == 700 -- Arts and recreation == == 800 -- Literature == == 900 -- History and geography == https://geology.com/stories/13/bear-areas/ === pre-2030 === 2022 "booked" by Hillsboro Police for sending email addressing "Christian Hate" and "Spiritual War" along with "exorcisms" and "Halloween Hysteria" in Marion County, OR where Salem Police Department abdicated from protecting some children in Salem from 2016-2021. 2021 Kaiser Permanente promised cash settlement to mitigate their abdication in Marion County. KP lawyer with intimate details about my vagina: terrence .j . loeber@kp.org 2012 "unliked" on FB by some Nazarene peers after openly questioning Alex Jones' allegation that Sandy Hook didn't happen and asking for compassion for parents who were called actors while they grieved publicly through no choice of their own. 2011 Lupron given by KP for menorrhagia as alternative to b/c pills first rxd in 86. How many women who have "mostly" been on b/c pills from 87-11 are obese? Why no menorrhagia while immersed in Japan? How close to a traditional Japanese diet can I get in the Willamette Valley and how close to no meat will my body allow? === TrumPutish War Against Humanity === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. === Ring of Fire === === Post "Roe v Wade" === Who did Roe v Wade protect? Why would a Nazarene raised pro-life support an "underground" network post Roe v Wade? === Our Contemporary "Underground Railroad" needs a submarine? === Why did Portland, OR close the Shanghai Tunnels recently? Human Trafficking through Astoria, OR has been going on "forever". How do we align an "underground railroad" with contemporary supports? == 10 -- A & + == == 11 -- B * x == == 12 -- C f(◯) == == 13 -- D Δ δ ƍ ≜ 𝜟 𝝳 == == 14 -- E 🐘 𓃰 == == 15 -- F == == 16 -- G == == 17 -- H == == 18 -- I == == 19 -- J == == 20 -- K == == 21 -- L == == 22 -- M == == 23 -- N == == 24 -- O == == 25 -- P == == 26 -- Q == == 27 -- R == == 28 -- S == == 29 -- T == == 30 -- U == == 31 -- V == == 32 -- W == == 33 -- X == == 34 -- Y == == 35 -- Z == 3nmg47pfuggxx8y0cc46r36e6u8k21c 2416397 2416396 2022-08-19T01:34:49Z 73.25.23.186 /* g */ wikitext text/x-wiki =0= == 00 == === 000 === ==== 0000 ==== ===== 00000 ===== ====== 000000 hex # ====== =🎱= [https://www.overleaf.com/learn/latex/TikZ_package TikZ¿¡⚽] [https://tikz.dev/ ฿ ₪ ₰] {| +BBwLTX Beaver Bank w LaTeX |- || 🎱 || main.pdf || destination || best distribution of purposeful print worthy content |- || ⛐ || readme.txt || 1st place to look for exceptions and exceptionaltions |- || 🈁 || main.aux || unknown |- || ɸ || Campus.jpeg || dot by dot picture w/o transparency |- || d͡ʒ || panther.jpeg || ligature |- || θ || mascot.jpg || the t a |- || ¶ || main.nav || unknown, but exciting place for an anchor icon |- || ‡.out || main.out || unknown |- ||『』.png || photo.png || dot by dot picture w alpha/transparency |- ||「」.snm || main.snm || wft? |- ||《》.tex || main.tex || this is the mistress J wants me to love |- || ⟨⟩ .log || main.log || sounds like somewhere stuff gets auto-logged |- || «» .toc || main.toc || table of contents: creative content starts here |} [https://www.overleaf.com/learn/latex/Learn_LaTeX_in_30_minutes ⛐] | git bash 🈁 | | pdflatex <your d🎱cument> | | [https://en.wikibooks.org/wiki/LaTeX/Basics#Compilation (ℝ\ℚ)] | | 1 = R\Q / (ℝ\ℚ) = ⬤ [https://www.latex-project.org/ 🎱 git a ⚽ clu & <3 🎮 LaTeX 🈁 👣 🏊 ] > https://ggf333ttu.blogspot.com/ https://www.youtube.com/watch?v=H3QmruFdfgo ==new habits== https://architecturenotes.co/redis/ 𝑓(◯) allostatis https://vim.rtorr.com/ https://github.com/rtorr/vim-cheat-sheet {| + 𝑓(◯) |- || ◯ | style="background:#000 ;" | <span style="color:#fff; font-size: 11px; "> <center>⟁ ◁ #fff ○ #000 △ ⟁ </span> || 人 || 人 | style="background:#fff ;" | <span style="color:#000; font-size: 11px; "> <center>⟁ ◁ #000 ○ #fff △ ⟁</span> || 人 || 人 | style="background:gray ;" | <span style="color:pink; font-size: 11px; "> <center>⟁ ◁ pink ○ gray △ ⟁</span> |- || 人 | style="background:#000;" | <span style="color:#fff; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#fff ;" | <span style="color:#000; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:gray;" | <span style="color:pink; font-size: 11px;"><center> ⟁ </center></span> || [http://www.amp-what.com/unicode/search/triangle |- || 1 | style="background:#111;" | <span style="color:#ff0; font-size: 11px; "> <center>◁ #ff0 ○ #111△</center></span> || 人 || 人 | style="background:#111;" | <span style="color:#F0f; font-size: 11px; "> <center>◁ #f0f ○ #111△</center></span> || 人 || 人 | style="background:#111;" | <span style="color:#0ff; font-size: 11px; "> <center>◁ #0ff ○ #111 △</center></span> |- || 人 | style="background:#111;" | <span style="color:#ff0; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#111;" | <span style="color:#f0f; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#111;" | <span style="color:#0ff; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#333;" | <span style="color:#f00; font-size: 11px; "> <center>◁ #ff0 ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#0f0; font-size: 11px; "> <center>◁ #f0f ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#00f; font-size: 11px; "> <center>◁ #0ff ○ #333 △</center></span> |- || 人 | style="background:#333;" | <span style="color:#f00; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#333;" | <span style="color:#0f0; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#333;" | <span style="color:#00f; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#555;" | <span style="color:black; font-size: 11px; "> ◁ black ≡ #000 △ </span> || 人 || 人 | style="background:#555;" | <span style="color:indigo; font-size: 9px; "> ◁ indigo ≡ #4B0082△</span> || 人 || 人 | style="background:#555;" | <span style="color:midnightblue; font-size: 11px; "> ◁ midnightblue ≡ #191970△</span> |- || 人 | style="background:#555;" | <span style="color:black; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#555;" | <span style="color:indigo; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#555;" | <span style="color:midnightblue; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#777;" | <span style="color:#fff9c4; font-size: 11px; "> ◁ #fff9c4 ○ #777 △ </span> || 人 || 人 | style="background:#777;" | <span style="color:#FFCDD2; font-size: 11px; "> ◁ #FFCDD2 ○ #777 △ </span> || 人 || 人 | style="background:#777;" | <span style="color:#BBDEFB; font-size: 11px; "> ◁ #BBDEFB ○ #777 △ </span> |- || 人 | style="background:#777;" | <span style="color:#fff9c4; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#777;" | <span style="color:#FFCDD2; font-size: 11px;"><center> ⟁ </center></span> || || | style="background:#777;" | <span style="color:#BBDEFB; font-size: 11px;"><center> ⟁ </center></span> || |- || ⟁ | style="background:#999;" | <span style="color:#ffe0b2; font-size: 11px; "> ◁ #ffe0b2 ○ #999 △ </span> || 人 || 人 | style="background:#999;" | <span style="color:#e1bee7; font-size: 11px; "> ◁ #e1bee7 ○ #999 △ </span> || 人 || 人 | style="background:#999;" | <span style="color:#C8E6C9; font-size: 11px; "> ◁ #C8E6C9 ○ #999 △ </span> |- || 人 | style="background:#999;" | <span style="color:#ffe0b2; font-size: 11px;"><center> ⌂ </center></span> || || | style="background:#999;" | <span style="color:#e1bee7; font-size: 11px;"><center> ⌂ </center></span> || || | style="background:#999;" | <span style="color:#C8E6C9; font-size: 11px;"><center> ⌂ </center></span> || |- || ⟁ | style="background:#aaa;" | <span style="color:#efefef; font-size: 11px; "> ◁ #efefef ○ #aaa △ </span> || 人 || 人 | style="background:#aaa;" | <span style="color:#222; font-size: 11px; "> ◁ #e1bee7 ○ #aaa △ </span> || 人 || 人 | style="background:#aaa;" | <span style="color:#777; font-size: 11px; "> ◁ #ababab ○ #aaa △ </span> |- || 人 | style="background:#aaa;" | <span style="color:#efefef ; font-size: 33px;"><center> ⚞人⚟ </center></span> || || | style="background:#aaa;" | <span style="color:#222; font-size: 33px;"><center> ⚞人⚟ </center></span> || || | style="background:#aaa;" | <span style="color:#777; font-size: 33px;"><center> ⚞人⚟ </center></span> || |} {| + font-size: 25px; |- || 3 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 25px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 4 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} ==WashingtonCounty.OR.US.case.22CN03334== [http://themetawiki.clu/w/index.php/WashingtonCounty.OR.US.case.22CN03334 http://themetawiki.clu/w/index.php/WashingtonCounty.OR.US.case.22CN03334] This is the case holding up plans for a physical wikiversity site in 97305-2644 Next court date is 9/8/2022 == Salem Accord == ===/əˈkôrd/ {phoenetic w _ } === ===アコード {phoenetic w kana} === ===符合 {kanji} === ===与える { hiragana | "one's own"? "tribal"? } === ==◯== <center style="background-color: indigo; color: yellow">⬛◯⬤⚪ ◁ yellow ○ indigo △ ⟁</center> <center style="background-color: indigo; color: magenta">⬛◯⬤⚪ ◁ magenta ○ indigo △ ⟁</center> <center style="background-color: indigo; color: cyan">⬛◯⬤⚪ ◁ cyan ○ indigo △ ⟁</center> <center style="background-color: indigo; color: lime">⬛◯⬤⚪ ◁ lime ○ indigo △ ⟁</center> <center style="background-color: midnightblue; color: indigo">⬛◯⬤⚪ ◁ indigo ○ midnightblue △ ⟁</center> <center style="background-color: midnightblue; color: black">⬛◯⬤⚪ ◁ black ○ midnightblue△ ⟁</center> <center style="background-color: indigo; color: midnightblue">⬛◯⬤⚪ ◁ midnightblue ○ indigo △ ⟁</center> <center style="background-color: indigo; color: black">⬛◯⬤⚪ ◁ black ○ indigo △ ⟁</center> <center style="background-color: black; color: indigo">⬛◯⬤⚪ ◁ indigo○ black △ ⟁</center> <center style="background-color: black; color: midnightblue">⬛◯⬤⚪ ◁ midnightblue ○ black△ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> <center style="background-color: indigo; color: white">⬛◯⬤⚪ ◁ white ○ indigo △ ⟁</center> ===人=== {| + font-size: 222px; primary |- || ◯ | style="background:#000 ;" | <span style="color:#fff "> <center>◁ #fff ○ #000 △ ⟁ </span> || 人 || 人 | style="background:#fff ;" | <span style="color:#000 "> <center>◁ #000 ○ #fff △ ⧋</span> || 人 || 人 | style="background:gray ;" | <span style="color:pink "> <center>◁ pink ○ gray △ 📐</span> |- || 人 | style="background:#000;" | <span style="color:#fff; font-size: 77px;"><center> ⟁ </center></span> || || | style="background:#fff ;" | <span style="color:#000; font-size: 77px;"><center> ⧋ </center></span> || || | style="background:gray;" | <span style="color:pink; font-size: 77px;"><center> 📐 </center></span> || [http://www.amp-what.com/unicode/search/triangle |- || 1 | style="background:#333;" | <span style="color:#ff0"> <center>◁ #ff0 ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#F0f"> <center>◁ #f0f ○ #333 △</center></span> || 人 || 人 | style="background:#333;" | <span style="color:#0ff"> <center>◁ #0ff ○ #333 △</center></span> |- || 人 | style="background:#333;" | <span style="color:#ff0; font-size: 277px;"><center> ⌂ </center></span> || || | style="background:#333;" | <span style="color:#f0f; font-size: 277px;"><center> ⌂ </center></span> || || | style="background:#333;" | <span style="color:#0ff; font-size: 277px;"><center> ⌂ </center></span> || |- || 2 | style="background:#808080;" | <span style="color:pink"> ◁ pink ○ #808080 △</span> || 人 || 人 | style="background:#808080;" | <span style="color:midnightblue"> ◁ midnightblue ○ #808080 △</span> || 人 || 人 | style="background:#808080;" | <span style="color:indigo"> ◁ indigo ○ #808080 △</span> |- || 人 | style="background:#808080;" | <span style="color:pink; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || || | style="background:#808080;" | <span style="color:midnightblue; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || || | style="background:#808080;" | <span style="color:indigo; font-size: 33px;"><center> ⚞🧿⚟ </center></span> || |} or ∨↯∧ & + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + {| + font-size: 100px; |- || ◯ | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 46px;"><center> ∨↯∧ </center></span> || |} {| + font-size: 222px; primary |- || 1 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ #fff9c4 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ #FFCDD2 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ #BBDEFB ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 222px;"><center> ⌂ </center></span> || |} {| + font-size: 111px; secondary |- || 2 | style="background:#808080;" | <span style="color:#ffe0b2"> ◁ #ffe0b2 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#e1bee7"> ◁ #e1bee7 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#C8E6C9"> ◁ #C8E6C9 ○</span> |- || 人 | style="background:#808080;" | <span style="color:#ffe0b2; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#e1bee7; font-size: 222px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#C8E6C9; font-size: 222px;"><center> ⌂ </center></span> || |} {| + font-size: 25px; |- || 3 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 25px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 25px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 4 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} {| + font-size: 100px; |- || 5 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} {| + font-size: 10px; |- || 6 | style="background:#808080;" | <span style="color:#fff9c4"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#FFCDD2"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:#BBDEFB"> ◁ ○</span> |- || 人 | style="background:#808080;" | <span style="color:#fff9c4; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 10px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 10px;"><center> ⌂ </center></span> || |} == 0 == {| |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家] || 小 |- || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] |} meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} {| + font-size: 100px; |- || 人 | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 || 人 | style="background:#808080;" | <span style="color:white"> ◁ ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 100px;"><center> ∨↯∧ </center></span> || || | style="background:#808080;" | <span style="color:#fff9c4; font-size: 100px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#FFCDD2; font-size: 100px;"><center> ⌂ </center></span> || || | style="background:#808080;" | <span style="color:#BBDEFB; font-size: 100px;"><center> ⌂ </center></span> || |- || |- || |} == or ∨↯∧ & + 🔥 十 ⨁ ⨂ ❂ ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○ ⌂ + .us == ==⌂== {| + font-size: 100px; |- || 人 | style="background:#333;" | <span style="color:white"> ◁ ⭕ 💮 ⚪ ⚫ 🔴 ○</span> || 人 |- || | style="background:#333;" | <span style="color:salmon; font-size: 100px;"><center> ⌂ </center></span> || |- || |- || |} [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ==✪⚪⬤🔥◍⚫⨁ 👀 ≡ odd → +1 [ { ( East ⚫🔴⚪○💮⭕ West ) } ] iff even ⇒ ÷2== ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] || ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪ || || ○人∘🧿⚪⬤◍⚫ || || ⨁⚉⨂❂✧✷✸✡✵ |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |} == ya == ◯ ○ ∘ ⬤ 大 【 ヤ / や 】 (ya) 人 【 ジン、 ニン / ひと 】(jean、he toe) 大家 (ya ya) Ya (hiragana: や, katakana: ヤ) ヤフー 屋 - Wiktionaryhttps://en.wiktionary.org › wiki › 屋 Semantic compound of 尸+至. 尸 does not represent the radical for death, but is a pictogram depicting a cloth draped. 至 means "dead end". Home (家) A home, or domicile, is a space used as a permanent or semi-permanent residence for one or many humans. It is a fully or semi sheltered space and can have both interior and exterior aspects to it. Ya (や) Kana Ya is one of the Japanese kana, each of which represents one mora. The hiragana is written in three strokes, while the katakana is written in two. Both represent. Their shapes have origins in the character 也. Wikipedia hiragana origin: 也 spelling kana: 大和のヤ Yamato no "ya" transliteration: ya unicode: U+3084, U+30E4 What is the pronunciation of Ya line? In historical kana orthography, it is written as "yau", "say", and "eu", and read as "yo", "you", and "yo", respectively. Even in modern times, "saying" and "going" are sometimes pronounced as "yu" and "yuku . " From the above, it can be said that Ya line is the yoon of that line . [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/Sandbox ⬤] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/plenary ◯] [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/inKind 大家][https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/specialdelivery ○ 人 ∘] = g = {| + |- || g || ⌂ || 時 |- || nono || ⌂ || ◁ || 前 || の || ノ || 名 || "Salmon" || #fa8072 || || || || || || rgb(250, 128, 114) || :: || hsl(6, 93%, 71%) |- || "のノ" || ⌂ || <span style="color:salmon;"><center> ∅ </center></span> | style="background:#000;"|<span style="color:salmon;"><center> #000 </center></span> | style="background:#808080;" | <span style="color:salmon;"><center> #808080 </center></span> | style="background:#fff;" | <span style="color:salmon;"><center> #fff </center></span> | style="background:#333;" | <span style="color:salmon;"><center> #333 </center></span> | style="background:#333;" | <span style="color:#fa8072;"><center> #fa8072 </center></span> | style="background:#333;" | <span style="color:#fa8072ff;"><center> #fa8072ff </center></span> | style="background:#333;" | <span style="color:#fa8072aa;"><center> #fa8072aa </center></span> | style="background:#333;" | <span style="color:#fa807280;"><center> #fa807280 </center></span> | style="background:#333;" | <span style="color:#fa807233;"><center> #fa807233 </center></span> | style="background:#333;" | <span style="color:#fa807200;"><center> #fa807200 </center></span> |- || know no ノノ || ⌂ || <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#000;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#808080;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#fff;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:salmon; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072ff; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa8072aa; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807280; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807233; font-size: 100px;"> ☂ </span> | style="background:#333;" | <span style="color:#fa807200; font-size: 100px;"> ☂ </span> |} {| + |- || hsl(0-33, 93%, 71%) || <span style="color:hsl(0, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(1, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(2, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(3, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(4, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(5, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(7, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(8, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(9, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(10, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(11, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(12, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(13, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(14, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(15, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(16, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(17, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(18, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(19, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(20, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(21, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(22, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(23, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(24, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(25, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(26, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(27, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(28, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(29, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(30, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(31, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(32, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(33, 93%, 71%);"><center>⌂ </center></span> |} {| + |- || hsl(6-606, 93%, 71%) || <span style="color:hsl(6, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(36, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(66, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(96, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(126, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(156, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(186, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(216, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(246, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(276, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(306, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(336, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(366, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(396, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(426, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(456, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(486, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(516, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(546, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(576, 93%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(606, 93%, 71%);"><center>⌂ </center></span> |} {| + |- || hsl(6, 93%, 0-100%) || <span style="color:hsl(6, 93%, 0%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 10%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 20%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 30%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 40%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 50%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 60%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 70%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 80%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 90%);"><center>⌂ </center></span> || <span style="color:hsl(6, 93%, 100%);"><center>⌂ </center></span> |} {| + |- || hsl(6, 0-100%, 71%) || <span style="color:hsl(6, 0%, 71%);"><center> ⌂ </center></span> || <span style="color:hsl(6, 10%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 20%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 30%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 40%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 50%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 60%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 70%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 80%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 90%, 71%);"><center>⌂ </center></span> || <span style="color:hsl(6, 100%, 71%);"><center>⌂ </center></span> |} meritorium . meritorious : merit .or.iou.us {{User alternative account|VeronicaJeanAnderson}} ∨⚡\🗲↯/ϟ∧ ✮☆⚝⛤🟊✰✭▲◂◁◀◢⍟◶✪⚪⬤🔥◍⚫⨁⚉⨂❂✧✷✸✡✵ http://slither.io/ https://www.thescienceofpsychotherapy.com/behaviour-affection-and-emotional-control/ {| class="wikitable" style="text-align: center;" |+ ⚞🧿⚟_◞◜↷◝◟_◞◜⚞🧿⚟🧿⚞🧿⚟◝◟_◞◜↶◝◟_⚞🧿⚟ |- || ✪⚪⬤🔥◍⚫⨁ || [https://www.twitch.tv/archie97305 👀] | style="background:pink;" | <span style="color:#808080"> ≡ odd → +1 </span> | style="background:pink;" | <span style="color:#808080"> [ { ( East </span> | style="background:black;" | <span style="color:white"> ⚫🔴⚪○💮⭕</span> | style="background:#808080;" | <span style="color:pink"> West ) } ] </span> | style="background:#808080;" | <span style="color:pink"> iff even ⇒ ÷2 </span> |- || Primary || [https://www.amnesty.org/en/ 1] |- || Secondary || [https://philosophy.stackexchange.com/questions/31029/why-was-the-horseshoe-symbol-%E2%8A%83-selected-for-material-implication 2] |- || Tertiary || [https://en.wikiversity.org/wiki/Wikiversity:Main_Page 3] |- || Quaternary || [http://localhost:8080/ 4] |- || Quinary || [https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson 5] |- || Senary || [https://en.wikiversity.org/wiki/User:Archie97305 6] |- || Septenary || [https://maritimearchaeological.org/beeswax-wreck/ 7] |- || Octenary || [https://www.youtube.com/freecodecamp 8] |- || nonary || [https://www.freecodecamp.org/ 9] |- ||Base Name ||[https://wordsmith.org/board/ubbthreads.php?ubb=showflat&Number=84101 `] |- || binary || 2 |- ||ternary ||3 |- ||quaternary ||4 |- ||quinary ||5 |- ||senary ||6 |- ||septenary ||7 |- ||octal ||8 |- ||nonary |- ||decimal |- ||undenary |- ||duodecimal |- ||hexadecimal ||16 |- ||vigesimal ||20 |- ||sexagesimal ||60 |} How do you want your water served when you get here? https://pubs.usgs.gov/sir/2005/5168/pdf/sir2005-5168.pdf Robert Lee Stinson %VOX "tautology club says hi" w 11am "Naturalist Society for the Humane Treatment of Monsters" from dnd game on twitter [https://www.youtube.com/watch?v=n7uNA5fO1iI rice ex in CA] https://www.oregonwild.org/about/blog/oregon-grizzly-country https://therevelator.org/yellowstone-grizzlies-unbearable-divides/ https://developer.mozilla.org/en-US/docs/Web/CSS/color_value/hsl https://www.researchgate.net/about Amare, Nicole & Manning, A.. (2012). Seeing typeface personality: Emotional responses to form as tone. IEEE International Professional Communication Conference. 1-9. 10.1109/IPCC.2012.6408605. Various studies have correlated specific visual characteristics of typefaces with specific overall emotional effects: curvilinear forms and open letter shapes generally feel “friendly” but also “formal” or “informal,” depending on other factors; large contrasts in stroke widths, cap height, and aspect ratio generally feel “interesting,” but also “attractive” or “aggressive,” depending on other factors; low-variety and low-contrast forms generally feel “professional” but also “reliable” or “boring.” Although the current findings on typeface personality are useful, they have not indicated a systematic explanation for why specific physical typeface forms have the specific emotion effects that they do. This paper will report results of an empirical study in which 102 participants indicated their immediate emotional responses to each of 36 distinct typeface designs. Results support correlation between specific typeface features (variety vs. contrast vs. pattern) and specific emotional parameters (amusement vs. agitation vs. focus), explaining findings of previous studies, suggesting various classroom approaches to purpose-driven typeface selection. {{User alternative account|VeronicaJeanAnderson}} {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( A B E ) } ] </span> | style="background:black;" | <span style="color:white"> [ { ( [https://www.twitch.tv/archie97305 👀] ) } ]</span> |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> index.html</span> || notepad/atom (atom is deprecated) |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> vue </span> || [https://www.vim.org/ vim] [https://github.com/vim/vim-win32-installer/releases installer] |- || Tertiary | style="background:white;" | <span style="color:black"> css </span> || global css @ || gg css @ || NPC css @ |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> pug </span> |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> |- || Senary || b |} https://en.wikiversity.org/wiki/User:VeronicaJeanAnderson/sandbox trying to create a 1 -> 2 -> 3 -> 4 -> 5 -> 6 system in the apartment here that can be copied from site to site using artistic threads to help a Nice And Proper NAP-er navigate between properties with ease while maintaining adequate supportive care that we all require to enable us to focus on whatever catches our fancy. sun; natural light; breathe; BGs carbs; hygiene; laundry away bedroom; needles; blood; garbage out kitchen/nutritional/study social/outreach/linking worlds back porch 0 -- Computer Science, information and general works {| class="wikitable" style="text-align: center;" |+ ᐪgenki-ness; +, -tachi . . . |- | style="background:black;" | <span style="color:white"> [ { ( T O P ) } ] </span> | style="background:black;" | <span style="color:white"> [ ℳ ] </span> | style="background:white;" | <span style="color:black"> { ¢ } </span> | style="background:#F9F9F9;" | <span style="color:pink"> ( ৳ ) </span> | style="background:black;" | <span style="color:white"> [ { ( I.n C.ase of E.mergency ) } ] </span> | style="background:teal;" | <span style="color:lime"> ᐪ l i p s c h i t z </span> || [https://www.youtube.com/watch?v=qrrz54UtkCc ᐪ] |- || Primary | style="background:#FFFFE6;" | <span style="color:black"> physical</span> | style="background:#FFE6E6;" | <span style="color:black"> emotional</span> | style="background:#E6EAFF;" | <span style="color:black"> social</span> || This reflects health enough to communicate with people intimately enough to address real immediate issues | style="background:#FFFFE6;" | <span style="color:teal"> ^ torikomu </span> ||[https://www.youtube.com/watch?v=YxvBPH4sArQ ^] |- || Secondary | style="background:#FFF2E6;" | <span style="color:black"> occupational</span> | style="background:#F2E6FF;" | <span style="color:black"> intellectual</span> | style="background:#E6FFEA;" | <span style="color:black"> environmental</span> || This reflects living somewhere promoting healthy reasoning | style="background:#FFE6E6;" | <span style="color: teal"> | kaizen | </span> || | |- || Tertiary | style="background:white;" | <span style="color:black"> spiritual</span> | style="background:#BFBFBF;" | <span style="color:white"> factual </span> | style="background:#F2F2F2;" | <span style="color:black"> nutritional</span> || This reflects healthy mindful every habits | style="background:#E6EAFF;" | <span style="color:teal"> . genkiness . .</span> || . |- || Quaternary | style="background:#FFE6FB;" | <span style="color:black"> generational</span> | style="background:#E6FFFF;" | <span style="color:black"> miscellaneal</span> | style="background:#F2E0CE;" | <span style="color:black"> punctuational</span> || This reflects having it all together enough to enjoy the holidays | style="background:#FFF2E6;" | <span style="color:lime"> # goblin </span> || # |- || Quinary | style="background:#F9F9F9;" | <span style="color:pink"> (direct object) </span> | style="background:white;" | <span style="color:black"> {verb} </span> | style="background:black;" | <span style="color:white"> [noun] </span> || This reflects deliberate professional progress | style="background:#F2E6FF;" | <span style="color:lime"> / tsugu /</span> || / |- || Senary || b || 〇 || x || This reflects influencing others | style="background:#E6FFEA;" | <span style="color:lime"> @ g @ g @ </span> || [https://www.youtube.com/watch?v=SYnVYJDxu2Q @] |} === 000 *itches=== == 100 -- Philosophy and psychology == How can I use color to manipulate behavior and improve communication? ===named=== ==== Re⋮Beccaδ#639 ==== ===== rebeccapurple :: #663399 ===== https://meyerweb.com/eric/thoughts/2014/06/19/rebeccapurple/ ====black==== ====white==== ====græy==== ====pink==== ====indigo==== ====midnightblue==== ===hex=== ====#fff==== ====#fff==== {| class="wikitable" style="text-align: center;" |+ |- | style="background:black;" | <span style="color:white"> [ white { on black ⚞🧿⚟ #fff on #000 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:pink"> [ pink { on 50% grey ⚞🧿⚟ #ffc0cb on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#191970"> [ midnightblue { on 50% grey ⚞🧿⚟ #191970 on #808080 ⚞🧿⚟ } ] </span> |- | style="background:#808080;" | <span style="color:#4b0082"> [ indigo { on 50% grey ⚞🧿⚟ #4b0082 on #808080 ⚞🧿⚟ } ] </span> |} ===cmyk=== https://colordesigner.io/convert/cmyktohex ====gg on cmyk(0,0,0,33.3) w|materializecss.com==== https://materializecss.com/color.html {| class="wikitable" style="text-align: center;" |+ |- | style="background:#ababab" | <span style="color:#fff9c4"> [ gg_yellow { on cmyk(0,0,0,33.3) ⚞🧿⚟ #fff9c4 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffe0b2"> [ gg_orange { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffe0b2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#ffcdd2"> [ gg_red { on cmyk(0,0,0,33.3) ⚞🧿⚟ #ffcdd2 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#e1bee7"> [ gg_purple { on cmyk(0,0,0,33.3) ⚞🧿⚟ #e1bee7 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#bbdefb"> [ gg_blue { on cmyk(0,0,0,33.3) ⚞🧿⚟ #bbdefb on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#c8e6c9"> [ gg_green { on cmyk(0,0,0,33.3) ⚞🧿⚟ #c8e6c9 on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#efefef"> [ gg_white { on cmyk(0,0,0,33.3) ⚞🧿⚟ #efefef on #ababab⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#111"> [ gg_black { on cmyk(0,0,0,33.3) ⚞🧿⚟ #111 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#808080"> [ gg_grey { on cmyk(0,0,0,33.3) ⚞🧿⚟ #808080 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#f8bbd0"> [ gg_pink { on cmyk(0,0,0,33.3) ⚞🧿⚟ #f8bbd0 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#b2ebf2"> [ gg_cyan { on cmyk(0,0,0,33.3) ⚞🧿⚟ #b2ebf2 on #ababab ⚞🧿⚟ } ] </span> |- | style="background:#ababab;" | <span style="color:#d7ccc8"> [ gg_brown { on cmyk(0,0,0,33.3) ⚞🧿⚟ #d7ccc8 on #ababab )⚞🧿⚟} ] </span> |} ===rgba=== === TrumPutin-ism === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. == 200 -- Religion == Royal We "spiritual warfare" 1000 things I did 1992-2022 other than lie my way onto the supreme court to overturn Roe v Wade {| |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ 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|| ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |- || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ || ɸ |} == 300 -- Social sciences == https://wattention.com/traditional-rice-harvesting-in-japan/ https://www.wwoofjapan.com/home/index.php?lang=en == 400 -- Language == === Programming === ==== .png ==== ==== Esperanto ==== ==== HTML ==== ==== PUG ==== == 500 -- Pure Science == == 600 -- Technology == === local hosts=== [http://localhost:8080/ 8080] file:///D:/index.html === Roland SP 404MKII === https://www.roland.com/global/products/sp-404mk2/ https://www.roland.com/global/support/by_product/sp-404mk2/owners_manuals/ @https://static.roland.com/manuals/sp-404mk2_app/eng/19610757.html === VIM === https://vim-adventures.com/ == 700 -- Arts and recreation == == 800 -- Literature == == 900 -- History and geography == https://geology.com/stories/13/bear-areas/ === pre-2030 === 2022 "booked" by Hillsboro Police for sending email addressing "Christian Hate" and "Spiritual War" along with "exorcisms" and "Halloween Hysteria" in Marion County, OR where Salem Police Department abdicated from protecting some children in Salem from 2016-2021. 2021 Kaiser Permanente promised cash settlement to mitigate their abdication in Marion County. KP lawyer with intimate details about my vagina: terrence .j . loeber@kp.org 2012 "unliked" on FB by some Nazarene peers after openly questioning Alex Jones' allegation that Sandy Hook didn't happen and asking for compassion for parents who were called actors while they grieved publicly through no choice of their own. 2011 Lupron given by KP for menorrhagia as alternative to b/c pills first rxd in 86. How many women who have "mostly" been on b/c pills from 87-11 are obese? Why no menorrhagia while immersed in Japan? How close to a traditional Japanese diet can I get in the Willamette Valley and how close to no meat will my body allow? === TrumPutish War Against Humanity === Trump has demonstrably alienated the USA from allies both foreign and domestic. While Oregon's AG works on Epstein and Weinstein, contemporaneous crimes go unabated and have created a new problem where otherwise law abiding folk find themselves on the wrong side of the law. Oregon doesn't have enough public defenders to fight violent crime, yet children are alienated from their church and families to hide atrocities they don't even know about. === Ring of Fire === === Post "Roe v Wade" === Who did Roe v Wade protect? Why would a Nazarene raised pro-life support an "underground" network post Roe v Wade? === Our Contemporary "Underground Railroad" needs a submarine? === Why did Portland, OR close the Shanghai Tunnels recently? Human Trafficking through Astoria, OR has been going on "forever". How do we align an "underground railroad" with contemporary supports? == 10 -- A & + == == 11 -- B * x == == 12 -- C f(◯) == == 13 -- D Δ δ ƍ ≜ 𝜟 𝝳 == == 14 -- E 🐘 𓃰 == == 15 -- F == == 16 -- G == == 17 -- H == == 18 -- I == == 19 -- J == == 20 -- K == == 21 -- L == == 22 -- M == == 23 -- N == == 24 -- O == == 25 -- P == == 26 -- Q == == 27 -- R == == 28 -- S == == 29 -- T == == 30 -- U == == 31 -- V == == 32 -- W == == 33 -- X == == 34 -- Y == == 35 -- Z == 99kqvou9kaf3luz1ofgc63vu1ofrxx3 0 262713 2416165 2416012 2022-08-18T14:38:09Z Bobamnertiopsis 24451 /* Pathogenesis */ <sub> in article title wikitext text/x-wiki {{Article info | journal = WikiJournal of Medicine | first1 = Siang Ching | last1 = Raymond Chieng | orcid1 = 0000-0003-1286-2196 |last2 = |first2 = |last3 = |first3 = |last4 = |first4 = |submitted = 2020-05-01 | affiliation1 = Klinik Kesihatan Bintangor, Sarawak, Malaysia | correspondence1 = by [[w:Special:EmailUser/Cerevisae|online form]] | w1 = Melioidosis | et_al = true | abstract = '''Melioidosis''' is an [[w:infectious disease|infectious disease]] caused by a [[w:gram-negative|gram-negative]] [[w:bacterium|bacterium]] called ''[[w:Burkholderia pseudomallei|Burkholderia pseudomallei]]''.<ref name="Joost 2018"/> Most people exposed to ''B. pseudomallei'' experience no symptoms; however, those who do experience symptoms have signs and symptoms that range from mild such as [[w:fever|fever]] and skin changes, to severe with [[w:pneumonia|pneumonia]], [[w:abscess|abscess]]es, and [[w:septic shock|septic shock]] that could cause death.<ref name="Joost 2018"/> Approximately 10% of people with melioidosis develop symptoms that last longer than two months, termed "chronic melioidosis".<ref name="Joost 2018"/> Humans are infected with ''B. pseudomallei'' by contact with contaminated soil or water. The bacteria enter the body through wounds, inhalation, or ingestion. Person-to-person or animal-to-human transmission is extremely rare.<ref name="Joost 2018"/> The infection is constantly present in Southeast Asia particularly in northeast [[w:Thailand|Thailand]] and northern Australia.<ref name="Joost 2018"/> In temperate countries such as Europe and the United States, melioidosis cases are usually imported from countries where melioidosis is endemic.<ref name="Currie 2015"/> The signs and symptoms of melioidosis resemble [[w:tuberculosis|tuberculosis]] and misdiagnosis is common.<ref name="Yi 2014"/> Diagnosis is usually confirmed by the growth of ''B. pseudomallei'' from an infected person's blood or other bodily fluid such as pus, sputum, and urine.<ref name="Joost 2018"/> Those with melioidosis are treated first with an "intensive phase" course of intravenous antibiotics (most commonly [[w:ceftazidime|ceftazidime]]) followed by a several-months treatment course of [[w:co-trimoxazole|co-trimoxazole]].<ref name="Joost 2018"/> In countries with the advanced healthcare system, approximately 10% of people with melioidosis die from the disease. In less developed countries, the death rate could reach 40%.<ref name="Joost 2018">{{cite journal |author= Wiersinga WJ |author2=Virk HS |author3=Torres AG |author4=Currie BJ et al |title=Melioidosis |journal=Nature Reviews Disease Primers |date=1 February 2018 |volume=4 |issue= |pages=17107 |doi=10.1038/nrdp.2017.107 |pmid=29388572 |pmc=6456913 }}</ref> Efforts to prevent melioidosis include: wearing protective gear while handling contaminated water or soil, practising hand hygiene, drinking boiled water, and avoiding direct contact with soil, water, or heavy rain.<ref name="Joost 2018"/> There is little evidence in supporting the use of melioidosis prophylaxis in humans. The [[w:antibiotic|antibiotic]] [[w:co-trimoxazole|co-trimoxazole]] is used as a preventative only for individuals at high risk for getting the disease after being exposed to the bacteria in laboratory settings.<ref name="Joost 2018"/> One study conducted in 2018 determined that the drug could be useful in preventing melioidosis in high-risk renal failure patients undergoing haemodylysis.<ref>{{cite journal |author=Majoni SW |author2=Hughes JT |author3=Heron B |author4=Currie BJ |title=Trimethoprim+Sulfamethoxazole Reduces Rates of Melioidosis in High-Risk Hemodialysis Patients |journal=Kidney International Reports |date=2018 |volume=3 |issue=1 |pages=160-167 |doi=10.1016/j.ekir.2017.09.005 |pmid=29340327 |pmcid=5762962 }}</ref> There is no approved vaccine for melioidosis.<ref name="Joost 2018"/> Approximately 165,000 people are infected by melioidosis per year, resulting in about 89,000 deaths, based on a mathematical model published in 2016.<ref name="Limmathurotsakul 2016">{{cite journal |author=Limmathurotsakul D |author2=Golding N |author3=Dance DAB |author4=Messina JP et al |title=Predicted global distribution of Burkholderia pseudomallei and burden of melioidosis |journal=Nature Microbiology |date=2016 |volume=1 |issue=1 |pages=15008 |doi=10.1038/nmicrobiol.2015.8 |pmid=26877885 |pmcid=4746747 }}</ref> [[w:Diabetes|Diabetes]] is a major risk factor for melioidosis; over half of melioidosis cases are in people with diabetes.<ref name="Joost 2018"/> Increased rainfall and severe weather events such as thunderstorm are associated with an increased number of melioidosis cases in [[w:Endemic (epidemiology)|endemic]] areas.<ref name="Yi 2014"/> }} ==Signs and symptoms== ===Acute=== {{fig|1 |Melioidosis PA and lateral X rays.jpg |Chest X-ray showing opacity of the left upper lobe of the lung of a patient with melioidosis complicated with pneumonia. |attribution=Samira Rahat Afroze et al, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0] |size=285px }} {{fig|2 |CT and MRI scan of the brain with melioidosis.jpg |CT and MRI scans showing lesion of the right frontal lobe of the brain of a patient with melioidosis. |attribution=Wei-yuan Huang et al, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0] |size=285px }} {{fig|3 |Septic arthritis of left hip joint with melioidosis.jpg |Septic arthritis of the left hip with joint destruction due to melioidosis. |attribution=N. P. Weerasinghe et al, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0] |size=285px }} Most people exposed to ''[[w:Burkholderia pseudomallei|B. pseudomallei]]'' experience no symptoms.<ref name="Yi 2014"/><ref>{{cite journal |author=Mustafa M |author2=Balingi J |author3=Torresi J |author4=Menon J |title=Clinical manifestations, diagnosis, and treatment of Melioidosis |journal=IOSR Journal of Pharmacy |date=February 2015 |volume=5 |issue=2 |pages=13-19 |doi= |pmid= |url=https://web.archive.org/web/20220626050909/https://www.ums.edu.my/fpsk/files/12.-Dr.-Fredie-Robinson-IOSR-Journal-of-Pharmacy.pdf }}</ref> The mean [[w:incubation period|incubation period]] of acute melioidosis is 9 days (range 1&ndash;21 days).<ref name="Joost 2018"/> Nevertheless, symptoms of melioidosis can appear in 24 hours for those who experienced near drowning in water.<ref name="Bennett 2015"/> Those affected present with symptoms of [[w:sepsis|sepsis]] (predominantly fever) with or without [[w:pneumonia|pneumonia]], or localised [[w:abscess|abscess]] or other focus of infection. The presence of non-specific signs and symptoms has caused melioidosis to be nicknamed "the great mimicker".<ref name="Joost 2018"/> [[w:diabetes mellitus|Diabetes mellitus]] is one of the most important risk factors in developing melioidosis. The disease should be considered in anyone who has spent time in endemic areas who develops a fever, pneumonia, or abscesses in their liver, spleen, prostate, or parotid gland.<ref name="Joost 2018"/> The clinical manifestation of the disease can range from simple skin changes such as abscesses or ulcerations to severe organ problems.<ref>{{cite journal |author=Fertitta L |author2=Monsel G |author3=Torresi J |author4=Caumes E |title=Cutaneous melioidosis: a review of the literature |journal=The International Society of Dermatology |date=2018 |volume=58 |issue=2 |pages=221–227 |doi=10.1111/ijd.14167 |pmid=30132827 }}</ref> The commonest organs affected are liver, spleen, lungs, prostate, and kidneys. Among the most common features are [[w:bacteremia|bacteremia]] (in 40 to 60% of cases), pneumonia (50%), and [[w:septic shock|septic shock]] (20%).<ref name="Joost 2018"/><ref name="Currie 2021">{{cite journal |author=Currie BJ |author2=Mayo M |author3=Ward LM et al |title=The Darwin Prospective Melioidosis Study: a 30-year prospective, observational investigation |journal=The Lancet Infectious Diseases |date=1 December 2021 |volume=21 |issue=2 |pages=1737-1746 |doi=10.1016/S1473-3099(21)00022-0 |pmid=34303419 |pmc= }}</ref> People with only pneumonia may have a prominent cough with sputum and shortness of breath. However, those with septic shock together with pneumonia may have minimal coughing.<ref name="Yi 2014"/> Results of a chest X-ray can range from diffuse nodular infiltrates in those with septic shock to progressive [[w:pulmonary consolidation|consolidation]] located most commonly in the [[w:Lung#Anatomy|upper lobes]] for those with pneumonia only. [[w:Pleural effusion|Pleural effusion]] and [[w:empyema|empyema]] are more common for melioidosis affecting lower lobes of the lungs.<ref name="Yi 2014"/> In 10% of cases, people develop secondary pneumonia caused by other bacteria after the primary infection.<ref name="Currie 2015"/> In northern Australia, 60% of the infected children presented with only skin lesions, while 20% presented with pneumonia.<ref name="Currie 2015"/> Depending on the course of infection, other severe manifestations develop. Approximately 1 to 5% of those infected develop [[w:encephalomyelitis|inflammation of the brain and brain covering]] or [[w:brain abscess|brain abscess]]; 14 to 28% develop [[w:acute pyelonephritis|pyelonephritis]], kidney abscess or prostatic abscesses; 0 to 30% develop neck or [[w:parotid gland|salivary gland]] abscesses; 10 to 33% develop liver, spleen, or paraintestinal abscesses; and 4 to 14% develop [[w:septic arthritis|septic arthritis]] and [[w:osteomyelitis|osteomyelitis]].<ref name="Joost 2018"/> Rare manifestations include [[w:lymphadenopathy|lymph node disease]] resembling tuberculosis,<ref name="Gassiep 2020">{{cite journal |author=Gassiep I |author2=Armstrong M |author3=Norton R |title=Human Melioidosis |journal=Clinical Microbiology Reviews |date=11 March 2020 |volume=33 |issue=2 |doi=10.1128/CMR.00006-19 |pmid=32161067 |pmc=7067580 }}</ref> [[w:mediastinum|mediastinal]] masses, [[w:pericardial effusion|pericardial effusion]],<ref name="Currie 2015"/> [[w:mycotic aneurysm|mycotic aneurysm]],<ref name="Joost 2018"/> and [[w:pancreatitis|inflammation of the pancreas]].<ref name="Currie 2015"/> In Australia, up to 20% of infected males develop prostatic abscess which may manifests clinically as [[w:dysuria|pain during urination]], difficulty in passing urine, and [[w:urinary retention|urinary retention]] requiring [[w:catheter|catheter]]isation.<ref name="Joost 2018"/> [[w:Rectal examination|Rectal examination]] may find enlarged [[w:prostate|prostate]].<ref name="Currie 2015"/> In Thailand, 30% of the infected children develop parotid abscesses.<ref name="Joost 2018"/> Encephalomyelitis not only happens in those with risk factors, but can also occur in healthy people without risk factors. Those with melioidosis encephomyelitis tend to have normal [[w:computed tomography|computed tomography]] (CT) scans but increased [[w:MRI_sequence#T1_and_T2|T2 signal]] by [[w:magnetic resonance imaging|magnetic resonance imaging]] (MRI), extending to the [[w:brain stem|brain stem]] and [[w:spinal cord|spinal cord]]. Clinical signs include: unilateral [[w:upper motor neuron|upper motor neuron]] limb weakness, [[w:focal neurological signs|cerebellar signs]], and cranial nerve palsies ([[w:Sixth nerve palsy|VI]], [[w:Facial nerve paralysis|VII]] nerve palsies and [[w:bulbar palsy|bulbar palsy]]). Some cases presented with [[w:flaccid paralysis|flaccid paralysis]] alone.<ref name="Currie 2015"/> In northern Australia, all melioidosis with encephalomyelitis cases had elevated white cells in the [[w:cerebrospinal fluid|cerebrospinal fluid]] (CSF), mostly [[w:mononuclear cell|mononuclear cell]]s with elevated CSF protein.<ref name="Gassiep 2020"/> ===Chronic=== Chronic melioidosis is usually defined by symptoms lasting greater than two months and occurs in about 10% of patients.<ref name="Joost 2018"/> Clinical presentations include fever, weight loss, productive cough with or without bloody sputum which may mimic [[w:tuberculosis|tuberculosis]]. Additionally, long-standing abscesses at multiple body sites may also present.<ref name="Yi 2014"/> Tuberculosis should be considered for lymph nodes enlargement at the [[w:root of the lung|root of the lung]]. Additionally, pneumonia caused by melioidosis rarely causes scarring and calcification of the lungs, unlike tuberculosis.<ref name="Gassiep 2020"/> ===Latent=== The potential for prolonged incubation was recognized in US servicemen involved in the Vietnam War, and was referred to as the "Vietnam time-bomb".<ref name="Yi 2014"/> Initially, it was thought that the longest period between presumed exposure and clinical presentation is 62 years in a [[w:prisoner of war|prisoner of war]] in Burma-Thailand-Malaysia.<ref name="Ngauy 2005">{{cite journal |author=Ngauy V |author2=Lemeshev Y |author3=Sadkowski L |author4=Crawford G |title=Cutaneous Melioidosis in a Man Who Was Taken as a Prisoner of War by the Japanese during World War II |year=2005 |journal=Journal of Clinical Microbiology |volume=43 |issue=2 |pages=970&ndash;972 |pmid=15695721 |doi=10.1128/JCM.43.2.970-972.2005 |pmc=548040 }}</ref> However, subsequent genotyping of the bacteria isolate from the Vietnam veteran showed that the isolate may not come from [[w:Southeast Asia|Southeast Asia]], but from South America.<ref>{{cite journal |author=Gee JE |author2=Gulvick CA |author3=Elrod MG et al |title=Phylogeography of Burkholderia pseudomallei Isolates, Western Hemisphere |journal=Emergency Infectious Disease |date=July 2017 |volume=23 |issue=7 |pages=1133–1138 |doi=10.3201/eid2307.161978 |pmid=28628442 |pmc=5512505 }}</ref> This reinstates another report that put the longest latency period for melioidosis as 29 years.<ref>{{cite journal |author=Chodimella U |author2=Hoppes WL |author3=Whalen S |title=Septicemia and suppuration in a Vietnam veteran |journal=Hospital Practice |date=15 May 1997 |volume=32 |issue=5 |pages=219–221 |pmid=9153149 |doi=10.1080/21548331.1997.11443493 }}</ref> Patients with latent melioidosis may be symptom-free for decades.<ref name="Ngauy 2005"/> Less than 5% of all melioidosis cases have activation after a period of latency.<ref name="Joost 2018"/> Various comorbidities such as diabetes, renal failure, and alcoholism can predispose to reactivation of melioidosis.<ref name="Yi 2014"/> ==Cause== ===Bacteria=== {{fig|4 |Burkholderia pseudomallei gram stain safety pin apperance.jpg |''B. pseudomallei'' with bipolar gram staining showing safety pin appearance. |attribution=Neha Shrestha, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0] |size=285px }} Melioidosis is caused by [[w:gram-negative|gram-negative]], [[w:motility|motile]], [[w:Saprotrophic nutrition|saprophytic]] bacteria named ''Burkholderia pseudomallei''.<ref name="Chakravorty 2019">{{cite journal |author= Chakravorty A |author2=Heath C |title=Melioidosis: An updated review |journal= Australian Journal of General Practice |date=5 May 2019 |volume=48 |issue=5 |pages=327-332 |doi=10.31128/AJGP-04-18-4558 |pmid=31129946 }}</ref> The bacteria are usually [[w:opportunistic infection|opportunistic]], [[w:Facultative parasite|facultative]] [[w:intracellular|intracellular]] pathogens.<ref name="Chakravorty 2019"/> It is also aerobic and [[w:oxidase test|oxidase test]] positive.<ref name="Yi 2014"/> A granule at the centre of the bacterium makes it resemble a “safety pin” when [[w:Gram stain|Gram stain]]ed.<ref name="Yi 2014"/> The bacteria emit a strong soil smell after 24 to 48 hours of growth in culture, however smelling for the identification of the bacteria is not recommended for routine laboratory practice. One of the factors causing ''B. pseudomallei'''s resistance to various kinds of antibiotics is because of its production of a [[w:glycocalyx|glycocalyx]] [[w:polysaccharide|polysaccharide]] capsule.<ref name="Allen C 2005">{{cite journal |author=Cheng AC |author2=Currie BJ |title=Melioidosis: Epidemiology, Pathophysiology, and Management |journal=Clinical Microbiology Reviews |date=April 2005 |volume=18 |issue=2 |pages=383–416 |doi=10.1128/CMR.18.2.383-416.2005 |pmid=15831829 |pmc=1082802 }}</ref> It is generally resistant to [[w:gentamicin|gentamicin]] and [[w:colistin|colistin]] but sensitive to [[w:amoxicillin/clavulanic acid|co-amoxiclav]]. ''B. pseudomallei'' is a [[w:biosafety level|biosafety level]] 3 pathogen which requires specialized laboratory handling.<ref name="Yi 2014"/> In humans and animals, another similar organism named ''[[w:Burkholderia mallei|Burkholderia mallei]]'' is the causative agent of the disease [[w:glanders|glanders]].<ref name="Joost 2018"/> ''B. pseudomallei'' can be differentiated from another closely related, but less pathogenic species ''B. thailandensis'' by its ability to assimilate [[w:arabinose|arabinose]].<ref name="Gassiep 2020"/> ''B. pseudomallei'' is highly adaptable to various host environments ranging from inside [[w:Mycorrhiza|mycorrhizal fungi]] spores to [[w:amoeba|amoeba]].<ref name="Yi 2014"/> Its adaptability may give it a survival advantage in the human body.<ref name="Joost 2018"/> The genome of ''B. pseudomallei'' consists of two [[w:replicon (genetics)|replicons]]: chromosome 1 encodes [[w:housekeeping gene|housekeeping functions]] of the bacteria such as cell wall synthesis, mobility, and metabolism; chromosome 2 encodes functions that allow the bacteria to adapt to various environments. [[w:Horizontal gene transfer|Horizontal gene transfer]] has resulted in highly variable genomes in ''B. pseudomallei''. Australia has been suggested as the origin for ''B. pseudomallei'' because of the high genetic variability of the bacteria found in this region. Bacteria that was introduced to Central and South America in the 17th to 19th centuries seem to have a common ancestor from Africa.<ref name="Chewapreecha 2017">{{cite journal |author= Chewapreecha C |author2=Holden MT |author3=Vehkala M |author4=Välimäki N et al |title=Global and regional dissemination and evolution of Burkholderia pseudomallei |journal= Nature Microbiology |date=23 January 2017 |volume=2 |issue=16263 |pages=1-8 |doi=10.1038/nmicrobiol.2016.263 |pmid=28112723 |pmcid=5300093 }}</ref> ''B. mallei'' is a clone of ''B. pseudomallei'' that has lost substantial portions of its genome as it adapted to live exclusively in mammals.<ref name="Currie 2015">{{cite journal |author=Currie BJ |title=Melioidosis: evolving concepts in epidemiology, pathogenesis, and treatment |journal=Seminars in Respiratory and Critical Care Medicine |date=February 2015 |volume=36 |issue=1 |pages=111–125 |doi=10.1055/s-0034-1398389 |pmid=25643275 }}</ref> This makes the ''B. mallei'' genome much smaller than ''B. pseudomallei''.<ref name="Nierman 2004">{{cite journal |author=Nierman WC |author2=DeShazer D |author3=Kim HS |title=Structural flexibility in the Burkholderia mallei genome |journal=Proceedings of National Academy of Science of United States of America |date=28 September 2004 |volume=101 |issue=39 |pages=14246-14251 |doi=10.1073/pnas.0403306101 |pmid=15377793 |pmc=521142 }}</ref> ===Transmission=== ''B. pseudomallei'' is normally found in soil and surface water, and is most abundant at soil depths of 10 to 90 cm.<ref name="Joost 2018"/> It has been found in soils, ponds, streams, pools, stagnant water, and rice paddy fields.<ref name="Yi 2014">{{cite journal |author=Foong YC |author2=Tan M |author3=Bradbury RS |title=Melioidosis: A Review |journal=Journal of Remote and Rural Health |date=30 October 2014 |volume=14 |issue=4 |pages=2763 |pmid=25359677|doi=10.22605/RRH2763 |url=https://www.rrh.org.au/journal/article/2763 }}</ref> ''B. pseudomallei'' can survive in nutrient-poor conditions such as distilled water, desert soil, and nutrient-depleted soil for more than 16 years.<ref name="Joost 2018"/> It can also survive in antiseptic and detergent solutions, acidic environments ([[w:pH|pH]] 4.5 for 70 days), and in environments at temperatures ranging from 24 °C (75.2 °F) to 32 °C (89.6 °F). However, the bacteria may be killed by the presence of ultraviolet light.<ref name="Joost 2018"/> Bacteria can enter the body through wounds, inhalation, and [[w:ingestion|ingestion]] of contaminated soil or water.<ref name="Joost 2018"/> Person-to-person transmission is extremely rare.<ref name="Yi 2014"/> Melioidosis is a recognised disease in animals including pigs, cats, dogs, goats, sheep, horses and others. Cattle, water buffalo, and crocodiles are considered to be relatively resistant to melioidosis despite their constant exposure to mud. Birds are also considered resistant to melioidosis although several cases had been reported in Australia and aquatic birds.<ref name="Gassiep 2020"/><ref name="Allen C 2005"/> [[w:zoonosis|Transmission from animals to humans]] is rare.<ref name="Joost 2018"/><ref name="Yi 2014"/> Inadequate [[w:Water chlorination|chlorination]] of water supply has been associated with ''B. pseudomallei'' outbreak in Northern and Western Australia.<ref name="Currie 2001">{{cite journal |author=Currie BJ |author2=Mayo M |author3=Anstey NM |author4=Donohoe P |author5=Haase A |author6=Kemp DJ |title=A cluster of melioidosis cases from an endemic region is clonal and is linked to the water supply using molecular typing of Burkholderia pseudomallei isolates |journal=American Society of Tropical Medicine and Hygiene |date=September 2001 |volume=65 |issue=3 |pages=177-179 |pmid=11561699 |doi=10.4269/ajtmh.2001.65.177 }}</ref><ref name="Inglis">{{cite journal |author=Inglis TJ |author2=Garrow SC |author3=Adams C |author4=Henderson M |author5=Mayo M |author6=Currie BJ |title=Acute melioidosis outbreak in Western Australia |journal=Epidemiology and Infections |date=December 1999 |volume=123 |issue=3 |pages=437-443 |pmid=10694154 |pmc= 2810777 | doi=10.1017/s0950268899002964 }}</ref> The were also several cases of where bacteria have also been found in unchlorinated water supply in rural Thailand.<ref name="Limmathurotsakul 2012">{{cite journal |author=Limmathurotsakul D |author2=Wongsuvan G |author3=Aanensen D |author4=Ngamwilai S |title=Melioidosis Caused by Burkholderia pseudomallei in Drinking Water, Thailand, 2012 |journal=Emerging Infectious Diseases |date=February 2014 |volume=20 |issue=2 |pages=265-268 |pmid=24447771 |pmc=3901481 |doi=10.3201/eid2002.121891 }}</ref> Based on the whole genome sequencing of the bacteria, the variety of the bacteria ''B. pseudomallei'' in Papua New Guinea is narrow due to limited movements of the indigenous people. This findings supports the hypothesis that humans play an important role in bacterial transmission.<ref>{{cite journal |author=Baker A |author2=Pearson T |author3=Price EP |author4=Dale J et al |title=Molecular Phylogeny of Burkholderia pseudomallei from a Remote Region of Papua New Guinea |journal=PLOS One |year=2011 |volume=6 |issue=3 |pages=e18343 |pmid=21483841 |pmc=3069084 |doi=10.1371/journal.pone.0018343 |bibcode=2011PLoSO...618343B }}</ref> ==Pathogenesis== {{fig|5 |Pathogenesis of melioidosis.svg |Ways of ''B. pseudomallei'' bacteria infecting human cells and blood stream. |attribution=Cerevisae, [https://creativecommons.org/licenses/by-sa/4.0/deed.en CC-BY-SA 4.0] |size=285px }} ''B. pseudomallei'' has the ability to infect various types of cells and to evade human immune responses. Bacteria first enter at a break in the skin or [[w:mucous membrane|mucous membrane]] and replicate in the epithelial cells. From there, they use [[w:flagella|flagella]]r motility to spread and infect various cell types.<ref name="Gassiep 2020"/> In the bloodstream, the bacteria can infect both [[w:phagocyte|phagocyte]]s and non-phagocytes.<ref name="Gassiep 2020"/> ''B. pseudomallei'' use their flagella to move near [[w:Host (biology)|host]] cells, then attach to the cells using various adhesion proteins, including the [[w:type IV pilus|type IV pilus]] protein PilA as well as adhesion proteins BoaA and BoaB.<ref name="Gassiep 2020"/> Additionally, adhesion of the bacteria partially depends on the presence of the host protein [[w:Coagulation factor II receptor|Protease-activated receptor-1]] which is present on the surface of [[w:endothelial cell|endothelial cell]]s, [[w:platelet|platelet]]s, and [[w:monocyte|monocyte]]s. Once bound, the bacteria enter host cells through [[w:endocytosis|endocytosis]], ending up inside an endocytic [[w:Vesicle (biology and chemistry)|vesicle]]. As the vesicle acidifies, ''B. pseudomallei'' uses its [[w:Bacterial secretion system#Type III secretion system (T3SS or TTSS)|Type 3 secretion system]] (T3SS) to inject effector proteins into the host cell, disrupting the vesicle and allowing the bacteria to escape into the host [[w:cytoplasm|cytoplasm]]. Within the host cytoplasm, the bacteria evade being killed by the host [[w:autophagy|autophagy]] using various T3SS effector proteins. The bacteria replicate in the host cytoplasm.<ref name="Joost 2018"/><ref name="Gassiep 2020"/> Inside the host cell, the bacteria move by inducing the polymerization of the host [[w:actin|actin]] behind them, propelling the bacteria forward.<ref name="Joost 2018"/> This actin-mediated motility is accomplished with the [[w:Secretion#Type V secretion system .28T5SS.29|autotransporter]] BimA which interacts with actin at the tail-end of the bacterium.<ref name="Joost 2018"/><ref name="Gassiep 2020"/> The bacteria that has BimABm [[w:allele|allele]] has higher possibility of causing neurological melioidosis, thus higher chance of death and residual disability to the host when compared to the bacteria that has BimABp variant.<ref>{{cite journal |author=Gora H |author2=Hasan T |author3=Smith S et al. |title=Melioidosis of the central nervous system; impact of the ''bimA_Bm'' allele on patient presentation and outcome |journal=Clinical Infectious Diseases |date=7 February 2022 |volume |issue= |pages=ciac111 |doi=10.1093/cid/ciac111 |pmid=35137005 }}</ref> Propelled by actin, the bacteria push against the host membrane, creating protrusions that extend into neighbouring cells. These protrusions cause neighboring cells to fuse, leading to the formation of [[w:multinucleated giant cells|multinucleated giant cells]] (MNGCs). When MNGCs lyse, they form plaques (a central clear area with a ring of fused cells) that provide shelter for the bacteria for further replication or [[w:latent infection|latent infection]]. This same process in infected neurons can allow bacteria to travel through nerve roots in the spinal cord and brain, leading to [[w:encephalomyelitis|inflammation of the brain and spinal cord]]. In addition to spreading from cell to cell, the bacteria can also spread through the bloodstream, causing sepsis. The bacteria can survive in [[w:antigen-presenting cell|antigen-presenting cell]]s and [[w:dendritic cell|dendritic cell]]s. Thus, these cells act as vehicles that transport the bacteria into the lymphatic system, causing widespread dissemination of the bacteria in the human body.<ref name="Joost 2018"/><ref name="Gassiep 2020"/> While ''B. pseudomallei'' can survive in phagocytic cells, these cells can kill ''B. pseudomallei'' by several mechanisms. Macrophages activated by [[w:interferon gamma|interferon gamma]] (IFN) have improved the killing of ''B. pseudomallei'' via the production of [[w:Nitric oxide synthase#iNOS|inducible nitric oxide synthase]]. Acidification of the endosome and degradation of the bacteria is also possible, however, the bacterial capsule and LPS makes ''B. pseudomallei'' resistant to lysosomal degradation. Once ''B. pseudomallei'' escapes into the host cytosol it can be recognized by [[w:pattern recognition receptor|pattern recognition receptor]]s such as [[w:NOD-like receptor|NOD-like receptor]]s, triggering the formation of the [[w:inflammasome|inflammasome]] and activation of [[w:caspase 1|caspase 1]], which induces death of the host cell by [[w:pyroptosis|pyroptosis]] and further activation of the immune system. Several systemic host defenses also contribute to the immune response. ''B. pseudomallei'' triggers both the [[w:complement system|complement system]] and [[w:coagulation cascade|coagulation cascade]], however the thick bacterial capsule prevent the action of the [[w:complement membrane attack complex|complement membrane attack complex]].<ref name="Joost 2018"/><ref name="Gassiep 2020"/> Additional elements of the immune system are activated by the host [[w:toll-like receptor|toll-like receptor]]s such as TLR2, TLR4, and TLR5 that recognize the [[w:pathogen-associated molecular pattern|conserved piece]]s of the bacteria such as LPS and flagella. This activation results in the production of [[w:cytokine|cytokine]]s such as [[w:Interleukin 1 beta|Interleukin 1 beta]] (IL-1β) and [[w:Interleukin 18|Interleukin 18]] (IL-18). IL-18 increases IFN production through [[w:natural killer cell|natural killer cell]]s while IL-1beta reduces the IFN production. These immune molecules drive the recruitment of other immune cells such as [[w:neutrophil|neutrophil]]s, [[w:dendritic cell|dendritic cell]]s, [[w:B cell|B cell]]s, and [[w:T cell|T cell]]s to the site of infection. T cells seem to be particularly important for controlling ''B. pseudomallei''; T cell numbers are increased in survivors, and low T cell numbers are associated with a high risk of death from melioidosis. Despite this, HIV infection is not a risk factor for melioidosis. Although macrophages show deregulated [[w:cytokine|cytokine]] responses in individuals with HIV infection, bacterial internalization and intracellular killing are still effective. People infected with ''B. pseudomallei'' may develop antibodies against the bacteria, and people that live in endemic areas tend to have antibodies in their blood that recognize ''B. pseudomallei''. However, the effectiveness of these antibodies at preventing melioidosis is unclear.<ref name="Joost 2018"/><ref name="Gassiep 2020"/> ''B. pseudomallei'' can remain latent in the human body for up to 29 years until it is reactivated during human [[w:immunosuppression|immunosuppression]] or stress response. However, the site of bacteria during latent infection and the mechanism by which they avoid immune recognition for years are both unclear. Amongst mechanisms suggested are: residing in the nucleus of the cell to prevent being digested, entering a stage of slower growth, antibiotic resistance, and genetic adaption to the host environment. [[w:Granuloma|Granuloma]]s (containing neutrophils, macrophages, lymphocytes, and multinucleated giant cells) formed at the infection site in melioidosis have been associated with latent infection in humans.<ref name="Joost 2018"/> ==Diagnosis== {{fig|6 |Bps close.JPG |Appearance of ''B. pseudomallei'' colonies on Ashdown's medium after four days of incubation. |attribution=Gavin Koh, [https://creativecommons.org/licenses/by-sa/4.0/ CC-BY-SA 4.0] |size=285px }} {{fig|7 |Immunofluorescent-stained sample of guinea pig tissue leads to positive diagnosis of melioidosis.png |Immunofluorescent microscopy showing the presence of rod-shaped ''B. pseudomallei'' (circled area) on a guinea pig tissue. |attribution=Dr. Biegeleisen, CDC, Public domain |size=285px }} {{fig|8 |Latex agglutination test positive for melioidosis.jpg |Right most slide showing positive latex agglutination for melioidosis. Latex agglutination uses latex coated antibodies to detect melioidosis antigens. |attribution=Neha Shrestha et al., [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0] |size=285px }} ===Culture=== Bacterial culture has 60% sensitivity in diagnosing melioidosis.<ref name="Limmathurotsakul 2010">{{cite journal |author=Limmathurotsakul D |author2=Jamsen K |author3=Arayawichanont A |author4=Simpson JA et al |title=Defining the True Sensitivity of Culture for the Diagnosis of Melioidosis Using Bayesian Latent Class Models |journal=PLOS One |date=30 August 2010 |volume=5 |issue=8 |pages=e12485 |pmid=20830194 | pmc=2932979 | doi=10.1371/journal.pone.0012485 }}</ref> ''B. pseudomallei'' is never part of human flora. Therefore, any growth of the bacteria is diagnostic of melioidosis. Other samples such as throat, rectal swabs, pus from abscesses, and sputum can also be used for culture.<ref name="Joost 2018"/> However, culture from CSF is difficult because in one case series, only 29% of the neuromelioidosis cases are culture positive.<ref name="Gassiep 2020"/> When bacteria do not grow from people strongly suspected of having melioidosis, repeated cultures should be taken as subsequent cultures can become positive.<ref name="Joost 2018"/> ''B. pseudomallei'' can be grown on any blood agar, [[w:MacConkey agar|MacConkey agar]], and agar containing antibiotics such as [[w:Ashdown's medium|Ashdown's medium]] (containing [[w:gentamicin|gentamicin]]),<ref name="Gassiep 2020"/> and Ashdown's broth (containing [[w:colistin|colistin]])<ref name="Currie 2015"/> for better isolation of ''B. pseudomallei'' from other types of bacteria.<ref name="Gassiep 2020"/> Agar plates for melioidosis should be incubated at 37 °C (98.6 °F) in air <ref name="Yi 2014"/> and inspected daily for four days. On the agar plates, ''B. pseudomallei'' forms creamy, [[w:Hemolysis (microbiology)|non-haemolytic]], colonies after 2 days of incubation. After 4 days of incubation, colonies appear dry and wrinkled.<ref name="Joost 2018"/> Colonies of ''B. pseudomallei'' that are grown on Francis medium (a modification of Ashdown medium with gentamicin concentration increased to 8&nbsp;mg/L and neutral red indicator replaced with 0.2% bromocresol purple) are yellow.<ref name="Francis 2006">{{cite journal |author=Francis A |author2=Aiyar S |author3=Yean Yean C |author4=Naing L |author5=Ravichandran M |title=An improved selective and differential medium for the isolation of Burkholderia pseudomallei from clinical specimens |journal=Diagnostic Microbiology and Infectious Disease |date=28 November 2005 |volume=55 |issue=2 |pages=95-99 |pmid=16626918 | doi=10.1016/j.diagmicrobio.2005.11.008}}</ref> For laboratories located outside endemic areas, ''Burkholderia cepacia'' selective agar can be used if Ashdown's medium is not available.<ref name="Yi 2014"/> It is important not misinterpret the bacterial growth as ''[[w:Pseudomonas|Pseudomonas]]'' or ''[[w:Bacillus|Bacillus]]'' spp. Other biochemical screening tools can also be used for detecting ''B. pseudomallei'', including the [[w:analytical profile index|API 20NE or 20E biochemical kit]] combined with Gram stain, [[w:oxidase test|oxidase test]], typical growth characteristics, and resistance to certain antibiotics of the bacteria.<ref name="Currie 2015"/> API 20NE biochemical kit is 99% sensitive in identifying ''B. pseudomallei''.<ref name="Gassiep 2020"/> Molecular methods such as 16S rDNA sequencing, multiplex [[w:polymerase chain reaction|polymerase chain reaction]] (PCR), and real-time PCR can also be used to identify ''B. pseudomallei'' in culture.<ref name="Joost 2018"/><ref name="Fook Koh 2012">{{cite journal |author=Koh SF |author2=Tay ST |author3=Sermswan R |author4=Wongratanacheewin S et al |title=Development of a multiplex PCR assay for rapid identification of Burkholderia pseudomallei, Burkholderia thailandensis, Burkholderia mallei and Burkholderia cepacia complex |journal=Journal of Microbiological Methods |date=September 2012 |volume=90 |issue=3 |pages=305-308 |pmid=22705921 |doi=10.1016/j.mimet.2012.06.002 }}</ref><ref name="Gee 2003">{{cite journal |author=Gee JE |author2=Sacchi CT |author3=Glass MB |author4=Dee BK et al |title=Use of 16S rRNA gene sequencing for rapid identification and differentiation of ''Burkholderia pseudomallei'' and ''B. mallei''|journal=Journal of Clinical Microbiology |date=October 2003 |volume=41 |issue=10 |pages=4547-4654 |pmid=14532197 |pmc= 254370 |doi=10.1128/jcm.41.10.4647-4654.2003 }}</ref><ref name="Wang 2022"/> Other bacterial genes such as fliC genes encoding flagellin, rpsU gene encoding for ribosomal protein, and TTS genes encoding Type III secretion systems has also been employed for detection. Another method of gene detection namely [[w:multiple displacement amplification|multiple cross displacement amplification]] for the bacterial TTS1 gene detection produces results within an hour.<ref name="Wang 2022">{{cite journal |author=Wang X |author2=Wang L |author3=Zhu H et al |title=Reliable detection of ''Burkholderia pseudomallei'' using multiple cross displacement amplification label-based biosensor|journal=BMC Microbiology |date=10 March 2022 |volume=22 |issue=1 |pages= |pmid=35272632 |pmc= PMC8908694 |doi=10.1186/s12866-022-02485-2 }}</ref> ===Hematological and biochemical tests=== General blood tests in people with melioidosis show low white blood cell counts (indicates infection), raised liver enzymes, increased [[w:bilirubin|bilirubin]] levels (indicates liver dysfunction), and raised urea and creatinine levels (indicates kidney dysfunction). [[w:Hypoglycemia|Low blood glucose]] and [[w:acidosis|acidosis]] predicts a poorer prognosis in those with melioidosis. However, other tests such as [[w:C-reactive protein|C-reactive protein]] and [[w:procalcitonin|procalcitonin]] levels are not reliable in predicting the severity of melioidosis infection.<ref name="Allen C 2005"/> ===Serological tests=== [[w:serology|Serological test]]s such as indirect [[w:haemagglutination assay|haemagglutination]] assay (IHA) have been used to detect the presence of antibodies against ''B. pseudomallei''. However, different groups of people have widely different levels of antibodies, so interpretation of these tests depends on location. In Australia, less than 5% of people have ''B. pseudomallei'' antibodies, so the presence of even relatively low amounts of antibody is unusual and could suggest melioidosis. In Thailand, many people have antibodies against ''B. pseudomallei'' so the diagnosis of melioidosis should not be reliant entirely on the serological tests done in endemic areas.<ref name="Joost 2018"/><ref name="Currie 2015"/> Indirect immunofluorescent test (IFAT) uses either ''B. pseudomallei'' or ''B. thailandensis'' antigens to look for the total number of antibodies in human serum. Using IFAT is labour intensive and is not used in large scale investigations.<ref name="Suat Moi 2013">{{cite journal |author=Puah SM |author2=Puthucheary SD |author3=Chua KH |title=Potential Immunogenic Polypeptides of Burkholderia pseudomallei Identified by Shotgun Expression Library and Evaluation of Their Efficacy for Serodiagnosis of Melioidosis |journal=International Journal of Medical Sciences |date=2013 |volume=10 |issue=5 |pages=539-547 |pmid=23532805 |pmc=3607239 |doi=10.7150/ijms.5516 }}</ref> Antigen detect tests allow rapid detection of melioidosis. Examples of antigen detection tests are: latex agglutination test and [[w:ELISA|ELISA]]. [[w:Latex agglutination|Latex agglutination]] uses antibodies coated on latex beads to detect ''B. pseudomallei'' antigens in solid or liquid media, although not all the assays can detect different species of ''Burkholderia''.<ref name="Amornchai 2007">{{cite journal |author=Amornchai P |author2=Chierakul W |author3=Wuthiekanun V |author4=Mahakhunkijcharoen Y et al |title=Accuracy of Burkholderia pseudomallei Identification Using the API 20NE System and a Latex Agglutination Test |journal=Journal of Clinical Microbiology |date=2007 |volume=45 |issue=11 |pages=3774-3776 |pmid=17804660 | pmc=2168515 | doi=10.1128/JCM.00935-07 }}</ref> Latex agglutination is useful in screening for suspected ''B. pseudomallei'' colonies.<ref name="Joost 2018"/> IgG and IgM ELISAs has been used to detect lipopolysaccharide (LPS) antigens of ''B. pseudomallei'', but plagued with low sensitivity.<ref name="KP Lau 2015">{{cite journal |author=KP Lau S |author2=Sridhar S |author3=Chun Ho C et al |title=Laboratory diagnosis of melioidosis: Past, present and future |journal=Experimental Biology and Medicine |date=June 2015 |volume=240 |issue=6 |pages=742-751 |pmid=25908634 |pmc=PMC4935216 |doi=10.1177/1535370215583801 }}</ref> Commercial ELISA kits for melioidosis no longer available in the market due to low sensitivity to human antibodies detection.<ref name="Gassiep 2020"/> Nevertheless, antigen detection tests may be useful in severely ill patients because the bacterial load is high enough for detection. Other methods of antigen detection such as [[w:direct fluorescent antibody|direct immunofluorescence]], antibody-sandwich ELISAs, and [[w:lateral flow test|lateral flow immunoassays]] using [[w:monoclonal antibody|monoclonal antibody]].<ref name="KP Lau 2015"/> ===Microscopy=== By microscopy, ''B. pseudomallei'' is seen as [[w:Gram stain|gram-negative]] and rod-shaped, with a bipolar staining similar in appearance to a safety pin. Bacteria can sometimes be seen directly in clinical samples from infected people; however, identification by light microscopy is neither [[w:Sensitivity and specificity|specific]] nor [[w:Sensitivity and specificity|sensitive]]. [[w:Immunofluorescence microscopy|Immunofluorescence microscopy]] is highly specific for detecting bacteria directly from clinical specimens, but has less than 50% sensitivity.<ref name="Joost 2018"/><ref name="Currie 2015"/> ===Imaging=== Various imaging modalities can also help with the diagnosis of melioidosis. In acute melioidosis with the spreading of the bacteria through the bloodstream, the chest X-ray shows multifocal nodular lesions. It may also show merging nodules or [[w:cavitary pneumonia|cavitations]]. For those with acute melioidosis without the spread to the bloodstream, chest x-ray most commonly shows upper lobe [[w:Pulmonary consolidation|consolidation]] or cavitations.<ref name="Gassiep 2020"/> In chronic melioidosis, the slowly progressing of upper lobe consolidation of the lungs resembles tuberculosis.<ref name="Gassiep 2020"/> For abscesses located in other parts of the body apart from the lungs, especially in the liver and spleen, [[w:computed tomography|CT]] scan has higher sensitivity when compared with an ultrasound scan. In liver and splenic abscesses, an ultrasound scan shows "target-like" lesions while CT scan shows "honeycomb sign" (abscess with loculations separated by thin septa) in liver abscesses.<ref name="Gassiep 2020"/> For melioidosis involving the brain, MRI have higher sensitivity than a CT scan in diagnosing the lesion. MRI shows ring-enhancing lesions for brain melioidosis.<ref name="Gassiep 2020"/> ==Prevention== Melioidosis is a notifiable disease in Australia<ref name="Gassiep 2020"/> which enables the country to monitor disease burden and contain outbreaks. On the other hand, melioidosis is only a notifiable condition in Thailand since June 2016.<ref name="Gassiep 2020"/><ref name="Hantrakun 2019">{{cite journal |author=Hantrakun V |author2=Kongyu S |author3=Klaytong P |author4=Rongsumlee S et al |title=Clinical Epidemiology of 7126 Melioidosis Patients in Thailand and the Implications for a National Notifiable Diseases Surveillance System |journal= Open Forum Infectious Diseases |date=19 November 2019 |volume=6 |issue=12 |pages=ofz498 |pmid=32083145 | pmc=7020769 | doi=10.1093/ofid/ofz498 }}</ref> However, until recently, the official notification system in Thailand has significantly underestimated the incidence of culture-positive melioidosis and its mortality.<ref name="Hantrakun 2019"/> Nevertheless, Australia also embarked on awareness campaigns to increase the community's understanding of the disease.<ref name="Gassiep 2020"/> In the United Kingdom, where reporting by laboratories is mandatory, 41.3% of cases imported since 2010 were not notified.<ref name="Conor 2020">{{cite journal |author=Conor CO |author2=Kenna D |author3=Walsh A et al |title=Imported melioidosis in the United Kingdom: Increasing incidence but continued under-reporting |journal= Journal of Clinical Microbiology |date=October 2020 |volume=7 |issue= |pages=100051 |pmid= | pmc= | doi=10.1016/j.clinpr.2020.100051 }}</ref> In the United States, lab workers can handle clinical specimens of ''B. pseudomallei'' under [[w:BSL-2|BSL-2]] conditions, while mass production of such organisms requires [[w:BSL-3|BSL-3]] precautions.<ref>{{cite book |author=Centers for Disease Control and Prevention |title=Biosafety in Microbiological and Biomedical Laboratories |publisher=National Institutes of Health |location=Atlanta, Georgia |year=2009 |edition=5th|isbn=9780160850424 }}</ref> On the other hand, in other endemic areas where the ''B. pseudomallei'' samples were handled less stringently, there has been no confirmed laboratory-acquired infection reported. This phenomenon may show that the risk of infection with ''B. pseudomallei'' is less than a typical biohazard type 3 agent.<ref name="Dance 2017">{{cite journal |author=Dance DAB |author2=Limmathurotsakul D |author3=Currie BJ |title=Burkholderia pseudomallei: Challenges for the Clinical Microbiology Laboratory—a Response from the Front Line |journal= Journal of Clinical Microbiology |date=February 2017 |volume=55 |issue=3 |pages=980-982 |pmid=28232503 | pmc=5328468 | doi=10.1128/JCM.02378-16 }}</ref> There are also several cases of hospital-acquired infection of melioidosis.<ref name="Joost 2018"/> Therefore, healthcare providers are recommended to practice hand hygiene and [[w:universal precautions|universal precautions]].<ref name="Joost 2018"/> Large-scale water chlorination has been successful at reducing ''B. pseudomallei'' in the water in Australia.<ref name="Howard 2003">{{cite journal |author=Howard K |author2=Inglis TJJ |title=The effect of free chlorine on Burkholderia pseudomallei in potable water |journal= Water Research |date=February 2017 |volume=37 |issue=18 |pages=4425-4432 |pmid=14511713 | doi=10.1016/S0043-1354(03)00440-8 }}</ref><ref name="Joost 2018"/> In middle to low-income countries, water should be boiled before consumption.<ref name="Joost 2018"/> In high income countries, water could be treated with ultraviolet light for those at risk of contracting melioidosis.<ref name="McRobb E 2013">{{cite journal |author=McRobb E |author2=Kaestli M |author3=Mayo M |author4=Price EP et al |title= Melioidosis from Contaminated Bore Water and Successful UV Sterilization |journal= American Journal of Tropical Medicine and Hygiene |date=7 August 2013 |volume=89 |issue=2 |pages=367-368 |pmid=23751401 | pmc=3741262 | doi=10.4269/ajtmh.13-0101 }}</ref><ref name="Joost 2018"/> Those who are at high risk of contact with the bacteria should wear protective gear (such as boots and gloves) during work.<ref name="Joost 2018"/> Those staying in endemic areas should avoid direct contact with soil, and outdoor exposure to heavy rain or dust clouds. Bottled water or boiled water are preferred as drinking water.<ref name="Limmathurotsakul 2013">{{cite journal |author=Limmathurotsakul D |author2=Kanoksil M |author3=Wuthiekanun V et al |title= Activities of daily living associated with acquisition of melioidosis in northeast Thailand: a matched case-control study |journal= PLOS Neglected Tropical Diseases |date= 2013 |volume=7 |issue=2 |pages=e2072 |pmid=23437412 | pmc=3578767 | doi=10.1371/journal.pntd.0002072 }}</ref><ref name="Joost 2018"/> A study conducted from 2014 to 2018, however showed no significant differences on whether behavioural changes can reduce the risk of contracting melioidosis. Modification of behavioural changes or more frequent interventions may be needed to ensure a definite reduction in risk of getting melioidosis.<ref name="Suntornsut 2021">{{cite journal |author=Suntornsut P |author2=Teparrukkul P |author3=Wongsuvan G et al |title= Effectiveness of a multifaceted prevention programme for melioidosis in diabetics (PREMEL): A stepped-wedge cluster-randomised controlled trial |date= June 2021|journal=PLOS Neglected Tropical Diseases |volume=15 |issue=6 |pages=e0009060 |pmid=34170931 | pmc=PMC8266097 | doi=10.1371/journal.pntd.0009060 }}</ref> ===Antibiotic prophylaxis=== Administering cotrimoxazole three times a week throughout a wet season for dialysis patients has no obvious benefit of preventing melioidosis. Besides, high cost and side effects of this drug limits its use to only those with high risk of getting melioidosis.<ref name="Chau 2018">{{cite journal |author=Chau KWT |author2=Smith S |author3=Kang K et al |title=Antibiotic Prophylaxis for Melioidosis in Patients Receiving Hemodialysis in the Tropics? One Size Does Not Fit All |journal= The American Journal of Tropical Medicine and Hygiene |date=16 July 2018 |volume=99 |issue=3 |pages=597-600 |pmid=30014827 |pmc=PMC6169155 |doi=10.4269/ajtmh.18-0421 }}</ref> After exposure to ''B. pseudomallei'' (particularly following a laboratory accident, penetrating injuries, exposure of mouth and eyes to contaminated materials or aerosols), treatment with antibiotics is only given when in highly selected individuals after weighing the risk of adverse effects of the drugs against the benefits from contracting melioidosis. Cotrimoxazole can be used in this context. Alternatively, [[w:co-amoxiclav|co-amoxiclav]] and [[w:doxycycline|doxycycline]] can be used for those who are intolerant to co-trimoxazole. Low-risk individuals would receive frequent monitoring instead.<ref name="Lipsitz 2012">{{cite journal |author=Lipsitz R |author2=Garges S |author3=Aurigemma R et al |title=Workshop on treatment of and postexposure prophylaxis for Burkholderia pseudomallei and B. mallei Infection, 2010 |journal= Emerging Infectious Diseases |date=18 December 2012 |volume=18 |issue=12 |pages=e2 |pmid=23171644 |pmc=3557896 |doi=10.3201/eid1812.120638 }}</ref> ===Vaccination=== {{Further|w:Burkholderia pseudomallei#Vaccine candidates}} Several vaccine candidates have been tested in animal models. Nevertheless, no vaccine candidates have been tried in humans. Major hurdles of the vaccines are limited efficacy in animal models, establishing the best method of vaccine administration in humans and logistical and financial issues in establishing human trials in endemic areas.<ref name="Gassiep 2020"/> ==Treatment== The treatment of melioidosis is divided into two stages: an intravenous intensive phase and an eradication phase to prevent recurrence. The choice of antibiotics depends upon the susceptibility of the bacteria to various antibiotics. ''B. pesudomallei'' are generally susceptible to ceftazidime, meropenem, imipenem, and co-amoxiclav. These drugs generally kill bacteria. ''B. pseudomallei'' is also susceptible to doyxcycline, chloramphenicol, and co-trimoxazole. These drugs generally inhibit the growth of the bacteria. However, the bacteria are resistant to penicillin, ampicillin, 1st and 2nd generation [[w:cephalosporin|cephalosporin]], gentamicin, streptomycin, tobramycin, macrolides, and polymyxins.<ref name="Joost 2018"/> On the other hand, 86% of the ''B. pseudomallei'' isolates from the region of [[w:Sarawak|Sarawak]], [[w:Malaysia|Malaysia]] are susceptible to gentamicin and this has not been found elsewhere in other parts of the world.<ref name="Podin 2014">{{cite journal |author=Podin Y |author2=Sarovich DS |author3=Price EP et al |title=Burkholderia pseudomallei Isolates from Sarawak, Malaysian Borneo, Are Predominantly Susceptible to Aminoglycosides and Macrolides |journal= Antimicrobial Agents and Chemotherapy |date=January 2014 |volume=58 |issue=1 |pages=162-166 |pmid=24145517 | pmc=3910780 | doi=10.1128/AAC.01842-13 }}</ref> Prior to 1989, the standard treatment for acute melioidosis was a three-drug combination of [[w:chloramphenicol|chloramphenicol]], [[w:co-trimoxazole|co-trimoxazole]] and [[w:doxycycline|doxycycline]]; this regimen is associated with a mortality rate of 80% and is no longer used unless no other alternatives are available.<ref name="CAZ">{{cite journal |author=White NJ |author2=Dance DA |author3=Chaowagul W |author4=et al | title = Halving of mortality of severe melioidosis by ceftazidime | journal =The Lancet | year = 1989 | volume = 2 | issue = 8665 | pages = 697&ndash;701 | pmid = 2570956 | doi = 10.1016/S0140-6736(89)90768-X }}</ref> All three drugs are bacteriostatic (they stop the bacterium from growing, but do not kill it) and the action of co-trimoxazole antagonizes both chloramphenicol and doxycycline.<ref name="Dance2006">{{cite journal |author=Dance DA |author2=Wuthiekanun V |author3=Chaowagul W |author4=White NJ |title=Interactions in vitro between agents used to treat melioidosis |journal=Journal of Antimicrobial Chemotherapy |year=1989 |volume=24 |pages=311&ndash;316 |pmid=2681117 |doi=10.1093/jac/24.3.311 |issue=3 }}</ref> ===Intensive phase=== [[w:Intravenous|Intravenous]] [[w:ceftazidime|ceftazidime]] is the current drug of choice for treatment of acute melioidosis and should be administered for at least 10 to 14 days. [[w:Meropenem|Meropenem]], [[w:imipenem|imipenem]] and the [[w:cefoperazone|cefoperazone]]-[[w:sulbactam|sulbactam]] combination (Sulperazone) are also effective.<ref name="Joost 2018"/> Intravenous amoxicillin-clavulanate ([[w:co-amoxiclav|co-amoxiclav]]) may be used if none of the above four drugs is available;<ref name="Joost 2018"/> co-amoxiclav prevents death from melioidosis as well as ceftazidime.<ref name="Bennett 2015">{{cite book |editor=Bennett JE; Dolin R; Blaser MJ |title=Mandell, Douglas, and Bennett's Principles and Practice of Infectious Diseases |chapter=''Burkholderia pseudomallei'' and ''Burkholderia mallei'': Melioidosis and Glanders |date=2015 |publisher=Elsevier |isbn=978-1-4557-4801-3 |pages=2541–2549|doi=10.1016/B978-1-4557-4801-3.00223-X|author=Currie BJ |edition=8th }}</ref> Co-amoxiclav is also used if patient has allergy towards [[w:Sulfonamide (medicine)|sulfonamide]], unable to tolerate co-trimaxazole, in pregnant patients or in children. High dose of co-amoxiclav (20 mg/kg for amoxicillin and 5mg/kg for clavulanate) is recommended to prevent treatment failures.<ref name="Cheng 2008">{{cite journal |author=Cheng AC |author2=Chierakul W |author3=Chaowagul W et al |title=Consensus Guidelines for Dosing of Amoxicillin-Clavulanate in Melioidosis |journal= American Journal of Tropical Medicine and Hygiene |date=February 2008 |volume=78 |issue=2 |pages=208-209 |pmid=18256414 | pmc=PMC3034162 | doi=10.4269/ajtmh.2008.78.208 }}</ref><ref name="Suputtamongkol 1994">{{cite journal |author=Suputtamongkol Y |author2=Rajchanuwong A |author3=Chaowagul W et al |title=Ceftazidime vs. amoxicillin/clavulanate in the treatment of severe melioidosis |journal= Clinical Infectious Diseases |date=November 1994 |volume=19 |issue=5 |pages=846-853 |pmid=7893868 | pmc= | doi=10.1093/clinids/19.5.846 }}</ref> Intravenous [[w:antibiotic|antibiotic]]s are given for a minimum of 10 to 14 days. The median fever clearance time in melioidosis is 9 days.<ref name="Joost 2018"/> The treatment duration is in accordance with Darwin melioidosis treatment guidelines where there is low rate of [[w:recrudescence|recrudescence]] and [[w:relapse|relapse]].<ref name="Sullivan 2020">{{cite journal |author=Sullivan RP |author2=Marshall CS |author3=Anstey NM |author4=Ward L |author5=Currie BJ |title=2020 Review and revision of the 2015 Darwin melioidosis treatment guideline; paradigm drift not shift |journal= PLOS Neglected Tropical Diseases |date=28 September 2020 |volume=14 |issue=9 |pages=e0008659 |pmid=32986699 | pmc=7544138 | doi=10.1371/journal.pntd.0008659 }}</ref> Meropenem is the preferred antibiotic therapy for neurological melioidosis and those with [[w:sepsis|septic shock]] admitted into [[w:intensive care unit|intensive care unit]]s. Co-trimoxazole is recommended in addition to ceftazidime for neurological melioidosis, osteomyelitis, septic arthritis, skin and gastrointestinal infection, and deeply seated abscess. For deep-seated infections such as abscesses of internal organs, osteomyelitis, septic arthritis, and neurological melioidosis, the duration of antibiotics given should be longer (up to 4 to 8 weeks). The time taken for the fever to be resolved can be more than 10 days in those with deep-seated infection. According to the 2020 Revised Royal Darwin Hospital Guideline, the dosage for intravenous ceftazidime is 2g 6-hourly in adults (50&nbsp;mg/kg up to 2g in children less than 15 years old). The dosage for meropenem is 1g 8-hourly in adults (25&nbsp;mg/kg up to 1g in children).<ref name="Sullivan 2020"/> Acquired resistance to ceftazidime, carbapenems, and co-amoxiclav is rare in the intensive phase but resistance to cotrimoxazole during eradication therapy is technically difficult to assess.<ref name="Dance 2014"/> There are no differences between using cefoperazone/sulbactam or ceftazidime to treat melioidosis as both shows similar death rates and disease progression following treatment. However, data are lacking to recommend cefoperazone/sulbactam usage.<ref name="Dance 2014">{{cite journal |author=Dance D |title=Treatment and prophylaxis of melioidosis |journal=International Journal of Antimicrobial Agents |date=April 2014 |volume=43 |issue=4 |pages=310-318 |pmid=24613038 |doi=10.1016/j.ijantimicag.2014.01.005}}</ref><ref name="Apisarnthanarak 2002">{{cite journal |author1=Apisarnthanarak A |author2= Little JR |title=The role of cefoperazone-sulbactam for treatment of severe melioidosis |journal=Clinical Infectious Diseases |date=1 March 2002 |volume=34 |issue=5 |pages=721-722 |pmid=11823963 |doi=10.1086/338722}}</ref> For those with kidney impairment, the dosage of ceftazidime, meropenem, and co-trimoxazole should be lowered.<ref name="Currie 2015"/> Once the clinical condition improved, meropenem can be switched back to ceftazidime.<ref name="Joost 2018"/> ===Eradication phase=== Following the treatment of the acute disease, eradication treatment with [[w:co-trimoxazole|co-trimoxazole]] is the drug of choice and should be used for 3 months (12 weeks) as all-cause mortality was lower in 12 weeks group when compared to those receiving treatment for 20 weeks.<ref name="Anunnatsiri 2021">{{cite journal |author1=Anunnatsiri S |author2=Chaowagul W |author3=Teparrukkul P |title=A Comparison Between 12 Versus 20 Weeks of Trimethoprim-sulfamethoxazole as Oral Eradication Treatment for Melioidosis: An Open-label, Pragmatic, Multicenter, Non-inferiority, Randomized Controlled Trial |journal= Clinical Infectious Diseases |date=2021 |volume=73 |issue=11 |pages=e3627-e3633 |pmid=32725199 | pmc=PMC8662794 | doi=10.1093/cid/ciaa1084 }}</ref> For those with neurological melioidosis and [[w:osteomyelitis|osteomyelitis]], drugs should be given for more than 6 months. Co-amoxiclav and [[w:doxycycline|doxycycline]] are drugs of second choice. Co-trimoxazole should not be used in those with [[w:glucose-6-phosphate dehydrogenase|glucose-6-phosphate dehydrogenase]] deficiency as it can cause [[w:haemolytic anemia|haemolytic anemia]]. However, in Thailand, usage of co-trimoxazole does not accompany G6PD screening.<ref name="Joost 2018"/> Other side effects such as rash, [[w:hyperkalemia|hyperkalemia]], renal dysfunction, and gastrointestinal symptoms should prompt the reduction of co-trimoxazole doses. [[w:Chloramphenicol|Chloramphenicol]] is no longer routinely recommended for this purpose. Co-amoxiclav is an alternative for patients unable to take co-trimoxazole and [[w:doxycycline|doxycycline]] (e.g. pregnant women and children under the age of 12), but is not as effective and has a higher relapse rate. Single-agent treatment with [[w:fluoroquinolone|fluoroquinolone]] (e.g., [[w:ciprofloxacin|ciprofloxacin]]) or doxycycline for the oral eradication phase is ineffective.<ref name="Joost 2018"/> In Australia, co-trimoxazole is used with children and pregnant mothers after the first 12 weeks of pregnancy. Meanwhile, in Thailand, co-amoxiclav is the drug of choice for children and pregnant women.<ref name="Joost 2018"/> ''B. pseudomallei'' rarely acquires resistance when co-amoxiclav is used.<ref name="Dance 2014"/> The dosing regimen for co-trimoxazole (trimethoprim/sulfamethoxazole) in eradication phase is 6/30&nbsp;mg/kg, up to maximum 240/1200&nbsp;mg in children, 240/1200&nbsp;mg in adults weighing 40 to 60&nbsp;kg, and 320/1600&nbsp;mg in adults weighing more than 60&nbsp;kg, taken orally every 12 hours.<ref name="Sullivan 2020"/> In both Thailand and Australia, co-trimoxazole is taken together with [[w:folic acid|folic acid]] (0.1&nbsp;mg/kg up to 5&nbsp;mg in children).<ref name="Joost 2018"/><ref name="Sullivan 2020"/> There are also cases where melioidosis is successfully treated with co-trimoxazole for 3 months without going through intensive therapy provided that there is only skin manifestations without the involvement of internal organs or sepsis.<ref name="Joost 2018"/> Resistance to cotrimoxazole is rare in Asia.<ref name="Dance 2014 resistance">{{cite journal |author1=Dance DAB |author2=Davong V |author3=Soeng S |author4=Phetsouvanh R |author5=Newton PN |author6=Turner P |title=Trimethoprim/sulfamethoxazole resistance in Burkholderia pseudomallei |journal= International Journal of Antimicrobial Agents |date=October 2014 |volume=44 |issue=4 |pages=368-369 |pmid=25245211 | pmc=4195405 | doi=10.1016/j.ijantimicag.2014.06.003 }}</ref> Besides that, it is difficult to determine the resistance reliably because resistance to cotrimoxazole is defined when [[w:minimum inhibitory concentration|minimum inhibitory concentration]] (MIC) of more than 4 mg/L is required to completely inhibit the growth of 80% of the bacteria (80% inhibition point). Interpretation of 80% inhibition point is subjective and prone to human error.<ref name="Saiprom 2015">{{cite journal |author1=Saiprom N |author2=Amornchai P |author3=Wuthiekanun V et al |title=Trimethoprim/sulfamethoxazole resistance in clinical isolates of Burkholderia pseudomallei from Thailand |journal= International Journal of Antimicrobial Agents |date=May 2015 |volume=45 |issue=5 |pages=557-559 |pmid=25758020 | pmc=4537509 | doi=10.1016/j.ijantimicag.2015.01.006 }}</ref> In 2021, European Committee on Antimicrobial Susceptibility Testing (EUCAST) released a new guideline on interpreting the susceptibility of ''B pseudomallei'' towards various antibiotics on [[w:antibiotic sensitivity testing|disc susceptibility testing]]. The new guideline includes "S" for susceptible organism, "I" for susceptible organism only after increased exposure (when dosage or concentration of the drug increases) and "R" for resistant organism.<ref name="Dance 2021">{{cite journal |author1=Dance DAB |author2=Wuthiekanun P |author3=Baird W et al |title=Interpreting Burkholderia pseudomallei disc diffusion susceptibility test results by the EUCAST method |journal= Clinical Microbiology and Infection |date=June 2021 |volume=27 |issue=6 |pages=827-829 |pmid=33636339 | pmc= | doi=10.1016/j.cmi.2021.02.017 }}</ref> ===Surgery=== Surgical drainage is indicated for single, large abscesses in the liver, muscle, and prostate. However, for multiple abscesses in the liver, spleen, and kidney, surgical drainage may not be possible or necessary. For septic arthritis, [[w:arthrotomy|arthrotomy]] washout and drainage are required. Surgical [[w:debridement|debridement]] may be necessary.<ref name="Joost 2018"/> For those with [[w:mycotic aneurysm|mycotic aneurysm]], urgent surgery is required for prosthetic vascular grafts. Lifelong therapy with co-trimoxazole may be needed for those with prosthetic vascular grafts according to a review of case reports in 2005.<ref name="Hong Low 2005">{{cite journal |author=Low JGH |author2=Quek AML |author3=Sin YK |author4=Ang BSP |title=Mycotic aneurysm due to Burkholderia pseudomallei infection: case reports and literature review |journal= Clinical Infectious Diseases |date=January 2005 |volume=40 |issue=1 |pages=193-198 |pmid=15614712 | pmc= | doi=10.1086/426590 }}</ref> Other abscesses rarely need to be drained because most resolve with antibiotic treatment.<ref name="Joost 2018"/> Prostate abscess may require routine imaging. Antibiotics treatment for prostatic abscess may be enough except for abscesses more than 10 to 15 mm where surgical drainage is required.<ref name="Kozlowska 2018">{{cite journal |author=Kozlowska J |author2=Smith S |author3=Roberts J |author4=Pridgeon S |author5=Hanson J |title=Prostatic Abscess due to Burkholderia pseudomallei: Facilitating Diagnosis to Optimize Management |journal= American Journal of Tropical Medicine |date=January 2018 |volume=98 |issue=1 |pages=227-230 |pmid=29141724 | pmc= 5928742 | doi=10.4269/ajtmh.17-0633 }}</ref><ref name="Ng 2009">{{cite journal |author=Ng TH |author2=How SH |author3=Amran AR |author4=Razali MR |author5=Kuan YC |title=Melioidotic prostatic abscess in Pahang |journal= American Journal of Tropical Medicine |date=April 2009 |volume=50 |issue=4 |pages=385-389 |pmid=19421682 | pmc= | doi=10.4269/ajtmh.17-0633 }}</ref><ref name="Morse 2009">{{cite journal |author=Morese LP |author2=Moller CCB |author3=Harvey E |author4=Ward L et al |title=Prostatic Abscess Due to Burkholderia pseudomallei: 81 Cases From a 19-Year Prospective Melioidosis Study |journal= The Journal of Urology |date=1 August 2009 |volume=182 |issue=2 |pages=542-547 |pmid=19524969 | pmc= | doi=10.1016/j.juro.2009.04.010 }}</ref> ===Others=== Several immunomodulating therapies are suggested to boost the human body immune function against the bacteria because the pathogenesis of melioidosis is thought to be contributed by defects in [[w:neutrophil|neutrophils]].<ref name="Joost 2018"/> The Royal Darwin Hospital 2014 guidelines recommended [[w:granulocyte colony-stimulating factor|granulocyte colony-stimulating factor]] (G-CSF) as immunomodulating therapy for those with septic shock at 300ug daily as soon as the bacteriological laboratory flag the culture as possibly ''Burkholderia pseudomallei''. The main contraindication of starting (G-CSF) is a heart event. The G-CSF is continued for ten days depends on clinical response or a contraindication develops such as white cell count greater than >50,000 X106/litre.<ref name="Sullivan 2020"/> Anti-PDI ([[w:programmed cell death|programmed cell death]]) agents could be useful in melioidosis treatment especially for those with septic shock. This is because ''Burkholderia pseudomallei'' bacteria increases the expression of PDI-1 that regulates and inhibits the formation of T-cells that are essential for fighting against melioidosis.<ref name="Buddhisa 2015">{{cite journal |author=Buddhisa S |author2=Rinchai D |author3=Ato M et al |title=Programmed Death Ligand 1 on Burkholderia pseudomallei–Infected Human Polymorphonuclear Neutrophils Impairs T Cell Functions |journal= The Journal of Immunology |date=May 2015 |volume=194 |issue=9 |pages=4413-4421 |pmid=25801435 | pmc=3910780 | doi=10.4049/jimmunol.1402417 }}</ref> ==Prognosis== In well-resourced settings, where the disease can be detected and treated early, the risk of death is 10%. In resource-poor settings, the risk of death from the disease is more than 40%.<ref name="Joost 2018"/> Recurrent melioidosis can occur either due to re-infection or relapse after the completion of eradication therapy. Re-infection is due to a new strain of ''B. pseudomallei'' bacteria. Meanwhile, relapse is due to failure to clear infections after the eradication therapy. Recurrent melioidosis is rare since 2014 due to improved antibiotic therapy and prolongation of the intensive phase of therapy as evident in Darwin Prospective Melioidosis Study.<ref name="Sarovich 2014">{{cite journal |author=Sarovich DS |author2=Ward L |author3=Price EP |author4=Mayo M et al |title=Recurrent Melioidosis in the Darwin Prospective Melioidosis Study: Improving Therapies Mean that Relapse Cases Are Now Rare |journal=Journal of Clinical Microbiology |date=February 2014 |volume=52 |issue=2 |pages=650-653 |pmid=24478504 |pmc=3911345 | doi=10.1128/JCM.02239-13 }}</ref> On the other hand, [[w:recrudescence|recrudescence]] are those who present with symptoms during the eradication therapy. Recrudescence rates may be improved by ensuring adherence to a full course of eradication therapy e.g. by reducing self-discharge [[w:against medical advice|against medical advice]].<ref name="Pitman 2015">{{cite journal |author=Pitman MC |author2=Luck T |author3=Marshall CS |author4=Anstey NM et al |title=Intravenous Therapy Duration and Outcomes in Melioidosis: A New Treatment Paradigm |journal=PLOS Neglected Tropical Diseases |date=26 March 2015 |volume=9 |issue=4 |pages=e0003737 |pmid=25811783 |pmc=4374799 | doi=10.1371/journal.pntd.0003586 }}</ref> Underlying medical conditions such as diabetes mellitus, chronic kidney disease, and cancer can worsen the long-term survival and disability of those who recover from infection. One of the complications of melioidosis is [[w:encephalomyelitis|encephalomyelitis]]. It can cause quadriparesis (muscle weakness in all the limbs), partial flaccid paraparesis (muscle weakness of both legs), or foot drop. For those with previous melioidosis-associated bone and joint infections, complications such as sinus tract infection, bone and joint deformities with limited range of motion can occur.<ref name="Joost 2018"/> ==Epidemiology== {{fig|9 |Melioidosis world map distribution.svg |Number of deaths by each country due to melioidosis in 2018. |attribution=Cerevisae, [https://creativecommons.org/licenses/by-sa/4.0/deed.en CC-BY-SA 4.0] |size=285px }} Melioidosis is an understudied disease that remains endemic in developing countries. In 2015, the International Melioidosis Society was formed to raise awareness of the disease.<ref name="Joost 2018"/> In 2016, a [[w:statistical model|statistical model]] was developed which predicted that the number is 165,000 cases per year with 138,000 of those occurring in East and South Asia and the Pacific.<ref>{{cite journal |author=Limmathurotsakul D |author2=Golding N |author3=Dance DA |author4=Messina JP |author5=Pigott DM |author6=Moyes CL |author7=Rolim DB |author8=Bertherat E |author9=Day NP |author10=Peacock SJ |author11=Hay SI |title=Predicted global distribution of Burkholderia pseudomallei and burden of melioidosis |journal=Nature Microbiology |date=11 January 2016 |volume=1 |issue=1 |pages=15008 |doi=10.1038/nmicrobiol.2015.8 |pmid=26877885 |pmc=4746747 }}</ref> In approximately half of those cases (54% or 89,000), people will die.<ref name="Joost 2018"/> Under-reporting is a common problem as only 1,300 cases were reported worldwide since 2010, which is less than 1% of the projected incidence based on the modelling.<ref name="Joost 2018"/> Lack of laboratory diagnostic capabilities and lack of disease awareness amongst health care providers also causes underdiagnosis. Even if bacterial cultures show positive result for ''B. pesudomallei'', they can be discarded as contaminants especially in laboratories in non-endemic areas.<ref name="Joost 2018"/> In 2015, it was estimated that the yearly [[w:disability-adjusted life year|disability-adjusted life year]] (DALY) was 84.3 per 100,000 people. As of 2022, melioidosis is not included in the WHO list of [[w:neglected tropical diseases|neglected tropical diseases]].<ref>{{cite journal |author=Birnie E |author2=Virk HS |author3=Savelkoel J |author4=Spijker R |author5=Bertherat E |title=Global burden of melioidosis in 2015: a systematic review and data synthesis |journal=The Lancet Infectious Diseases |date=August 2019 |volume=19 |issue=8 |pages=892-902 |doi=10.1016/S1473-3099(19)30157-4 |pmid=31285144 |pmc=6867904 }}</ref><ref>{{cite journal |author=Savelkoel J |author2=Dance DAB |author3=Currie BJ et al |title=A call to action: time to recognise melioidosis as a neglected tropical disease |journal=The Lancet Infectious Diseases |date=22 June 2022 |volume=22 |issue=6 |pages=e176-e182 |doi=10.1016/S1473-3099(21)00394-7 |pmid=34953519 |pmc= }}</ref> Melioidosis is endemic in parts of southeast Asia (including Thailand,<ref name="Hinjoy 2018">{{cite journal |author=Hinjoy S |author2=Hantrakun V |author3=Kongyu S |author4=Kaewrakmuk J |author5=Wangrangsimakul T et al |title=Melioidosis in Thailand: Present and Future |journal=Tropical Medicine and Infectious Disease |date=8 April 2018 |volume=3 |issue=2 |pages=38 |doi=10.3390/tropicalmed3020038 |pmid=29725623 |pmc=5928800 }}</ref> Laos,<ref>{{cite journal |author=Dance DAB |author2=Luangraj M |author3=Rattanavong S |author4=Sithivong N et al |title=Melioidosis in the Lao People's Democratic Republic |journal=Tropical Medicine and Infectious Disease |date=19 February 2018 |volume=3 |issue=1 |pages=21 |doi=10.3390/tropicalmed3010021 |pmid=30274419 |pmc=6136615 }}</ref> Singapore,<ref>{{cite journal |author=Sim SH |author2=Ong CEL |author3=Gan YH |author4=Wang D et al |title=Melioidosis in Singapore: Clinical, Veterinary, and Environmental Perspectives |journal=Tropical Medicine and Infectious Disease |date=12 March 2018 |volume=3 |issue=1 |pages=31 |doi=10.3390/tropicalmed3010031 |pmid=30274428 |pmc=6136607 }}</ref> Brunei,<ref>{{cite journal |author=Pande K |author2=Kadir KAA |author3=Asli R |author4=Chong VH |title=Melioidosis in Brunei Darussalam |journal=Tropical Medicine and Infectious Disease |date=19 February 2018 |volume=3 |issue=1 |pages=20 |doi=10.3390/tropicalmed3010020 |pmid=30274418 |pmc=6136610 }}</ref> Malaysia,<ref name="Nathan 2018">{{cite journal |author=Nathan S |author2=Chieng S |author3=Kingsley PV |author4=Mohan A |author5=Podin Y et al |title=Melioidosis in Malaysia: Incidence, Clinical Challenges, and Advances in Understanding Pathogenesis |journal=Tropical Medicine and Infectious Disease |date=27 February 2018 |volume=3 |issue=1 |pages=25 |doi=10.3390/tropicalmed3010025 |pmid=30274422 |pmc=6136604 }}</ref> Myanmar<ref>{{cite journal |author=Win MM |author2=Ashley EA |author3=Zin KN |author4=Aung MT |author5=Swee MMM et al |title=Melioidosis in Myanmar |journal=Tropical Medicine and Infectious Disease |date=1 March 2018 |volume=3 |issue=1 |pages=28 |doi=10.3390/tropicalmed3010028 |pmid=30274425 |pmc=6136617 }}</ref> and Vietnam<ref>{{cite journal |author=Trinh TT |author2=Nguyen LDN |author3=Nguyen TV |author4=Tran CX |author5=Le AV et al |title=Melioidosis in Vietnam: Recently Improved Recognition but still an Uncertain Disease Burden after Almost a Century of Reporting |journal=Tropical Medicine and Infectious Disease |date=9 April 2018 |volume=3 |issue=2 |pages=39 |doi=10.3390/tropicalmed3020039 |pmid=30274435 |pmc=6073866 }}</ref>), southern China,<ref>{{cite journal |author=Zheng X |author2=Xia Q |author3=Xia L |author4=Li W |title=Endemic Melioidosis in Southern China: Past and Present |journal=Tropical Medicine and Infectious Disease |date=25 February 2019 |volume=4 |issue=1 |pages=39 |doi=10.3390/tropicalmed4010039 |pmid=30823573 |pmc=6473618 }}</ref> Taiwan<ref>{{cite journal |author=Hsueh PT |author2=Huang WT |author3=Huseueh HK |author4=Chen YL |author5=Chen SY et al |title=Transmission Modes of Melioidosis in Taiwan |journal=Tropical Medicine and Infectious Disease |date=28 February 2018 |volume=3 |issue=1 |pages=26 |doi=10.3390/tropicalmed3010026 |pmid=30274423 |pmc=6136622 }}</ref> northern Australia.<ref name="Smith 2018">{{cite journal |author=Smith S |author2=Hanson J |author3=Currie BJ |title=Melioidosis: An Australian Perspective |journal=Tropical Medicine and Infectious Disease |date=1 March 2018 |volume=3 |issue=27 |pages=1 |doi=10.3390/tropicalmed3010027 |pmid=30274424 |pmc=6136632 }}</ref> India,<ref name="Mukhopadhyay 2018">{{cite journal |author=Mukhopadhyay C |author2=Shaw T |author3=Varghese GM |author4=Dance DAB |title=Melioidosis in South Asia (India, Nepal, Pakistan, Bhutan and Afghanistan) |journal=Tropical Medicine and Infectious Disease |date=22 May 2018 |volume=3 |issue=2 |pages=51 |doi=10.3390/tropicalmed3020051 |pmid=30274447 |pmc=6073985 }}</ref> and South America.<ref>{{cite journal |author=Rolim DB |author2=Lima RX |author3=Ribeiro AK |author4=Colares RM et al |title=Melioidosis in South America |journal=Tropical Medicine and Infectious Disease |date=5 June 2018 |volume=3 |issue=2 |pages=60 |doi=10.3390/tropicalmed3020060 |pmid=30274456 |pmc=6073846 }}</ref> Since 1991, a total of 583 cases were reported in India. Most Indian cases are located in [[w:Karnataka|Karnataka]] and [[w:Tamil Nadu|Tamil Nadu]].<ref name="Mukhopadhyay 2018"/> Fifty-one cases of melioidosis were reported in Bangladesh from 1961–2017. Nonetheless, lack of awareness and resources gives rise to underdiagnosis of the disease in the country.<ref>{{cite journal |author=Chowdhury FR |author2=Jilani MSA |author3=Barai L |author4=Rahman T et al |title=Melioidosis in Bangladesh: A Clinical and Epidemiological Analysis of Culture-Confirmed Cases |journal=Tropical Medicine and Infectious Disease |date=9 April 2018 |volume=3 |issue=2 |pages=40 |doi=10.3390/tropicalmed3020040 |pmid=30274436 |pmc=6073520 }}</ref> The true burden of melioidosis in Africa and Middle East remain unknown due to low amount of data. Several melioidosis cases were reported over the years. Although 24 African countries and three Middle Eastern countries predicted to be endemic with melioidosis, however not a single case was reported from these specific countries.<ref>{{cite journal |author=Steinmetz I |author2=Wagner GE |author3=Kanyala E et al |title=Melioidosis in Africa: Time to Uncover the True Disease Load |journal=Tropical Medicine and Infectious Disease |date=10 June 2018 |volume=3 |issue=2 |pages=62 |doi=10.3390/tropicalmed3020062 |pmid=30274458 |pmc=6073667 }}</ref> In the United States, two historical cases (1950 and 1971) and four recent cases (2010, 2011, 2013, 2020) have been reported amongst people that did not travel overseas.<ref name="Currie 2015"/><ref>{{cite journal |author=Cossaboom CM |author2=Marinova-Petkova A |author3=Strysko J et al |title=Melioidosis in a Resident of Texas with No Recent Travel History, United States |journal=Emerging Infectious Diseases |date=June 2020 |volume=26 |issue=6 |pages=1295-1299 |doi=10.3201/eid2606.190975 |pmid=32442394 |pmc=7258475 }}</ref> Despite extensive investigations, the source of melioidosis was never confirmed. One possible explanation is that importation of medicinal plant products or exotic reptiles could have resulted in the introduction of melioidosis in the United States.<ref name="Currie 2015"/> In 2021, there was a melioidosis outbreak in several states in the United States due to usage of contaminated aromatherapy spray imported from India.<ref>{{cite journal |author=Gee JE |author2=Bower WA |author3=Kunkel A et al |title=Multistate Outbreak of Melioidosis Associated with Imported Aromatherapy Spray |journal=New England Journal of Medicine |date=March 2022 |volume=386 |issue=9 |pages=861-868 |doi=10.1056/NEJMoa2116130 |pmid=35235727 }}</ref> There are also cases of infection through imported tropical fishes in home aquariums.<ref>{{cite journal |author=Dawson P |author2=Duwell MM |author3=Elrod MG et al |title=Human Melioidosis Caused by Novel Transmission of Burkholderia pseudomallei from Freshwater Home Aquarium, United States |journal=Emerging Infectious Diseases |date=December 2021 |volume=27 |issue=12 |pages=3030-3035 |doi=10.3201/eid2712.211756 |pmid=34570693 |pmc=8632198 }}</ref> In Europe, more than half of the melioidosis cases are imported from Thailand.<ref>{{cite journal |author=Tohic SL |author2=Montana M |author3=Koch L et al |title=A review of melioidosis cases imported into Europe |journal=European Journal of Clinical Microbiology & Infectious Diseases |date=25 March 2019 |volume=38 |issue=8 |pages=1395–1408 |doi=10.1007/s10096-019-03548-5 |pmid=30949898 }}</ref> Melioidosis is found in all age groups.<ref name="Joost 2018"/> For Australia and Thailand, the median age of infection is at 50 years; 5 to 10% of the patients are under 15 years.<ref name="Joost 2018"/> The single most important risk factor for developing melioidosis is [[w:diabetes mellitus|diabetes mellitus]], followed by hazardous alcohol use, chronic kidney disease, and chronic lung disease.<ref name="Suputtamongkol 1999">{{cite journal |author=Suputtamongkol Y |author2=Chaowagul W |author3=Chetchotisakd P |author4=Lertpatanasuwun N et al |year=1999 |title=Risk factors for melioidosis and bacteremic melioidosis |journal=Clinical Infectious Diseases |pmid=10476750 |volume=29 |issue=2 |pages=408&ndash;413 |doi=10.1086/520223 }}</ref> More than 50% of people with melioidosis have diabetes; diabetics have a 12-fold increased risk of contracting melioidosis. Diabetes decreases the ability of macrophages to fight the bacteria and reduced the [[w:T helper cell|T helper cell]] production. Excessive release of [[w:Tumor necrosis factor alpha|Tumor necrosis factor alpha]] and [[w:Interleukin 12|Interleukin 12]] by [[w:peripheral blood mononuclear cell|mononuclear cells]] increases the risk of septic shock.<ref name="Joost 2018"/> Other risk factors include [[w:thalassaemia|thalassaemia]], occupational exposure (e.g. [[w:rice paddy|rice paddy]] farmers),<ref name="Gassiep 2020"/> recreational exposure to soil, water, being male, age greater than 45 years, and prolonged steroid use/immunosuppression.<ref name="Joost 2018"/> However, 8% of children and 20% of adults with melioidosis have no risk factors.<ref name="Joost 2018"/> [[w:HIV|HIV]] infection does not appear to predispose to melioidosis, although several other co-infections have been reported.<ref name="Gassiep 2020"/> [[w:Infant|Infant]] cases have been reported possibly due to mother-to-child transmission, community-acquired infection, or healthcare-associated infection.<ref name="Joost 2018"/> Those who are well may also be infected with ''B. pseudomallei''. For example, 25% of children started [[w:seroconversion|producing antibodies]] against ''B. pseudomallei'' between 6 months to 4 years of staying in [[w:endemic (epidemiology)|endemic]] areas although they did not experience any melioidosis symptoms; suggesting they were exposed to it over this time. This means that many people without symptoms will test positive in serology tests in endemic areas.<ref name="Yi 2014"/> In Thailand, the seropositivity rate exceeds 50%, while in Australia the seropositivity rate is only 5%.<ref name="Currie 2015"/> The disease is clearly associated with increased rainfall, with the number of cases rising following increased precipitation. Severe rainfall increases the concentration of the bacteria in the topsoil, thus increasing thus of transmitting the bacteria through the air.<ref name="Gassiep 2020"/> A recent CDC Advisory indicated that the recent detection of the organism in the environment in Mississippi following the occurrence of two indigenous cases of melioidosis, confirms that parts of the southern USA should now be regarded as melioidosis-endemic.<ref>{{Cite web|url=https://www.emergency.cdc.gov/han/2022/pdf/CDC_HAN_470.pdf |title=Melioidosis Locally Endemic in Areas of the Mississippi Gulf Coast after Burkholderia pseudomallei Isolated in Soil and Water and Linked to Two Cases – Mississippi, 2020 and 2022|publisher=CDC|access-date=2022-08-05}}</ref> ==History== Pathologist [[w:Alfred Whitmore|Alfred Whitmore]] and his assistant Krishnaswami first reported melioidosis among beggars and morphine addicts at autopsy in Rangoon, present-day [[w:Burma|Myanmar]], in a report published in 1912.<ref name="Whitmore 1912">{{cite journal |author=Whitmore A |author2=Krishnaswami CS |title=A Hitherto Undescribed Infective Disease in Rangoon |journal=Indian Medical Gazette |date=1912 |volume=47 |issue=7 |pages=262–267 |pmid=29005374 |pmc=5168169 |doi= }}</ref> Whitmore was able to grow the organism in culture and its showed similarity with ''B. mallei'', another bacteria that was known causing glanders in animals. Therefore, he named the new organism ''Bacillus pseudomallei''. He did no further work on the organism.<ref name="Brightman 2020">{{cite journal |author=Brightman C |author2=Locum |title=Melioidosis, the Vietnamese time bomb |journal=Trends in Urology and Men's Health |year=4 June 2020 |volume=11 |issue=3 |pages=30-32 |pmid= |pmc= |doi=10.1002/tre.753 }}</ref> [[w:Arthur Conan Doyle|Arthur Conan Doyle]] may have read Whitmore's report before writing a short story that involved the fictitious tropical disease "Tapanuli fever" in a [[w:Sherlock Holmes|Sherlock Holmes]] story<ref name="pmid11823558">{{cite journal |author=Vora SK |title=Sherlock Holmes and a biological weapon |journal=Journal of the Royal Society of Medicine |volume=95 |issue=2 |pages=101–103 |date=February 2002 |pmid=11823558 |pmc=1279324 |doi=10.1258/jrsm.95.2.101 |url=http://www.jrsm.org/cgi/pmidlookup?view=long&pmid=11823558 }}</ref> titled “[[w:The Adventure of the Dying Detective|The Adventure of the Dying Detective]]” published in 1913.<ref name="Allen C 2005"/> In the same year, melioidosis outbreak occurred inside the Institute for Medical Research (IMR), [[w:Kuala Lumpur|Kuala Lumpur]], [[w:Federated Malay States|Malaya]] after its laboratory animals such as guinea pigs and rabbits were infected.<ref name="Nathan 2018"/> William Fletcher and Ambrose Thomas Stanton, doctors who worked at the IMR, were the next ones to study the organism. They were unable identify the organism that caused the outbreak. It was only in 1917, when Fletcher isolated an organism similar to Whitmore's bacillus from a Tamil rubber estate worker, the presence of the new species of bacteria was confirmed.<ref name="Brightman 2020"/> The term “melioidosis” was first coined in 1921. The name melioidosis is derived from the Greek ''melis'' (μηλις) meaning "a distemper of asses" with the suffixes -oid meaning "similar to" and -osis meaning "a condition", that is, a condition similar to glanders.<ref name="Stanton 1921">{{cite book |author=Stanton AT |author2=Fletcher W |chapter=Melioidosis, a new disease of the tropics |title=Far Eastern Association of Tropical Medicine: Transactions of the Fourth Congress |location=Batavia, Dutch East Indies |publisher=Javasche Boekhandel en Drukkerij |year=1921 }}</ref> ''B pseudomallei'' is similar in clinical presentation and genome make-up with ''B. mallei''<ref name="Godoy 2003">{{cite journal |author=Godoy D |author2=Randle G |author3=Simpson AJ |author4=Aanensen DM et al |title=Multilocus sequence typing and evolutionary relationships among the causative agents of melioidosis and glanders, Burkholderia pseudomallei and Burkholderia mallei |journal=Journal of Clinical Microbiology |year=May 2003 |volume=41 |issue=5 |pages=2068-2079 |pmid=12734250 |pmc=154742 |doi=10.1128/jcm.41.5.2068-2079.2003 }}</ref> but is distinguished from it due to epidemiological and [[w:zoonosis|zoonotic]] characteristics.<ref name="Yabuuci 1992"/> The first human case of melioidosis in South Asia was reported in Sri Lanka in 1927.<ref name="Mukhopadhyay 2018"/> In 1932, Thomas and Fletcher collected 83 cases of melioidosis from literature. In this [[w:cases series|case series]], there were only two survivals. Since then, more case series of melioidosis were reported.<ref name="Khaira 1959">{{cite journal |author=Khaira BS |author2=Young WB |author3=Hart PdeV |title=Melioidosis |journal=British Medical Journal |volume=1 |issue=5127 |pages=949-952 |date=11 April 1959 |pmid=13638596 |pmc=1993315 |doi=10.1136/bmj.1.5127.949 }}</ref> Thomas and Fletcher also pioneered the use of serological methods in diagnosing the disease. Thomas and Fletcher incorrectly believed that melioidosis infection came from human contact with [[w:rondent|rodents]]. However, observations on the disease noted that humans usually got it after exposure to mud or contaminated water. Besides, the organism was never grown from rats. This led to a search of the bacteria in the environment.<ref name="Ketheesan 2012">{{cite book |editor=Ketheesan N |chapter= Melioidosis in animals |title=Melioidosis: A Century of Observation and Research|author=Rush CM|author2=Thomas AD|isbn=978-0-444-53479-8 |location=Townsville, Australia |publisher=Elsevier B.V. |year=2012 |page=313 }}</ref> In 1936, the first animal (pig) case of melioidosis in Africa was reported in Madagascar.<ref name="Garin 2014">{{cite journal |author=Garin B |author2=Djaomazala I |author3=Dubois-Cauwelaert N et al |title=Autochthonous Melioidosis in Humans, Madagascar, 2012 and 2013 |journal=Emerging Infectious Diseases |volume=20 |issue=10 |pages=1739-1741 |date=October 2014 |pmid=25272365 |pmc=4193270 |doi=10.3201/eid2010.131524 }}</ref> In 1937, water was first identified as the habitat of ''B. pseudomallei''.<ref name="Limmathurotsakul review 2013 ">{{cite journal |author=Limmathurotsakul D |author2=Dance DAB |author3=Wuthiekanun V et al |title=Systematic Review and Consensus Guidelines for Environmental Sampling of Burkholderia pseudomallei |journal=PLOS Neglected Tropical Diseases |volume=7 |issue=3 |pages=e2105 |date=2013 |pmid=23556010 |pmc=3605150 |doi=10.1371/journal.pntd.0002105 }}</ref> The first case of Australian melioidosis was described in an outbreak in sheep in 1949 at [[w:North Queensland|North Queensland]]. This was followed by the first case of human melioidosis at [[w:Townsville|Townsville]] in 1950.<ref name="Currie 2000">{{cite journal |author=Currie BJ |author2=Fisher DA |author3=Howard DM et al |title=The epidemiology of melioidosis in Australia and Papua New Guinea |journal=Acta Tropica |volume=74 |issue=2 |pages=121-127 |date=February 2000 |pmid=10674639 |pmc= |doi=10.1016/s0001-706x(99)00060-1 }}</ref> Initially, the discovery of melioidosis in Australia had led to a debate on when and how the disease spread from Southeast Asia to a new distant environment.<ref name="Ketheesan 2012"/> However, this hypothesis was later disproved in 2017 when [[w:whole genome sequencing|whole genome sequencing]] of ''B. pseudomallei'' over 30 countries collected over 79 years suggested Australia as the early reservoir for melioidosis.<ref name="Chewapreecha 2017"/> In 1955, first case of local human melioidosis was reported in Thailand.<ref name="Hinjoy 2018"/> During the [[w:Vietnam War|Vietnam War]] from 1967 to 1973, 343 American soldiers were reported with melioidosis, with about 50 cases transmitted through inhalation.<ref name="Rubin 1963">{{cite journal |author=Rubin HL |author2=Alexander AD |author3=Yager RH |title=Melioidosis—a military medical problem? |journal=Military Medicine |volume=128 |issue= |pages=538-642 |date=1963 |pmid=13983001 |pmc= |doi=10.1093/milmed/128.6.538 }}</ref> An outbreak of melioidosis at the Paris Zoo in the 1970s (known as ''L'affaire du jardin des plantes'') was thought to have originated from an imported [[w:panda|panda]] or horses from Iran.<ref name="Allen C 2005"/><ref>{{cite journal |author=Mollaret HH |title='L'affaire du Jardin des plantes' ou comment le mélioïdose fit son apparition en France |journal=Médecine et Maladies Infectieuses |year=1988 |volume=18 |issue=Suppl 4 |pages=643&ndash;654 |doi=10.1016/S0399-077X(88)80175-6 }}</ref> It is unclear how imported melioidosis is able to persist in a completely new environment. Eventually, the outbreak terminated by itself after a period of time.<ref name="Ketheesan 2012"/> It was only during the 1980s, Infectious Disease Association of Thailand started took notice of this disease. First conference on melioidosis was held in 1985 in Thailand. It was during this meeting that collaboration between Sappasitprasong Hospital, Thailand, and Wellcome-Mahido-Oxford Tropical Medicine Research Programme was established. Such collaboration made Thailand a world leader in clinical and epidemiology research on melioidosis.<ref name="Ketheesan 2012"/> In 1989, several studies conducted in Thailand demonstrated ceftazidime as an effective antibiotic against melioidosis.<ref name="Ketheesan 2012"/> [[w:Ceftazidime|Ceftazidime]] had been shown to reduce the risk of death of melioidosis from 74% to 37%.<ref name="White 1989">{{cite journal |author=White NJ |author2=Dance DA |author3=Chaowagul W |author4=Wattanagoon Y et al |title=Halving of mortality of severe melioidosis by ceftazidime |journal=The Lancet |year=1989 |volume=23 |issue=2 |pages=697-701 |pmid=2570956 |pmc= |doi=10.1016/s0140-6736(89)90768-x }}</ref> In 1990, a non-virulent ‘arabinose-positive B. pseudomallei’ was found by Vanaporn Wuthiekanun. The organism was later reclassified into a new species called ''B. thailandensis''. This species has become a useful tool in the laboratory for the studies of the pathogenesis of ''B. pseudomallei''.<ref name="Ketheesan 2012"/> ''B. pseudomallei'' was previously classified as part of the genus ''[[w:Pseudomonas|Pseudomonas]]''. In 1992, the pathogen was formally named ''B. pseudomallei''.<ref name="Yabuuci 1992">{{cite journal |author=Yabuuci E |author2=Kosako Y |author3=Oyaizu H |author4=Yano I et al |title=Proposal of Burkholderia gen. nov. and transfer of seven species of the genus Pseudomonas homology group II to the new genus, with the type species Burkholderia cepacia (Palleroni and Holmes 1981) comb. nov |journal=Microbiology and Immunology |year=1993 |volume=36 |issue=12 |pages=1251-1275 |pmid=1283774 |pmc=154742 |doi=10.1111/j.1348-0421.1992.tb02129.x }}</ref> In 1994, First International Symposium on melioidosis was held in Kuala Lumpur where 80 delegates attended. Papers were presented and later published as a book.<ref name="Ketheesan 2012"/> Subsequent congresses were held in Thailand, Australia, and Singapore once every three years.<ref name="Ketheesan 2012"/> In 2002, ''B. pseudomallei'' was classified as a "Category B agent".<ref name="Rotz 2002">{{cite journal |author=Rotz LD |author2=Khan AS |author3=Lillibridge SR |author4=Ostroff SM |author5=Hughes JM |title=Public Health Assessment of Potential Biological Terrorism Agents |journal=Emerging Infectious Diseases |year=February 2002 |volume=8 |issue=2 |pages=225-230 |pmid=11897082 |pmc=2732458 |doi=10.3201/eid0802.010164 }}</ref> In 2004, the complete genome of ''B. pseudomallei'' was published.<ref name="Ketheesan 2012"/> In 2012, ''B pseudomallei'' was classified as a "Tier 1 select agent" by the U.S. Centers for Disease Control.<ref name="McRobb 2015">{{cite journal |author=McRobb E |author2=Sarovich DS |author3=Price EP |author4=Kaestli M et al |title=Tracing melioidosis back to the source: using whole-genome sequencing to investigate an outbreak originating from a contaminated domestic water supply |journal=Journal of Clinical Microbiology |year=April 2015 |volume=53 |issue=4 |pages=1144-1148 |pmid=25631791 |pmc=4365233 |doi=10.1128/JCM.03453-14 }}</ref> In 2014, [[w:co-trimoxazole|co-trimoxazole]] was established as the only oral eradication therapy rather than combination therapy of co-trimoxazole with doxycycline.<ref name="Chetchotisakd 2014">{{cite journal |author=Chetchotisakd P |author2=Chierakul W |author3=Chaowagul W |author4=Anunnatsiri S et al |title=Trimethoprim-sulfamethoxazole versus trimethoprim-sulfamethoxazole plus doxycycline as oral eradicative treatment for melioidosis (MERTH): a multicentre, double-blind, non-inferiority, randomised controlled trial |journal=The Lancet |year=1 March 2014 |volume=383 |issue=9919 |pages=807-814 |pmid=24284287 |pmc=3939931 |doi=10.1016/S0140-6736(13)61951-0 }}</ref> In 2016, a statistical model was developed to predict the occurrence of global melioidosis per year.<ref name="Limmathurotsakul 2016"/> ==Synonyms== *Pseudoglanders<ref>{{cite journal |author=Chai LYA |author2=Fisher D |title=Earth, wind, rain, and melioidosis |journal=The Lancet Planetary Health |date=1 August 2018 |volume=2 |issue=8 |pages=329–330 |doi=10.1016/S2542-5196(18)30165-7 |pmid=30082045 }}</ref> *Whitmore's disease (after Captain [[w:Alfred Whitmore|Alfred Whitmore]], who first described the disease)<ref name="Whitmore 1912"/> *Nightcliff gardener's disease ([[w:Nightcliff|Nightcliff]] is a suburb of [[w:Darwin, Northern Territory|Darwin, Australia]] where melioidosis is endemic)<ref>{{cite web |url=http://www.abc.net.au/pm/content/2005/s1285014.htm |title=Rise in melioidosis rates in NT |author=Barker A |date=19 June 2005 |accessdate=2007-06-24 |publisher=Australian Broadcasting Corporation }}</ref> *Paddy-field disease<ref>{{cite journal |author=Orellana C |title=Melioidosis strikes Singapore |journal=The Lancet Infectious Diseases |date=November 2004 |volume=4 |issue=11 |pages=655 |pmid=15534940 |doi=10.1016/S1473-3099(04)01190-9 }}</ref> *Morphia injector's septicaemia<ref name="Stevenson 1916">{{cite journal |author=Stevenson AC |title=Morphia injector's septicæmia (Whitmore's Disease) |journal=Transactions of The Royal Society of Tropical Medicine and Hygiene |date=June 1916 |volume=9 |issue=7 |pages=218-219 |pmid= |pmc= |doi=10.1016/S0035-9203(16)90035-X }}</ref> ==Biological warfare== Interest in melioidosis has been expressed because it has the potential to be developed as a [[w:biological warfare|biological weapon]]. Another similar bacterium, ''Burkholderia mallei'' was used by the Germans in [[w:World War I|World War I]] to infect livestock shipped to Allied countries.<ref name="Matthew 2018"/> Deliberate infection of human [[w:prisoners of war|prisoners of war]] and animals using ''B. mallei'' were carried out in China's [[w:Pingfang District|Pingfang District]] by the Japanese during World War II.<ref name="Allen C 2005"/> The [[w:Soviet Union|Soviet Union]] reportedly used ''B. mallei'' during the [[w:Soviet–Afghan War|Soviet–Afghan War]] in 1982 and 1984.<ref name="Matthew 2018"/> ''B. pseudomallei'', like ''B. mallei'', was studied by both the US<ref>{{cite book |editor=Withers MR |title=USAMRIID's Medical Management of Biological Casualties Handbook |edition=8th |publisher=U.S. Army Medical Institute of Infectious Diseases |location=Fort Detrick, Maryland|isbn=978-0-16-093126-0 |year=2014 |url=https://usamriid.health.mil/education/bluebookpdf/USAMRIID%20BlueBook%208th%20Edition%20-%20Sep%202014.pdf |pages=53 }}</ref> and Soviet Union as a potential biological warfare agent, but never weaponized.<ref name="Matthew 2018">{{cite journal |author=Nguyen HVN |author2=Smith ME |author3=Hayoun MA |title=Glanders and Melioidosis |date=27 October 2018 |pmid=28846298 |url=https://www.ncbi.nlm.nih.gov/books/NBK448110/ |accessdate=15 February 2019 |journal=StatPearls |publisher=StatPearls Publishing |id=NBK448110 }}</ref> Other countries such as Iran, Iraq, North Korea, and Syria may have investigated the properties of ''B. pseudomallei'' for biological weapons.<ref name="Yi 2014"/> The bacterium is readily available in the environment. It can also be aerosolized and transmitted via inhalation.<ref name="Yi 2014"/> However, the ''B. pseudomallei'' has never been used in biological warfare.<ref name="Yi 2014"/> The actual risk of the deliberate release of ''B. pseudomallei'' or ''B. mallei'' is unknown.<ref name="Gilad 1997">{{cite journal |author=Gilad J |author2=Harary I |author3=Dushnitsky T |author4=Schwartz D |author5=Amsalem Y |title=Burkholderia mallei and Burkholderia pseudomallei as bioterrorism agents: national aspects of emergency preparedness |journal=The Israel Medical Association Journal |year=1997 |volume=9 |issue=7 |pages=499-503 |pmid=17710778 }}</ref> ==Additional information== ===Acknowledgements=== All authors contributed to the article. ===Conflict of interest=== No funding was received for the preparation of this article. No financial competing interests declared. ===Ethics statement=== No ethical permission required. ==References== {{Reflist}} f4zefxcfjcybhyikg914qom9bk096n8 2416168 2416165 2022-08-18T14:46:19Z Bobamnertiopsis 24451 ndashes wikitext text/x-wiki {{Article info | journal = WikiJournal of Medicine | first1 = Siang Ching | last1 = Raymond Chieng | orcid1 = 0000-0003-1286-2196 |last2 = |first2 = |last3 = |first3 = |last4 = |first4 = |submitted = 2020-05-01 | affiliation1 = Klinik Kesihatan Bintangor, Sarawak, Malaysia | correspondence1 = by [[w:Special:EmailUser/Cerevisae|online form]] | w1 = Melioidosis | et_al = true | abstract = '''Melioidosis''' is an [[w:infectious disease|infectious disease]] caused by a [[w:gram-negative|gram-negative]] [[w:bacterium|bacterium]] called ''[[w:Burkholderia pseudomallei|Burkholderia pseudomallei]]''.<ref name="Joost 2018"/> Most people exposed to ''B. pseudomallei'' experience no symptoms; however, those who do experience symptoms have signs and symptoms that range from mild such as [[w:fever|fever]] and skin changes, to severe with [[w:pneumonia|pneumonia]], [[w:abscess|abscess]]es, and [[w:septic shock|septic shock]] that could cause death.<ref name="Joost 2018"/> Approximately 10% of people with melioidosis develop symptoms that last longer than two months, termed "chronic melioidosis".<ref name="Joost 2018"/> Humans are infected with ''B. pseudomallei'' by contact with contaminated soil or water. The bacteria enter the body through wounds, inhalation, or ingestion. Person-to-person or animal-to-human transmission is extremely rare.<ref name="Joost 2018"/> The infection is constantly present in Southeast Asia particularly in northeast [[w:Thailand|Thailand]] and northern Australia.<ref name="Joost 2018"/> In temperate countries such as Europe and the United States, melioidosis cases are usually imported from countries where melioidosis is endemic.<ref name="Currie 2015"/> The signs and symptoms of melioidosis resemble [[w:tuberculosis|tuberculosis]] and misdiagnosis is common.<ref name="Yi 2014"/> Diagnosis is usually confirmed by the growth of ''B. pseudomallei'' from an infected person's blood or other bodily fluid such as pus, sputum, and urine.<ref name="Joost 2018"/> Those with melioidosis are treated first with an "intensive phase" course of intravenous antibiotics (most commonly [[w:ceftazidime|ceftazidime]]) followed by a several-months treatment course of [[w:co-trimoxazole|co-trimoxazole]].<ref name="Joost 2018"/> In countries with the advanced healthcare system, approximately 10% of people with melioidosis die from the disease. In less developed countries, the death rate could reach 40%.<ref name="Joost 2018">{{cite journal |author= Wiersinga WJ |author2=Virk HS |author3=Torres AG |author4=Currie BJ et al |title=Melioidosis |journal=Nature Reviews Disease Primers |date=1 February 2018 |volume=4 |issue= |pages=17107 |doi=10.1038/nrdp.2017.107 |pmid=29388572 |pmc=6456913 }}</ref> Efforts to prevent melioidosis include: wearing protective gear while handling contaminated water or soil, practising hand hygiene, drinking boiled water, and avoiding direct contact with soil, water, or heavy rain.<ref name="Joost 2018"/> There is little evidence in supporting the use of melioidosis prophylaxis in humans. The [[w:antibiotic|antibiotic]] [[w:co-trimoxazole|co-trimoxazole]] is used as a preventative only for individuals at high risk for getting the disease after being exposed to the bacteria in laboratory settings.<ref name="Joost 2018"/> One study conducted in 2018 determined that the drug could be useful in preventing melioidosis in high-risk renal failure patients undergoing haemodylysis.<ref>{{cite journal |author=Majoni SW |author2=Hughes JT |author3=Heron B |author4=Currie BJ |title=Trimethoprim+Sulfamethoxazole Reduces Rates of Melioidosis in High-Risk Hemodialysis Patients |journal=Kidney International Reports |date=2018 |volume=3 |issue=1 |pages=160–167 |doi=10.1016/j.ekir.2017.09.005 |pmid=29340327 |pmcid=5762962 }}</ref> There is no approved vaccine for melioidosis.<ref name="Joost 2018"/> Approximately 165,000 people are infected by melioidosis per year, resulting in about 89,000 deaths, based on a mathematical model published in 2016.<ref name="Limmathurotsakul 2016">{{cite journal |author=Limmathurotsakul D |author2=Golding N |author3=Dance DAB |author4=Messina JP et al |title=Predicted global distribution of Burkholderia pseudomallei and burden of melioidosis |journal=Nature Microbiology |date=2016 |volume=1 |issue=1 |pages=15008 |doi=10.1038/nmicrobiol.2015.8 |pmid=26877885 |pmcid=4746747 }}</ref> [[w:Diabetes|Diabetes]] is a major risk factor for melioidosis; over half of melioidosis cases are in people with diabetes.<ref name="Joost 2018"/> Increased rainfall and severe weather events such as thunderstorm are associated with an increased number of melioidosis cases in [[w:Endemic (epidemiology)|endemic]] areas.<ref name="Yi 2014"/> }} ==Signs and symptoms== ===Acute=== {{fig|1 |Melioidosis PA and lateral X rays.jpg |Chest X-ray showing opacity of the left upper lobe of the lung of a patient with melioidosis complicated with pneumonia. |attribution=Samira Rahat Afroze et al, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0] |size=285px }} {{fig|2 |CT and MRI scan of the brain with melioidosis.jpg |CT and MRI scans showing lesion of the right frontal lobe of the brain of a patient with melioidosis. |attribution=Wei-yuan Huang et al, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0] |size=285px }} {{fig|3 |Septic arthritis of left hip joint with melioidosis.jpg |Septic arthritis of the left hip with joint destruction due to melioidosis. |attribution=N. P. Weerasinghe et al, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0] |size=285px }} Most people exposed to ''[[w:Burkholderia pseudomallei|B. pseudomallei]]'' experience no symptoms.<ref name="Yi 2014"/><ref>{{cite journal |author=Mustafa M |author2=Balingi J |author3=Torresi J |author4=Menon J |title=Clinical manifestations, diagnosis, and treatment of Melioidosis |journal=IOSR Journal of Pharmacy |date=February 2015 |volume=5 |issue=2 |pages=13–19 |doi= |pmid= |url=https://web.archive.org/web/20220626050909/https://www.ums.edu.my/fpsk/files/12.-Dr.-Fredie-Robinson-IOSR-Journal-of-Pharmacy.pdf }}</ref> The mean [[w:incubation period|incubation period]] of acute melioidosis is 9 days (range 1&ndash;21 days).<ref name="Joost 2018"/> Nevertheless, symptoms of melioidosis can appear in 24 hours for those who experienced near drowning in water.<ref name="Bennett 2015"/> Those affected present with symptoms of [[w:sepsis|sepsis]] (predominantly fever) with or without [[w:pneumonia|pneumonia]], or localised [[w:abscess|abscess]] or other focus of infection. The presence of non-specific signs and symptoms has caused melioidosis to be nicknamed "the great mimicker".<ref name="Joost 2018"/> [[w:diabetes mellitus|Diabetes mellitus]] is one of the most important risk factors in developing melioidosis. The disease should be considered in anyone who has spent time in endemic areas who develops a fever, pneumonia, or abscesses in their liver, spleen, prostate, or parotid gland.<ref name="Joost 2018"/> The clinical manifestation of the disease can range from simple skin changes such as abscesses or ulcerations to severe organ problems.<ref>{{cite journal |author=Fertitta L |author2=Monsel G |author3=Torresi J |author4=Caumes E |title=Cutaneous melioidosis: a review of the literature |journal=The International Society of Dermatology |date=2018 |volume=58 |issue=2 |pages=221–227 |doi=10.1111/ijd.14167 |pmid=30132827 }}</ref> The commonest organs affected are liver, spleen, lungs, prostate, and kidneys. Among the most common features are [[w:bacteremia|bacteremia]] (in 40 to 60% of cases), pneumonia (50%), and [[w:septic shock|septic shock]] (20%).<ref name="Joost 2018"/><ref name="Currie 2021">{{cite journal |author=Currie BJ |author2=Mayo M |author3=Ward LM et al |title=The Darwin Prospective Melioidosis Study: a 30-year prospective, observational investigation |journal=The Lancet Infectious Diseases |date=1 December 2021 |volume=21 |issue=2 |pages=1737–1746 |doi=10.1016/S1473-3099(21)00022-0 |pmid=34303419 |pmc= }}</ref> People with only pneumonia may have a prominent cough with sputum and shortness of breath. However, those with septic shock together with pneumonia may have minimal coughing.<ref name="Yi 2014"/> Results of a chest X-ray can range from diffuse nodular infiltrates in those with septic shock to progressive [[w:pulmonary consolidation|consolidation]] located most commonly in the [[w:Lung#Anatomy|upper lobes]] for those with pneumonia only. [[w:Pleural effusion|Pleural effusion]] and [[w:empyema|empyema]] are more common for melioidosis affecting lower lobes of the lungs.<ref name="Yi 2014"/> In 10% of cases, people develop secondary pneumonia caused by other bacteria after the primary infection.<ref name="Currie 2015"/> In northern Australia, 60% of the infected children presented with only skin lesions, while 20% presented with pneumonia.<ref name="Currie 2015"/> Depending on the course of infection, other severe manifestations develop. Approximately 1 to 5% of those infected develop [[w:encephalomyelitis|inflammation of the brain and brain covering]] or [[w:brain abscess|brain abscess]]; 14 to 28% develop [[w:acute pyelonephritis|pyelonephritis]], kidney abscess or prostatic abscesses; 0 to 30% develop neck or [[w:parotid gland|salivary gland]] abscesses; 10 to 33% develop liver, spleen, or paraintestinal abscesses; and 4 to 14% develop [[w:septic arthritis|septic arthritis]] and [[w:osteomyelitis|osteomyelitis]].<ref name="Joost 2018"/> Rare manifestations include [[w:lymphadenopathy|lymph node disease]] resembling tuberculosis,<ref name="Gassiep 2020">{{cite journal |author=Gassiep I |author2=Armstrong M |author3=Norton R |title=Human Melioidosis |journal=Clinical Microbiology Reviews |date=11 March 2020 |volume=33 |issue=2 |doi=10.1128/CMR.00006-19 |pmid=32161067 |pmc=7067580 }}</ref> [[w:mediastinum|mediastinal]] masses, [[w:pericardial effusion|pericardial effusion]],<ref name="Currie 2015"/> [[w:mycotic aneurysm|mycotic aneurysm]],<ref name="Joost 2018"/> and [[w:pancreatitis|inflammation of the pancreas]].<ref name="Currie 2015"/> In Australia, up to 20% of infected males develop prostatic abscess which may manifests clinically as [[w:dysuria|pain during urination]], difficulty in passing urine, and [[w:urinary retention|urinary retention]] requiring [[w:catheter|catheter]]isation.<ref name="Joost 2018"/> [[w:Rectal examination|Rectal examination]] may find enlarged [[w:prostate|prostate]].<ref name="Currie 2015"/> In Thailand, 30% of the infected children develop parotid abscesses.<ref name="Joost 2018"/> Encephalomyelitis not only happens in those with risk factors, but can also occur in healthy people without risk factors. Those with melioidosis encephomyelitis tend to have normal [[w:computed tomography|computed tomography]] (CT) scans but increased [[w:MRI_sequence#T1_and_T2|T2 signal]] by [[w:magnetic resonance imaging|magnetic resonance imaging]] (MRI), extending to the [[w:brain stem|brain stem]] and [[w:spinal cord|spinal cord]]. Clinical signs include: unilateral [[w:upper motor neuron|upper motor neuron]] limb weakness, [[w:focal neurological signs|cerebellar signs]], and cranial nerve palsies ([[w:Sixth nerve palsy|VI]], [[w:Facial nerve paralysis|VII]] nerve palsies and [[w:bulbar palsy|bulbar palsy]]). Some cases presented with [[w:flaccid paralysis|flaccid paralysis]] alone.<ref name="Currie 2015"/> In northern Australia, all melioidosis with encephalomyelitis cases had elevated white cells in the [[w:cerebrospinal fluid|cerebrospinal fluid]] (CSF), mostly [[w:mononuclear cell|mononuclear cell]]s with elevated CSF protein.<ref name="Gassiep 2020"/> ===Chronic=== Chronic melioidosis is usually defined by symptoms lasting greater than two months and occurs in about 10% of patients.<ref name="Joost 2018"/> Clinical presentations include fever, weight loss, productive cough with or without bloody sputum which may mimic [[w:tuberculosis|tuberculosis]]. Additionally, long-standing abscesses at multiple body sites may also present.<ref name="Yi 2014"/> Tuberculosis should be considered for lymph nodes enlargement at the [[w:root of the lung|root of the lung]]. Additionally, pneumonia caused by melioidosis rarely causes scarring and calcification of the lungs, unlike tuberculosis.<ref name="Gassiep 2020"/> ===Latent=== The potential for prolonged incubation was recognized in US servicemen involved in the Vietnam War, and was referred to as the "Vietnam time-bomb".<ref name="Yi 2014"/> Initially, it was thought that the longest period between presumed exposure and clinical presentation is 62 years in a [[w:prisoner of war|prisoner of war]] in Burma-Thailand-Malaysia.<ref name="Ngauy 2005">{{cite journal |author=Ngauy V |author2=Lemeshev Y |author3=Sadkowski L |author4=Crawford G |title=Cutaneous Melioidosis in a Man Who Was Taken as a Prisoner of War by the Japanese during World War II |year=2005 |journal=Journal of Clinical Microbiology |volume=43 |issue=2 |pages=970–972 |pmid=15695721 |doi=10.1128/JCM.43.2.970-972.2005 |pmc=548040 }}</ref> However, subsequent genotyping of the bacteria isolate from the Vietnam veteran showed that the isolate may not come from [[w:Southeast Asia|Southeast Asia]], but from South America.<ref>{{cite journal |author=Gee JE |author2=Gulvick CA |author3=Elrod MG et al |title=Phylogeography of Burkholderia pseudomallei Isolates, Western Hemisphere |journal=Emergency Infectious Disease |date=July 2017 |volume=23 |issue=7 |pages=1133–1138 |doi=10.3201/eid2307.161978 |pmid=28628442 |pmc=5512505 }}</ref> This reinstates another report that put the longest latency period for melioidosis as 29 years.<ref>{{cite journal |author=Chodimella U |author2=Hoppes WL |author3=Whalen S |title=Septicemia and suppuration in a Vietnam veteran |journal=Hospital Practice |date=15 May 1997 |volume=32 |issue=5 |pages=219–221 |pmid=9153149 |doi=10.1080/21548331.1997.11443493 }}</ref> Patients with latent melioidosis may be symptom-free for decades.<ref name="Ngauy 2005"/> Less than 5% of all melioidosis cases have activation after a period of latency.<ref name="Joost 2018"/> Various comorbidities such as diabetes, renal failure, and alcoholism can predispose to reactivation of melioidosis.<ref name="Yi 2014"/> ==Cause== ===Bacteria=== {{fig|4 |Burkholderia pseudomallei gram stain safety pin apperance.jpg |''B. pseudomallei'' with bipolar gram staining showing safety pin appearance. |attribution=Neha Shrestha, [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0] |size=285px }} Melioidosis is caused by [[w:gram-negative|gram-negative]], [[w:motility|motile]], [[w:Saprotrophic nutrition|saprophytic]] bacteria named ''Burkholderia pseudomallei''.<ref name="Chakravorty 2019">{{cite journal |author= Chakravorty A |author2=Heath C |title=Melioidosis: An updated review |journal= Australian Journal of General Practice |date=5 May 2019 |volume=48 |issue=5 |pages=327–332 |doi=10.31128/AJGP-04-18-4558 |pmid=31129946 }}</ref> The bacteria are usually [[w:opportunistic infection|opportunistic]], [[w:Facultative parasite|facultative]] [[w:intracellular|intracellular]] pathogens.<ref name="Chakravorty 2019"/> It is also aerobic and [[w:oxidase test|oxidase test]] positive.<ref name="Yi 2014"/> A granule at the centre of the bacterium makes it resemble a “safety pin” when [[w:Gram stain|Gram stain]]ed.<ref name="Yi 2014"/> The bacteria emit a strong soil smell after 24 to 48 hours of growth in culture, however smelling for the identification of the bacteria is not recommended for routine laboratory practice. One of the factors causing ''B. pseudomallei'''s resistance to various kinds of antibiotics is because of its production of a [[w:glycocalyx|glycocalyx]] [[w:polysaccharide|polysaccharide]] capsule.<ref name="Allen C 2005">{{cite journal |author=Cheng AC |author2=Currie BJ |title=Melioidosis: Epidemiology, Pathophysiology, and Management |journal=Clinical Microbiology Reviews |date=April 2005 |volume=18 |issue=2 |pages=383–416 |doi=10.1128/CMR.18.2.383-416.2005 |pmid=15831829 |pmc=1082802 }}</ref> It is generally resistant to [[w:gentamicin|gentamicin]] and [[w:colistin|colistin]] but sensitive to [[w:amoxicillin/clavulanic acid|co-amoxiclav]]. ''B. pseudomallei'' is a [[w:biosafety level|biosafety level]] 3 pathogen which requires specialized laboratory handling.<ref name="Yi 2014"/> In humans and animals, another similar organism named ''[[w:Burkholderia mallei|Burkholderia mallei]]'' is the causative agent of the disease [[w:glanders|glanders]].<ref name="Joost 2018"/> ''B. pseudomallei'' can be differentiated from another closely related, but less pathogenic species ''B. thailandensis'' by its ability to assimilate [[w:arabinose|arabinose]].<ref name="Gassiep 2020"/> ''B. pseudomallei'' is highly adaptable to various host environments ranging from inside [[w:Mycorrhiza|mycorrhizal fungi]] spores to [[w:amoeba|amoeba]].<ref name="Yi 2014"/> Its adaptability may give it a survival advantage in the human body.<ref name="Joost 2018"/> The genome of ''B. pseudomallei'' consists of two [[w:replicon (genetics)|replicons]]: chromosome 1 encodes [[w:housekeeping gene|housekeeping functions]] of the bacteria such as cell wall synthesis, mobility, and metabolism; chromosome 2 encodes functions that allow the bacteria to adapt to various environments. [[w:Horizontal gene transfer|Horizontal gene transfer]] has resulted in highly variable genomes in ''B. pseudomallei''. Australia has been suggested as the origin for ''B. pseudomallei'' because of the high genetic variability of the bacteria found in this region. Bacteria that was introduced to Central and South America in the 17th to 19th centuries seem to have a common ancestor from Africa.<ref name="Chewapreecha 2017">{{cite journal |author= Chewapreecha C |author2=Holden MT |author3=Vehkala M |author4=Välimäki N et al |title=Global and regional dissemination and evolution of Burkholderia pseudomallei |journal= Nature Microbiology |date=23 January 2017 |volume=2 |issue=16263 |pages=1–8 |doi=10.1038/nmicrobiol.2016.263 |pmid=28112723 |pmcid=5300093 }}</ref> ''B. mallei'' is a clone of ''B. pseudomallei'' that has lost substantial portions of its genome as it adapted to live exclusively in mammals.<ref name="Currie 2015">{{cite journal |author=Currie BJ |title=Melioidosis: evolving concepts in epidemiology, pathogenesis, and treatment |journal=Seminars in Respiratory and Critical Care Medicine |date=February 2015 |volume=36 |issue=1 |pages=111–125 |doi=10.1055/s-0034-1398389 |pmid=25643275 }}</ref> This makes the ''B. mallei'' genome much smaller than ''B. pseudomallei''.<ref name="Nierman 2004">{{cite journal |author=Nierman WC |author2=DeShazer D |author3=Kim HS |title=Structural flexibility in the Burkholderia mallei genome |journal=Proceedings of National Academy of Science of United States of America |date=28 September 2004 |volume=101 |issue=39 |pages=14246–14251 |doi=10.1073/pnas.0403306101 |pmid=15377793 |pmc=521142 }}</ref> ===Transmission=== ''B. pseudomallei'' is normally found in soil and surface water, and is most abundant at soil depths of 10 to 90 cm.<ref name="Joost 2018"/> It has been found in soils, ponds, streams, pools, stagnant water, and rice paddy fields.<ref name="Yi 2014">{{cite journal |author=Foong YC |author2=Tan M |author3=Bradbury RS |title=Melioidosis: A Review |journal=Journal of Remote and Rural Health |date=30 October 2014 |volume=14 |issue=4 |page=2763 |pmid=25359677|doi=10.22605/RRH2763 |url=https://www.rrh.org.au/journal/article/2763 }}</ref> ''B. pseudomallei'' can survive in nutrient-poor conditions such as distilled water, desert soil, and nutrient-depleted soil for more than 16 years.<ref name="Joost 2018"/> It can also survive in antiseptic and detergent solutions, acidic environments ([[w:pH|pH]] 4.5 for 70 days), and in environments at temperatures ranging from 24 °C (75.2 °F) to 32 °C (89.6 °F). However, the bacteria may be killed by the presence of ultraviolet light.<ref name="Joost 2018"/> Bacteria can enter the body through wounds, inhalation, and [[w:ingestion|ingestion]] of contaminated soil or water.<ref name="Joost 2018"/> Person-to-person transmission is extremely rare.<ref name="Yi 2014"/> Melioidosis is a recognised disease in animals including pigs, cats, dogs, goats, sheep, horses and others. Cattle, water buffalo, and crocodiles are considered to be relatively resistant to melioidosis despite their constant exposure to mud. Birds are also considered resistant to melioidosis although several cases had been reported in Australia and aquatic birds.<ref name="Gassiep 2020"/><ref name="Allen C 2005"/> [[w:zoonosis|Transmission from animals to humans]] is rare.<ref name="Joost 2018"/><ref name="Yi 2014"/> Inadequate [[w:Water chlorination|chlorination]] of water supply has been associated with ''B. pseudomallei'' outbreak in Northern and Western Australia.<ref name="Currie 2001">{{cite journal |author=Currie BJ |author2=Mayo M |author3=Anstey NM |author4=Donohoe P |author5=Haase A |author6=Kemp DJ |title=A cluster of melioidosis cases from an endemic region is clonal and is linked to the water supply using molecular typing of Burkholderia pseudomallei isolates |journal=American Society of Tropical Medicine and Hygiene |date=September 2001 |volume=65 |issue=3 |pages=177–179 |pmid=11561699 |doi=10.4269/ajtmh.2001.65.177 }}</ref><ref name="Inglis">{{cite journal |author=Inglis TJ |author2=Garrow SC |author3=Adams C |author4=Henderson M |author5=Mayo M |author6=Currie BJ |title=Acute melioidosis outbreak in Western Australia |journal=Epidemiology and Infections |date=December 1999 |volume=123 |issue=3 |pages=437–443 |pmid=10694154 |pmc= 2810777 | doi=10.1017/s0950268899002964 }}</ref> The were also several cases of where bacteria have also been found in unchlorinated water supply in rural Thailand.<ref name="Limmathurotsakul 2012">{{cite journal |author=Limmathurotsakul D |author2=Wongsuvan G |author3=Aanensen D |author4=Ngamwilai S |title=Melioidosis Caused by Burkholderia pseudomallei in Drinking Water, Thailand, 2012 |journal=Emerging Infectious Diseases |date=February 2014 |volume=20 |issue=2 |pages=265–268 |pmid=24447771 |pmc=3901481 |doi=10.3201/eid2002.121891 }}</ref> Based on the whole genome sequencing of the bacteria, the variety of the bacteria ''B. pseudomallei'' in Papua New Guinea is narrow due to limited movements of the indigenous people. This findings supports the hypothesis that humans play an important role in bacterial transmission.<ref>{{cite journal |author=Baker A |author2=Pearson T |author3=Price EP |author4=Dale J et al |title=Molecular Phylogeny of Burkholderia pseudomallei from a Remote Region of Papua New Guinea |journal=PLOS One |year=2011 |volume=6 |issue=3 |pages=e18343 |pmid=21483841 |pmc=3069084 |doi=10.1371/journal.pone.0018343 |bibcode=2011PLoSO...618343B }}</ref> ==Pathogenesis== {{fig|5 |Pathogenesis of melioidosis.svg |Ways of ''B. pseudomallei'' bacteria infecting human cells and blood stream. |attribution=Cerevisae, [https://creativecommons.org/licenses/by-sa/4.0/deed.en CC-BY-SA 4.0] |size=285px }} ''B. pseudomallei'' has the ability to infect various types of cells and to evade human immune responses. Bacteria first enter at a break in the skin or [[w:mucous membrane|mucous membrane]] and replicate in the epithelial cells. From there, they use [[w:flagella|flagella]]r motility to spread and infect various cell types.<ref name="Gassiep 2020"/> In the bloodstream, the bacteria can infect both [[w:phagocyte|phagocyte]]s and non-phagocytes.<ref name="Gassiep 2020"/> ''B. pseudomallei'' use their flagella to move near [[w:Host (biology)|host]] cells, then attach to the cells using various adhesion proteins, including the [[w:type IV pilus|type IV pilus]] protein PilA as well as adhesion proteins BoaA and BoaB.<ref name="Gassiep 2020"/> Additionally, adhesion of the bacteria partially depends on the presence of the host protein [[w:Coagulation factor II receptor|Protease-activated receptor-1]] which is present on the surface of [[w:endothelial cell|endothelial cell]]s, [[w:platelet|platelet]]s, and [[w:monocyte|monocyte]]s. Once bound, the bacteria enter host cells through [[w:endocytosis|endocytosis]], ending up inside an endocytic [[w:Vesicle (biology and chemistry)|vesicle]]. As the vesicle acidifies, ''B. pseudomallei'' uses its [[w:Bacterial secretion system#Type III secretion system (T3SS or TTSS)|Type 3 secretion system]] (T3SS) to inject effector proteins into the host cell, disrupting the vesicle and allowing the bacteria to escape into the host [[w:cytoplasm|cytoplasm]]. Within the host cytoplasm, the bacteria evade being killed by the host [[w:autophagy|autophagy]] using various T3SS effector proteins. The bacteria replicate in the host cytoplasm.<ref name="Joost 2018"/><ref name="Gassiep 2020"/> Inside the host cell, the bacteria move by inducing the polymerization of the host [[w:actin|actin]] behind them, propelling the bacteria forward.<ref name="Joost 2018"/> This actin-mediated motility is accomplished with the [[w:Secretion#Type V secretion system .28T5SS.29|autotransporter]] BimA which interacts with actin at the tail-end of the bacterium.<ref name="Joost 2018"/><ref name="Gassiep 2020"/> The bacteria that has BimABm [[w:allele|allele]] has higher possibility of causing neurological melioidosis, thus higher chance of death and residual disability to the host when compared to the bacteria that has BimABp variant.<ref>{{cite journal |author=Gora H |author2=Hasan T |author3=Smith S et al. |title=Melioidosis of the central nervous system; impact of the ''bimA_Bm'' allele on patient presentation and outcome |journal=Clinical Infectious Diseases |date=7 February 2022 |volume |issue= |pages=ciac111 |doi=10.1093/cid/ciac111 |pmid=35137005 }}</ref> Propelled by actin, the bacteria push against the host membrane, creating protrusions that extend into neighbouring cells. These protrusions cause neighboring cells to fuse, leading to the formation of [[w:multinucleated giant cells|multinucleated giant cells]] (MNGCs). When MNGCs lyse, they form plaques (a central clear area with a ring of fused cells) that provide shelter for the bacteria for further replication or [[w:latent infection|latent infection]]. This same process in infected neurons can allow bacteria to travel through nerve roots in the spinal cord and brain, leading to [[w:encephalomyelitis|inflammation of the brain and spinal cord]]. In addition to spreading from cell to cell, the bacteria can also spread through the bloodstream, causing sepsis. The bacteria can survive in [[w:antigen-presenting cell|antigen-presenting cell]]s and [[w:dendritic cell|dendritic cell]]s. Thus, these cells act as vehicles that transport the bacteria into the lymphatic system, causing widespread dissemination of the bacteria in the human body.<ref name="Joost 2018"/><ref name="Gassiep 2020"/> While ''B. pseudomallei'' can survive in phagocytic cells, these cells can kill ''B. pseudomallei'' by several mechanisms. Macrophages activated by [[w:interferon gamma|interferon gamma]] (IFN) have improved the killing of ''B. pseudomallei'' via the production of [[w:Nitric oxide synthase#iNOS|inducible nitric oxide synthase]]. Acidification of the endosome and degradation of the bacteria is also possible, however, the bacterial capsule and LPS makes ''B. pseudomallei'' resistant to lysosomal degradation. Once ''B. pseudomallei'' escapes into the host cytosol it can be recognized by [[w:pattern recognition receptor|pattern recognition receptor]]s such as [[w:NOD-like receptor|NOD-like receptor]]s, triggering the formation of the [[w:inflammasome|inflammasome]] and activation of [[w:caspase 1|caspase 1]], which induces death of the host cell by [[w:pyroptosis|pyroptosis]] and further activation of the immune system. Several systemic host defenses also contribute to the immune response. ''B. pseudomallei'' triggers both the [[w:complement system|complement system]] and [[w:coagulation cascade|coagulation cascade]], however the thick bacterial capsule prevent the action of the [[w:complement membrane attack complex|complement membrane attack complex]].<ref name="Joost 2018"/><ref name="Gassiep 2020"/> Additional elements of the immune system are activated by the host [[w:toll-like receptor|toll-like receptor]]s such as TLR2, TLR4, and TLR5 that recognize the [[w:pathogen-associated molecular pattern|conserved piece]]s of the bacteria such as LPS and flagella. This activation results in the production of [[w:cytokine|cytokine]]s such as [[w:Interleukin 1 beta|Interleukin 1 beta]] (IL-1β) and [[w:Interleukin 18|Interleukin 18]] (IL-18). IL-18 increases IFN production through [[w:natural killer cell|natural killer cell]]s while IL-1beta reduces the IFN production. These immune molecules drive the recruitment of other immune cells such as [[w:neutrophil|neutrophil]]s, [[w:dendritic cell|dendritic cell]]s, [[w:B cell|B cell]]s, and [[w:T cell|T cell]]s to the site of infection. T cells seem to be particularly important for controlling ''B. pseudomallei''; T cell numbers are increased in survivors, and low T cell numbers are associated with a high risk of death from melioidosis. Despite this, HIV infection is not a risk factor for melioidosis. Although macrophages show deregulated [[w:cytokine|cytokine]] responses in individuals with HIV infection, bacterial internalization and intracellular killing are still effective. People infected with ''B. pseudomallei'' may develop antibodies against the bacteria, and people that live in endemic areas tend to have antibodies in their blood that recognize ''B. pseudomallei''. However, the effectiveness of these antibodies at preventing melioidosis is unclear.<ref name="Joost 2018"/><ref name="Gassiep 2020"/> ''B. pseudomallei'' can remain latent in the human body for up to 29 years until it is reactivated during human [[w:immunosuppression|immunosuppression]] or stress response. However, the site of bacteria during latent infection and the mechanism by which they avoid immune recognition for years are both unclear. Amongst mechanisms suggested are: residing in the nucleus of the cell to prevent being digested, entering a stage of slower growth, antibiotic resistance, and genetic adaption to the host environment. [[w:Granuloma|Granuloma]]s (containing neutrophils, macrophages, lymphocytes, and multinucleated giant cells) formed at the infection site in melioidosis have been associated with latent infection in humans.<ref name="Joost 2018"/> ==Diagnosis== {{fig|6 |Bps close.JPG |Appearance of ''B. pseudomallei'' colonies on Ashdown's medium after four days of incubation. |attribution=Gavin Koh, [https://creativecommons.org/licenses/by-sa/4.0/ CC-BY-SA 4.0] |size=285px }} {{fig|7 |Immunofluorescent-stained sample of guinea pig tissue leads to positive diagnosis of melioidosis.png |Immunofluorescent microscopy showing the presence of rod-shaped ''B. pseudomallei'' (circled area) on a guinea pig tissue. |attribution=Dr. Biegeleisen, CDC, Public domain |size=285px }} {{fig|8 |Latex agglutination test positive for melioidosis.jpg |Right most slide showing positive latex agglutination for melioidosis. Latex agglutination uses latex coated antibodies to detect melioidosis antigens. |attribution=Neha Shrestha et al., [https://creativecommons.org/licenses/by/4.0/deed.en CC-BY 4.0] |size=285px }} ===Culture=== Bacterial culture has 60% sensitivity in diagnosing melioidosis.<ref name="Limmathurotsakul 2010">{{cite journal |author=Limmathurotsakul D |author2=Jamsen K |author3=Arayawichanont A |author4=Simpson JA et al |title=Defining the True Sensitivity of Culture for the Diagnosis of Melioidosis Using Bayesian Latent Class Models |journal=PLOS One |date=30 August 2010 |volume=5 |issue=8 |pages=e12485 |pmid=20830194 | pmc=2932979 | doi=10.1371/journal.pone.0012485 }}</ref> ''B. pseudomallei'' is never part of human flora. Therefore, any growth of the bacteria is diagnostic of melioidosis. Other samples such as throat, rectal swabs, pus from abscesses, and sputum can also be used for culture.<ref name="Joost 2018"/> However, culture from CSF is difficult because in one case series, only 29% of the neuromelioidosis cases are culture positive.<ref name="Gassiep 2020"/> When bacteria do not grow from people strongly suspected of having melioidosis, repeated cultures should be taken as subsequent cultures can become positive.<ref name="Joost 2018"/> ''B. pseudomallei'' can be grown on any blood agar, [[w:MacConkey agar|MacConkey agar]], and agar containing antibiotics such as [[w:Ashdown's medium|Ashdown's medium]] (containing [[w:gentamicin|gentamicin]]),<ref name="Gassiep 2020"/> and Ashdown's broth (containing [[w:colistin|colistin]])<ref name="Currie 2015"/> for better isolation of ''B. pseudomallei'' from other types of bacteria.<ref name="Gassiep 2020"/> Agar plates for melioidosis should be incubated at 37 °C (98.6 °F) in air <ref name="Yi 2014"/> and inspected daily for four days. On the agar plates, ''B. pseudomallei'' forms creamy, [[w:Hemolysis (microbiology)|non-haemolytic]], colonies after 2 days of incubation. After 4 days of incubation, colonies appear dry and wrinkled.<ref name="Joost 2018"/> Colonies of ''B. pseudomallei'' that are grown on Francis medium (a modification of Ashdown medium with gentamicin concentration increased to 8&nbsp;mg/L and neutral red indicator replaced with 0.2% bromocresol purple) are yellow.<ref name="Francis 2006">{{cite journal |author=Francis A |author2=Aiyar S |author3=Yean Yean C |author4=Naing L |author5=Ravichandran M |title=An improved selective and differential medium for the isolation of Burkholderia pseudomallei from clinical specimens |journal=Diagnostic Microbiology and Infectious Disease |date=28 November 2005 |volume=55 |issue=2 |pages=95–99 |pmid=16626918 | doi=10.1016/j.diagmicrobio.2005.11.008}}</ref> For laboratories located outside endemic areas, ''Burkholderia cepacia'' selective agar can be used if Ashdown's medium is not available.<ref name="Yi 2014"/> It is important not misinterpret the bacterial growth as ''[[w:Pseudomonas|Pseudomonas]]'' or ''[[w:Bacillus|Bacillus]]'' spp. Other biochemical screening tools can also be used for detecting ''B. pseudomallei'', including the [[w:analytical profile index|API 20NE or 20E biochemical kit]] combined with Gram stain, [[w:oxidase test|oxidase test]], typical growth characteristics, and resistance to certain antibiotics of the bacteria.<ref name="Currie 2015"/> API 20NE biochemical kit is 99% sensitive in identifying ''B. pseudomallei''.<ref name="Gassiep 2020"/> Molecular methods such as 16S rDNA sequencing, multiplex [[w:polymerase chain reaction|polymerase chain reaction]] (PCR), and real-time PCR can also be used to identify ''B. pseudomallei'' in culture.<ref name="Joost 2018"/><ref name="Fook Koh 2012">{{cite journal |author=Koh SF |author2=Tay ST |author3=Sermswan R |author4=Wongratanacheewin S et al |title=Development of a multiplex PCR assay for rapid identification of Burkholderia pseudomallei, Burkholderia thailandensis, Burkholderia mallei and Burkholderia cepacia complex |journal=Journal of Microbiological Methods |date=September 2012 |volume=90 |issue=3 |pages=305–308 |pmid=22705921 |doi=10.1016/j.mimet.2012.06.002 }}</ref><ref name="Gee 2003">{{cite journal |author=Gee JE |author2=Sacchi CT |author3=Glass MB |author4=Dee BK et al |title=Use of 16S rRNA gene sequencing for rapid identification and differentiation of ''Burkholderia pseudomallei'' and ''B. mallei''|journal=Journal of Clinical Microbiology |date=October 2003 |volume=41 |issue=10 |pages=4547–4654 |pmid=14532197 |pmc= 254370 |doi=10.1128/jcm.41.10.4647-4654.2003 }}</ref><ref name="Wang 2022"/> Other bacterial genes such as fliC genes encoding flagellin, rpsU gene encoding for ribosomal protein, and TTS genes encoding Type III secretion systems has also been employed for detection. Another method of gene detection namely [[w:multiple displacement amplification|multiple cross displacement amplification]] for the bacterial TTS1 gene detection produces results within an hour.<ref name="Wang 2022">{{cite journal |author=Wang X |author2=Wang L |author3=Zhu H et al |title=Reliable detection of ''Burkholderia pseudomallei'' using multiple cross displacement amplification label-based biosensor|journal=BMC Microbiology |date=10 March 2022 |volume=22 |issue=1 |pages= |pmid=35272632 |pmc= PMC8908694 |doi=10.1186/s12866-022-02485-2 }}</ref> ===Hematological and biochemical tests=== General blood tests in people with melioidosis show low white blood cell counts (indicates infection), raised liver enzymes, increased [[w:bilirubin|bilirubin]] levels (indicates liver dysfunction), and raised urea and creatinine levels (indicates kidney dysfunction). [[w:Hypoglycemia|Low blood glucose]] and [[w:acidosis|acidosis]] predicts a poorer prognosis in those with melioidosis. However, other tests such as [[w:C-reactive protein|C-reactive protein]] and [[w:procalcitonin|procalcitonin]] levels are not reliable in predicting the severity of melioidosis infection.<ref name="Allen C 2005"/> ===Serological tests=== [[w:serology|Serological test]]s such as indirect [[w:haemagglutination assay|haemagglutination]] assay (IHA) have been used to detect the presence of antibodies against ''B. pseudomallei''. However, different groups of people have widely different levels of antibodies, so interpretation of these tests depends on location. In Australia, less than 5% of people have ''B. pseudomallei'' antibodies, so the presence of even relatively low amounts of antibody is unusual and could suggest melioidosis. In Thailand, many people have antibodies against ''B. pseudomallei'' so the diagnosis of melioidosis should not be reliant entirely on the serological tests done in endemic areas.<ref name="Joost 2018"/><ref name="Currie 2015"/> Indirect immunofluorescent test (IFAT) uses either ''B. pseudomallei'' or ''B. thailandensis'' antigens to look for the total number of antibodies in human serum. Using IFAT is labour intensive and is not used in large scale investigations.<ref name="Suat Moi 2013">{{cite journal |author=Puah SM |author2=Puthucheary SD |author3=Chua KH |title=Potential Immunogenic Polypeptides of Burkholderia pseudomallei Identified by Shotgun Expression Library and Evaluation of Their Efficacy for Serodiagnosis of Melioidosis |journal=International Journal of Medical Sciences |date=2013 |volume=10 |issue=5 |pages=539–547 |pmid=23532805 |pmc=3607239 |doi=10.7150/ijms.5516 }}</ref> Antigen detect tests allow rapid detection of melioidosis. Examples of antigen detection tests are: latex agglutination test and [[w:ELISA|ELISA]]. [[w:Latex agglutination|Latex agglutination]] uses antibodies coated on latex beads to detect ''B. pseudomallei'' antigens in solid or liquid media, although not all the assays can detect different species of ''Burkholderia''.<ref name="Amornchai 2007">{{cite journal |author=Amornchai P |author2=Chierakul W |author3=Wuthiekanun V |author4=Mahakhunkijcharoen Y et al |title=Accuracy of Burkholderia pseudomallei Identification Using the API 20NE System and a Latex Agglutination Test |journal=Journal of Clinical Microbiology |date=2007 |volume=45 |issue=11 |pages=3774–3776 |pmid=17804660 | pmc=2168515 | doi=10.1128/JCM.00935-07 }}</ref> Latex agglutination is useful in screening for suspected ''B. pseudomallei'' colonies.<ref name="Joost 2018"/> IgG and IgM ELISAs has been used to detect lipopolysaccharide (LPS) antigens of ''B. pseudomallei'', but plagued with low sensitivity.<ref name="KP Lau 2015">{{cite journal |author=KP Lau S |author2=Sridhar S |author3=Chun Ho C et al |title=Laboratory diagnosis of melioidosis: Past, present and future |journal=Experimental Biology and Medicine |date=June 2015 |volume=240 |issue=6 |pages=742–751 |pmid=25908634 |pmc=PMC4935216 |doi=10.1177/1535370215583801 }}</ref> Commercial ELISA kits for melioidosis no longer available in the market due to low sensitivity to human antibodies detection.<ref name="Gassiep 2020"/> Nevertheless, antigen detection tests may be useful in severely ill patients because the bacterial load is high enough for detection. Other methods of antigen detection such as [[w:direct fluorescent antibody|direct immunofluorescence]], antibody-sandwich ELISAs, and [[w:lateral flow test|lateral flow immunoassays]] using [[w:monoclonal antibody|monoclonal antibody]].<ref name="KP Lau 2015"/> ===Microscopy=== By microscopy, ''B. pseudomallei'' is seen as [[w:Gram stain|gram-negative]] and rod-shaped, with a bipolar staining similar in appearance to a safety pin. Bacteria can sometimes be seen directly in clinical samples from infected people; however, identification by light microscopy is neither [[w:Sensitivity and specificity|specific]] nor [[w:Sensitivity and specificity|sensitive]]. [[w:Immunofluorescence microscopy|Immunofluorescence microscopy]] is highly specific for detecting bacteria directly from clinical specimens, but has less than 50% sensitivity.<ref name="Joost 2018"/><ref name="Currie 2015"/> ===Imaging=== Various imaging modalities can also help with the diagnosis of melioidosis. In acute melioidosis with the spreading of the bacteria through the bloodstream, the chest X-ray shows multifocal nodular lesions. It may also show merging nodules or [[w:cavitary pneumonia|cavitations]]. For those with acute melioidosis without the spread to the bloodstream, chest x-ray most commonly shows upper lobe [[w:Pulmonary consolidation|consolidation]] or cavitations.<ref name="Gassiep 2020"/> In chronic melioidosis, the slowly progressing of upper lobe consolidation of the lungs resembles tuberculosis.<ref name="Gassiep 2020"/> For abscesses located in other parts of the body apart from the lungs, especially in the liver and spleen, [[w:computed tomography|CT]] scan has higher sensitivity when compared with an ultrasound scan. In liver and splenic abscesses, an ultrasound scan shows "target-like" lesions while CT scan shows "honeycomb sign" (abscess with loculations separated by thin septa) in liver abscesses.<ref name="Gassiep 2020"/> For melioidosis involving the brain, MRI have higher sensitivity than a CT scan in diagnosing the lesion. MRI shows ring-enhancing lesions for brain melioidosis.<ref name="Gassiep 2020"/> ==Prevention== Melioidosis is a notifiable disease in Australia<ref name="Gassiep 2020"/> which enables the country to monitor disease burden and contain outbreaks. On the other hand, melioidosis is only a notifiable condition in Thailand since June 2016.<ref name="Gassiep 2020"/><ref name="Hantrakun 2019">{{cite journal |author=Hantrakun V |author2=Kongyu S |author3=Klaytong P |author4=Rongsumlee S et al |title=Clinical Epidemiology of 7126 Melioidosis Patients in Thailand and the Implications for a National Notifiable Diseases Surveillance System |journal= Open Forum Infectious Diseases |date=19 November 2019 |volume=6 |issue=12 |pages=ofz498 |pmid=32083145 | pmc=7020769 | doi=10.1093/ofid/ofz498 }}</ref> However, until recently, the official notification system in Thailand has significantly underestimated the incidence of culture-positive melioidosis and its mortality.<ref name="Hantrakun 2019"/> Nevertheless, Australia also embarked on awareness campaigns to increase the community's understanding of the disease.<ref name="Gassiep 2020"/> In the United Kingdom, where reporting by laboratories is mandatory, 41.3% of cases imported since 2010 were not notified.<ref name="Conor 2020">{{cite journal |author=Conor CO |author2=Kenna D |author3=Walsh A et al |title=Imported melioidosis in the United Kingdom: Increasing incidence but continued under-reporting |journal= Journal of Clinical Microbiology |date=October 2020 |volume=7 |issue= |pages=100051 |pmid= | pmc= | doi=10.1016/j.clinpr.2020.100051 }}</ref> In the United States, lab workers can handle clinical specimens of ''B. pseudomallei'' under [[w:BSL-2|BSL-2]] conditions, while mass production of such organisms requires [[w:BSL-3|BSL-3]] precautions.<ref>{{cite book |author=Centers for Disease Control and Prevention |title=Biosafety in Microbiological and Biomedical Laboratories |publisher=National Institutes of Health |location=Atlanta, Georgia |year=2009 |edition=5th|isbn=9780160850424 }}</ref> On the other hand, in other endemic areas where the ''B. pseudomallei'' samples were handled less stringently, there has been no confirmed laboratory-acquired infection reported. This phenomenon may show that the risk of infection with ''B. pseudomallei'' is less than a typical biohazard type 3 agent.<ref name="Dance 2017">{{cite journal |author=Dance DAB |author2=Limmathurotsakul D |author3=Currie BJ |title=Burkholderia pseudomallei: Challenges for the Clinical Microbiology Laboratory—a Response from the Front Line |journal= Journal of Clinical Microbiology |date=February 2017 |volume=55 |issue=3 |pages=980–982 |pmid=28232503 | pmc=5328468 | doi=10.1128/JCM.02378-16 }}</ref> There are also several cases of hospital-acquired infection of melioidosis.<ref name="Joost 2018"/> Therefore, healthcare providers are recommended to practice hand hygiene and [[w:universal precautions|universal precautions]].<ref name="Joost 2018"/> Large-scale water chlorination has been successful at reducing ''B. pseudomallei'' in the water in Australia.<ref name="Howard 2003">{{cite journal |author=Howard K |author2=Inglis TJJ |title=The effect of free chlorine on Burkholderia pseudomallei in potable water |journal= Water Research |date=February 2017 |volume=37 |issue=18 |pages=4425–4432 |pmid=14511713 | doi=10.1016/S0043-1354(03)00440-8 }}</ref><ref name="Joost 2018"/> In middle to low-income countries, water should be boiled before consumption.<ref name="Joost 2018"/> In high income countries, water could be treated with ultraviolet light for those at risk of contracting melioidosis.<ref name="McRobb E 2013">{{cite journal |author=McRobb E |author2=Kaestli M |author3=Mayo M |author4=Price EP et al |title= Melioidosis from Contaminated Bore Water and Successful UV Sterilization |journal= American Journal of Tropical Medicine and Hygiene |date=7 August 2013 |volume=89 |issue=2 |pages=367–368 |pmid=23751401 | pmc=3741262 | doi=10.4269/ajtmh.13-0101 }}</ref><ref name="Joost 2018"/> Those who are at high risk of contact with the bacteria should wear protective gear (such as boots and gloves) during work.<ref name="Joost 2018"/> Those staying in endemic areas should avoid direct contact with soil, and outdoor exposure to heavy rain or dust clouds. Bottled water or boiled water are preferred as drinking water.<ref name="Limmathurotsakul 2013">{{cite journal |author=Limmathurotsakul D |author2=Kanoksil M |author3=Wuthiekanun V et al |title= Activities of daily living associated with acquisition of melioidosis in northeast Thailand: a matched case-control study |journal= PLOS Neglected Tropical Diseases |date= 2013 |volume=7 |issue=2 |pages=e2072 |pmid=23437412 | pmc=3578767 | doi=10.1371/journal.pntd.0002072 }}</ref><ref name="Joost 2018"/> A study conducted from 2014 to 2018, however showed no significant differences on whether behavioural changes can reduce the risk of contracting melioidosis. Modification of behavioural changes or more frequent interventions may be needed to ensure a definite reduction in risk of getting melioidosis.<ref name="Suntornsut 2021">{{cite journal |author=Suntornsut P |author2=Teparrukkul P |author3=Wongsuvan G et al |title= Effectiveness of a multifaceted prevention programme for melioidosis in diabetics (PREMEL): A stepped-wedge cluster-randomised controlled trial |date= June 2021|journal=PLOS Neglected Tropical Diseases |volume=15 |issue=6 |pages=e0009060 |pmid=34170931 | pmc=PMC8266097 | doi=10.1371/journal.pntd.0009060 }}</ref> ===Antibiotic prophylaxis=== Administering cotrimoxazole three times a week throughout a wet season for dialysis patients has no obvious benefit of preventing melioidosis. Besides, high cost and side effects of this drug limits its use to only those with high risk of getting melioidosis.<ref name="Chau 2018">{{cite journal |author=Chau KWT |author2=Smith S |author3=Kang K et al |title=Antibiotic Prophylaxis for Melioidosis in Patients Receiving Hemodialysis in the Tropics? One Size Does Not Fit All |journal= The American Journal of Tropical Medicine and Hygiene |date=16 July 2018 |volume=99 |issue=3 |pages=597–600 |pmid=30014827 |pmc=PMC6169155 |doi=10.4269/ajtmh.18-0421 }}</ref> After exposure to ''B. pseudomallei'' (particularly following a laboratory accident, penetrating injuries, exposure of mouth and eyes to contaminated materials or aerosols), treatment with antibiotics is only given when in highly selected individuals after weighing the risk of adverse effects of the drugs against the benefits from contracting melioidosis. Cotrimoxazole can be used in this context. Alternatively, [[w:co-amoxiclav|co-amoxiclav]] and [[w:doxycycline|doxycycline]] can be used for those who are intolerant to co-trimoxazole. Low-risk individuals would receive frequent monitoring instead.<ref name="Lipsitz 2012">{{cite journal |author=Lipsitz R |author2=Garges S |author3=Aurigemma R et al |title=Workshop on treatment of and postexposure prophylaxis for Burkholderia pseudomallei and B. mallei Infection, 2010 |journal= Emerging Infectious Diseases |date=18 December 2012 |volume=18 |issue=12 |pages=e2 |pmid=23171644 |pmc=3557896 |doi=10.3201/eid1812.120638 }}</ref> ===Vaccination=== {{Further|w:Burkholderia pseudomallei#Vaccine candidates}} Several vaccine candidates have been tested in animal models. Nevertheless, no vaccine candidates have been tried in humans. Major hurdles of the vaccines are limited efficacy in animal models, establishing the best method of vaccine administration in humans and logistical and financial issues in establishing human trials in endemic areas.<ref name="Gassiep 2020"/> ==Treatment== The treatment of melioidosis is divided into two stages: an intravenous intensive phase and an eradication phase to prevent recurrence. The choice of antibiotics depends upon the susceptibility of the bacteria to various antibiotics. ''B. pesudomallei'' are generally susceptible to ceftazidime, meropenem, imipenem, and co-amoxiclav. These drugs generally kill bacteria. ''B. pseudomallei'' is also susceptible to doyxcycline, chloramphenicol, and co-trimoxazole. These drugs generally inhibit the growth of the bacteria. However, the bacteria are resistant to penicillin, ampicillin, 1st and 2nd generation [[w:cephalosporin|cephalosporin]], gentamicin, streptomycin, tobramycin, macrolides, and polymyxins.<ref name="Joost 2018"/> On the other hand, 86% of the ''B. pseudomallei'' isolates from the region of [[w:Sarawak|Sarawak]], [[w:Malaysia|Malaysia]] are susceptible to gentamicin and this has not been found elsewhere in other parts of the world.<ref name="Podin 2014">{{cite journal |author=Podin Y |author2=Sarovich DS |author3=Price EP et al |title=Burkholderia pseudomallei Isolates from Sarawak, Malaysian Borneo, Are Predominantly Susceptible to Aminoglycosides and Macrolides |journal= Antimicrobial Agents and Chemotherapy |date=January 2014 |volume=58 |issue=1 |pages=162–166 |pmid=24145517 | pmc=3910780 | doi=10.1128/AAC.01842-13 }}</ref> Prior to 1989, the standard treatment for acute melioidosis was a three-drug combination of [[w:chloramphenicol|chloramphenicol]], [[w:co-trimoxazole|co-trimoxazole]] and [[w:doxycycline|doxycycline]]; this regimen is associated with a mortality rate of 80% and is no longer used unless no other alternatives are available.<ref name="CAZ">{{cite journal |author=White NJ |author2=Dance DA |author3=Chaowagul W |author4=et al | title = Halving of mortality of severe melioidosis by ceftazidime | journal =The Lancet | year = 1989 | volume = 2 | issue = 8665 | pages = 697–701 | pmid = 2570956 | doi = 10.1016/S0140-6736(89)90768-X }}</ref> All three drugs are bacteriostatic (they stop the bacterium from growing, but do not kill it) and the action of co-trimoxazole antagonizes both chloramphenicol and doxycycline.<ref name="Dance2006">{{cite journal |author=Dance DA |author2=Wuthiekanun V |author3=Chaowagul W |author4=White NJ |title=Interactions in vitro between agents used to treat melioidosis |journal=Journal of Antimicrobial Chemotherapy |year=1989 |volume=24 |pages=311–316 |pmid=2681117 |doi=10.1093/jac/24.3.311 |issue=3 }}</ref> ===Intensive phase=== [[w:Intravenous|Intravenous]] [[w:ceftazidime|ceftazidime]] is the current drug of choice for treatment of acute melioidosis and should be administered for at least 10 to 14 days. [[w:Meropenem|Meropenem]], [[w:imipenem|imipenem]] and the [[w:cefoperazone|cefoperazone]]-[[w:sulbactam|sulbactam]] combination (Sulperazone) are also effective.<ref name="Joost 2018"/> Intravenous amoxicillin-clavulanate ([[w:co-amoxiclav|co-amoxiclav]]) may be used if none of the above four drugs is available;<ref name="Joost 2018"/> co-amoxiclav prevents death from melioidosis as well as ceftazidime.<ref name="Bennett 2015">{{cite book |editor=Bennett JE; Dolin R; Blaser MJ |title=Mandell, Douglas, and Bennett's Principles and Practice of Infectious Diseases |chapter=''Burkholderia pseudomallei'' and ''Burkholderia mallei'': Melioidosis and Glanders |date=2015 |publisher=Elsevier |isbn=978-1-4557-4801-3 |pages=2541–2549|doi=10.1016/B978-1-4557-4801-3.00223-X|author=Currie BJ |edition=8th }}</ref> Co-amoxiclav is also used if patient has allergy towards [[w:Sulfonamide (medicine)|sulfonamide]], unable to tolerate co-trimaxazole, in pregnant patients or in children. High dose of co-amoxiclav (20 mg/kg for amoxicillin and 5mg/kg for clavulanate) is recommended to prevent treatment failures.<ref name="Cheng 2008">{{cite journal |author=Cheng AC |author2=Chierakul W |author3=Chaowagul W et al |title=Consensus Guidelines for Dosing of Amoxicillin-Clavulanate in Melioidosis |journal= American Journal of Tropical Medicine and Hygiene |date=February 2008 |volume=78 |issue=2 |pages=208–209 |pmid=18256414 | pmc=PMC3034162 | doi=10.4269/ajtmh.2008.78.208 }}</ref><ref name="Suputtamongkol 1994">{{cite journal |author=Suputtamongkol Y |author2=Rajchanuwong A |author3=Chaowagul W et al |title=Ceftazidime vs. amoxicillin/clavulanate in the treatment of severe melioidosis |journal= Clinical Infectious Diseases |date=November 1994 |volume=19 |issue=5 |pages=846–853 |pmid=7893868 | pmc= | doi=10.1093/clinids/19.5.846 }}</ref> Intravenous [[w:antibiotic|antibiotic]]s are given for a minimum of 10 to 14 days. The median fever clearance time in melioidosis is 9 days.<ref name="Joost 2018"/> The treatment duration is in accordance with Darwin melioidosis treatment guidelines where there is low rate of [[w:recrudescence|recrudescence]] and [[w:relapse|relapse]].<ref name="Sullivan 2020">{{cite journal |author=Sullivan RP |author2=Marshall CS |author3=Anstey NM |author4=Ward L |author5=Currie BJ |title=2020 Review and revision of the 2015 Darwin melioidosis treatment guideline; paradigm drift not shift |journal= PLOS Neglected Tropical Diseases |date=28 September 2020 |volume=14 |issue=9 |pages=e0008659 |pmid=32986699 | pmc=7544138 | doi=10.1371/journal.pntd.0008659 }}</ref> Meropenem is the preferred antibiotic therapy for neurological melioidosis and those with [[w:sepsis|septic shock]] admitted into [[w:intensive care unit|intensive care unit]]s. Co-trimoxazole is recommended in addition to ceftazidime for neurological melioidosis, osteomyelitis, septic arthritis, skin and gastrointestinal infection, and deeply seated abscess. For deep-seated infections such as abscesses of internal organs, osteomyelitis, septic arthritis, and neurological melioidosis, the duration of antibiotics given should be longer (up to 4 to 8 weeks). The time taken for the fever to be resolved can be more than 10 days in those with deep-seated infection. According to the 2020 Revised Royal Darwin Hospital Guideline, the dosage for intravenous ceftazidime is 2g 6-hourly in adults (50&nbsp;mg/kg up to 2g in children less than 15 years old). The dosage for meropenem is 1g 8-hourly in adults (25&nbsp;mg/kg up to 1g in children).<ref name="Sullivan 2020"/> Acquired resistance to ceftazidime, carbapenems, and co-amoxiclav is rare in the intensive phase but resistance to cotrimoxazole during eradication therapy is technically difficult to assess.<ref name="Dance 2014"/> There are no differences between using cefoperazone/sulbactam or ceftazidime to treat melioidosis as both shows similar death rates and disease progression following treatment. However, data are lacking to recommend cefoperazone/sulbactam usage.<ref name="Dance 2014">{{cite journal |author=Dance D |title=Treatment and prophylaxis of melioidosis |journal=International Journal of Antimicrobial Agents |date=April 2014 |volume=43 |issue=4 |pages=310–318 |pmid=24613038 |doi=10.1016/j.ijantimicag.2014.01.005}}</ref><ref name="Apisarnthanarak 2002">{{cite journal |author1=Apisarnthanarak A |author2= Little JR |title=The role of cefoperazone-sulbactam for treatment of severe melioidosis |journal=Clinical Infectious Diseases |date=1 March 2002 |volume=34 |issue=5 |pages=721–722 |pmid=11823963 |doi=10.1086/338722}}</ref> For those with kidney impairment, the dosage of ceftazidime, meropenem, and co-trimoxazole should be lowered.<ref name="Currie 2015"/> Once the clinical condition improved, meropenem can be switched back to ceftazidime.<ref name="Joost 2018"/> ===Eradication phase=== Following the treatment of the acute disease, eradication treatment with [[w:co-trimoxazole|co-trimoxazole]] is the drug of choice and should be used for 3 months (12 weeks) as all-cause mortality was lower in 12 weeks group when compared to those receiving treatment for 20 weeks.<ref name="Anunnatsiri 2021">{{cite journal |author1=Anunnatsiri S |author2=Chaowagul W |author3=Teparrukkul P |title=A Comparison Between 12 Versus 20 Weeks of Trimethoprim-sulfamethoxazole as Oral Eradication Treatment for Melioidosis: An Open-label, Pragmatic, Multicenter, Non-inferiority, Randomized Controlled Trial |journal= Clinical Infectious Diseases |date=2021 |volume=73 |issue=11 |pages=e3627–e3633 |pmid=32725199 | pmc=PMC8662794 | doi=10.1093/cid/ciaa1084 }}</ref> For those with neurological melioidosis and [[w:osteomyelitis|osteomyelitis]], drugs should be given for more than 6 months. Co-amoxiclav and [[w:doxycycline|doxycycline]] are drugs of second choice. Co-trimoxazole should not be used in those with [[w:glucose-6-phosphate dehydrogenase|glucose-6-phosphate dehydrogenase]] deficiency as it can cause [[w:haemolytic anemia|haemolytic anemia]]. However, in Thailand, usage of co-trimoxazole does not accompany G6PD screening.<ref name="Joost 2018"/> Other side effects such as rash, [[w:hyperkalemia|hyperkalemia]], renal dysfunction, and gastrointestinal symptoms should prompt the reduction of co-trimoxazole doses. [[w:Chloramphenicol|Chloramphenicol]] is no longer routinely recommended for this purpose. Co-amoxiclav is an alternative for patients unable to take co-trimoxazole and [[w:doxycycline|doxycycline]] (e.g. pregnant women and children under the age of 12), but is not as effective and has a higher relapse rate. Single-agent treatment with [[w:fluoroquinolone|fluoroquinolone]] (e.g., [[w:ciprofloxacin|ciprofloxacin]]) or doxycycline for the oral eradication phase is ineffective.<ref name="Joost 2018"/> In Australia, co-trimoxazole is used with children and pregnant mothers after the first 12 weeks of pregnancy. Meanwhile, in Thailand, co-amoxiclav is the drug of choice for children and pregnant women.<ref name="Joost 2018"/> ''B. pseudomallei'' rarely acquires resistance when co-amoxiclav is used.<ref name="Dance 2014"/> The dosing regimen for co-trimoxazole (trimethoprim/sulfamethoxazole) in eradication phase is 6/30&nbsp;mg/kg, up to maximum 240/1200&nbsp;mg in children, 240/1200&nbsp;mg in adults weighing 40 to 60&nbsp;kg, and 320/1600&nbsp;mg in adults weighing more than 60&nbsp;kg, taken orally every 12 hours.<ref name="Sullivan 2020"/> In both Thailand and Australia, co-trimoxazole is taken together with [[w:folic acid|folic acid]] (0.1&nbsp;mg/kg up to 5&nbsp;mg in children).<ref name="Joost 2018"/><ref name="Sullivan 2020"/> There are also cases where melioidosis is successfully treated with co-trimoxazole for 3 months without going through intensive therapy provided that there is only skin manifestations without the involvement of internal organs or sepsis.<ref name="Joost 2018"/> Resistance to cotrimoxazole is rare in Asia.<ref name="Dance 2014 resistance">{{cite journal |author1=Dance DAB |author2=Davong V |author3=Soeng S |author4=Phetsouvanh R |author5=Newton PN |author6=Turner P |title=Trimethoprim/sulfamethoxazole resistance in Burkholderia pseudomallei |journal= International Journal of Antimicrobial Agents |date=October 2014 |volume=44 |issue=4 |pages=368–369 |pmid=25245211 | pmc=4195405 | doi=10.1016/j.ijantimicag.2014.06.003 }}</ref> Besides that, it is difficult to determine the resistance reliably because resistance to cotrimoxazole is defined when [[w:minimum inhibitory concentration|minimum inhibitory concentration]] (MIC) of more than 4 mg/L is required to completely inhibit the growth of 80% of the bacteria (80% inhibition point). Interpretation of 80% inhibition point is subjective and prone to human error.<ref name="Saiprom 2015">{{cite journal |author1=Saiprom N |author2=Amornchai P |author3=Wuthiekanun V et al |title=Trimethoprim/sulfamethoxazole resistance in clinical isolates of Burkholderia pseudomallei from Thailand |journal= International Journal of Antimicrobial Agents |date=May 2015 |volume=45 |issue=5 |pages=557–559 |pmid=25758020 | pmc=4537509 | doi=10.1016/j.ijantimicag.2015.01.006 }}</ref> In 2021, European Committee on Antimicrobial Susceptibility Testing (EUCAST) released a new guideline on interpreting the susceptibility of ''B pseudomallei'' towards various antibiotics on [[w:antibiotic sensitivity testing|disc susceptibility testing]]. The new guideline includes "S" for susceptible organism, "I" for susceptible organism only after increased exposure (when dosage or concentration of the drug increases) and "R" for resistant organism.<ref name="Dance 2021">{{cite journal |author1=Dance DAB |author2=Wuthiekanun P |author3=Baird W et al |title=Interpreting Burkholderia pseudomallei disc diffusion susceptibility test results by the EUCAST method |journal= Clinical Microbiology and Infection |date=June 2021 |volume=27 |issue=6 |pages=827–829 |pmid=33636339 | pmc= | doi=10.1016/j.cmi.2021.02.017 }}</ref> ===Surgery=== Surgical drainage is indicated for single, large abscesses in the liver, muscle, and prostate. However, for multiple abscesses in the liver, spleen, and kidney, surgical drainage may not be possible or necessary. For septic arthritis, [[w:arthrotomy|arthrotomy]] washout and drainage are required. Surgical [[w:debridement|debridement]] may be necessary.<ref name="Joost 2018"/> For those with [[w:mycotic aneurysm|mycotic aneurysm]], urgent surgery is required for prosthetic vascular grafts. Lifelong therapy with co-trimoxazole may be needed for those with prosthetic vascular grafts according to a review of case reports in 2005.<ref name="Hong Low 2005">{{cite journal |author=Low JGH |author2=Quek AML |author3=Sin YK |author4=Ang BSP |title=Mycotic aneurysm due to Burkholderia pseudomallei infection: case reports and literature review |journal= Clinical Infectious Diseases |date=January 2005 |volume=40 |issue=1 |pages=193–198 |pmid=15614712 | pmc= | doi=10.1086/426590 }}</ref> Other abscesses rarely need to be drained because most resolve with antibiotic treatment.<ref name="Joost 2018"/> Prostate abscess may require routine imaging. Antibiotics treatment for prostatic abscess may be enough except for abscesses more than 10 to 15 mm where surgical drainage is required.<ref name="Kozlowska 2018">{{cite journal |author=Kozlowska J |author2=Smith S |author3=Roberts J |author4=Pridgeon S |author5=Hanson J |title=Prostatic Abscess due to Burkholderia pseudomallei: Facilitating Diagnosis to Optimize Management |journal= American Journal of Tropical Medicine |date=January 2018 |volume=98 |issue=1 |pages=227–230 |pmid=29141724 | pmc= 5928742 | doi=10.4269/ajtmh.17-0633 }}</ref><ref name="Ng 2009">{{cite journal |author=Ng TH |author2=How SH |author3=Amran AR |author4=Razali MR |author5=Kuan YC |title=Melioidotic prostatic abscess in Pahang |journal= American Journal of Tropical Medicine |date=April 2009 |volume=50 |issue=4 |pages=385–389 |pmid=19421682 | pmc= | doi=10.4269/ajtmh.17-0633 }}</ref><ref name="Morse 2009">{{cite journal |author=Morese LP |author2=Moller CCB |author3=Harvey E |author4=Ward L et al |title=Prostatic Abscess Due to Burkholderia pseudomallei: 81 Cases From a 19-Year Prospective Melioidosis Study |journal= The Journal of Urology |date=1 August 2009 |volume=182 |issue=2 |pages=542–547 |pmid=19524969 | pmc= | doi=10.1016/j.juro.2009.04.010 }}</ref> ===Others=== Several immunomodulating therapies are suggested to boost the human body immune function against the bacteria because the pathogenesis of melioidosis is thought to be contributed by defects in [[w:neutrophil|neutrophils]].<ref name="Joost 2018"/> The Royal Darwin Hospital 2014 guidelines recommended [[w:granulocyte colony-stimulating factor|granulocyte colony-stimulating factor]] (G-CSF) as immunomodulating therapy for those with septic shock at 300ug daily as soon as the bacteriological laboratory flag the culture as possibly ''Burkholderia pseudomallei''. The main contraindication of starting (G-CSF) is a heart event. The G-CSF is continued for ten days depends on clinical response or a contraindication develops such as white cell count greater than >50,000 X106/litre.<ref name="Sullivan 2020"/> Anti-PDI ([[w:programmed cell death|programmed cell death]]) agents could be useful in melioidosis treatment especially for those with septic shock. This is because ''Burkholderia pseudomallei'' bacteria increases the expression of PDI-1 that regulates and inhibits the formation of T-cells that are essential for fighting against melioidosis.<ref name="Buddhisa 2015">{{cite journal |author=Buddhisa S |author2=Rinchai D |author3=Ato M et al |title=Programmed Death Ligand 1 on Burkholderia pseudomallei–Infected Human Polymorphonuclear Neutrophils Impairs T Cell Functions |journal= The Journal of Immunology |date=May 2015 |volume=194 |issue=9 |pages=4413–4421 |pmid=25801435 | pmc=3910780 | doi=10.4049/jimmunol.1402417 }}</ref> ==Prognosis== In well-resourced settings, where the disease can be detected and treated early, the risk of death is 10%. In resource-poor settings, the risk of death from the disease is more than 40%.<ref name="Joost 2018"/> Recurrent melioidosis can occur either due to re-infection or relapse after the completion of eradication therapy. Re-infection is due to a new strain of ''B. pseudomallei'' bacteria. Meanwhile, relapse is due to failure to clear infections after the eradication therapy. Recurrent melioidosis is rare since 2014 due to improved antibiotic therapy and prolongation of the intensive phase of therapy as evident in Darwin Prospective Melioidosis Study.<ref name="Sarovich 2014">{{cite journal |author=Sarovich DS |author2=Ward L |author3=Price EP |author4=Mayo M et al |title=Recurrent Melioidosis in the Darwin Prospective Melioidosis Study: Improving Therapies Mean that Relapse Cases Are Now Rare |journal=Journal of Clinical Microbiology |date=February 2014 |volume=52 |issue=2 |pages=650–653 |pmid=24478504 |pmc=3911345 | doi=10.1128/JCM.02239-13 }}</ref> On the other hand, [[w:recrudescence|recrudescence]] are those who present with symptoms during the eradication therapy. Recrudescence rates may be improved by ensuring adherence to a full course of eradication therapy e.g. by reducing self-discharge [[w:against medical advice|against medical advice]].<ref name="Pitman 2015">{{cite journal |author=Pitman MC |author2=Luck T |author3=Marshall CS |author4=Anstey NM et al |title=Intravenous Therapy Duration and Outcomes in Melioidosis: A New Treatment Paradigm |journal=PLOS Neglected Tropical Diseases |date=26 March 2015 |volume=9 |issue=4 |pages=e0003737 |pmid=25811783 |pmc=4374799 | doi=10.1371/journal.pntd.0003586 }}</ref> Underlying medical conditions such as diabetes mellitus, chronic kidney disease, and cancer can worsen the long-term survival and disability of those who recover from infection. One of the complications of melioidosis is [[w:encephalomyelitis|encephalomyelitis]]. It can cause quadriparesis (muscle weakness in all the limbs), partial flaccid paraparesis (muscle weakness of both legs), or foot drop. For those with previous melioidosis-associated bone and joint infections, complications such as sinus tract infection, bone and joint deformities with limited range of motion can occur.<ref name="Joost 2018"/> ==Epidemiology== {{fig|9 |Melioidosis world map distribution.svg |Number of deaths by each country due to melioidosis in 2018. |attribution=Cerevisae, [https://creativecommons.org/licenses/by-sa/4.0/deed.en CC-BY-SA 4.0] |size=285px }} Melioidosis is an understudied disease that remains endemic in developing countries. In 2015, the International Melioidosis Society was formed to raise awareness of the disease.<ref name="Joost 2018"/> In 2016, a [[w:statistical model|statistical model]] was developed which predicted that the number is 165,000 cases per year with 138,000 of those occurring in East and South Asia and the Pacific.<ref>{{cite journal |author=Limmathurotsakul D |author2=Golding N |author3=Dance DA |author4=Messina JP |author5=Pigott DM |author6=Moyes CL |author7=Rolim DB |author8=Bertherat E |author9=Day NP |author10=Peacock SJ |author11=Hay SI |title=Predicted global distribution of Burkholderia pseudomallei and burden of melioidosis |journal=Nature Microbiology |date=11 January 2016 |volume=1 |issue=1 |pages=15008 |doi=10.1038/nmicrobiol.2015.8 |pmid=26877885 |pmc=4746747 }}</ref> In approximately half of those cases (54% or 89,000), people will die.<ref name="Joost 2018"/> Under-reporting is a common problem as only 1,300 cases were reported worldwide since 2010, which is less than 1% of the projected incidence based on the modelling.<ref name="Joost 2018"/> Lack of laboratory diagnostic capabilities and lack of disease awareness amongst health care providers also causes underdiagnosis. Even if bacterial cultures show positive result for ''B. pesudomallei'', they can be discarded as contaminants especially in laboratories in non-endemic areas.<ref name="Joost 2018"/> In 2015, it was estimated that the yearly [[w:disability-adjusted life year|disability-adjusted life year]] (DALY) was 84.3 per 100,000 people. As of 2022, melioidosis is not included in the WHO list of [[w:neglected tropical diseases|neglected tropical diseases]].<ref>{{cite journal |author=Birnie E |author2=Virk HS |author3=Savelkoel J |author4=Spijker R |author5=Bertherat E |title=Global burden of melioidosis in 2015: a systematic review and data synthesis |journal=The Lancet Infectious Diseases |date=August 2019 |volume=19 |issue=8 |pages=892–902 |doi=10.1016/S1473-3099(19)30157-4 |pmid=31285144 |pmc=6867904 }}</ref><ref>{{cite journal |author=Savelkoel J |author2=Dance DAB |author3=Currie BJ et al |title=A call to action: time to recognise melioidosis as a neglected tropical disease |journal=The Lancet Infectious Diseases |date=22 June 2022 |volume=22 |issue=6 |pages=e176–e182 |doi=10.1016/S1473-3099(21)00394-7 |pmid=34953519 |pmc= }}</ref> Melioidosis is endemic in parts of southeast Asia (including Thailand,<ref name="Hinjoy 2018">{{cite journal |author=Hinjoy S |author2=Hantrakun V |author3=Kongyu S |author4=Kaewrakmuk J |author5=Wangrangsimakul T et al |title=Melioidosis in Thailand: Present and Future |journal=Tropical Medicine and Infectious Disease |date=8 April 2018 |volume=3 |issue=2 |pages=38 |doi=10.3390/tropicalmed3020038 |pmid=29725623 |pmc=5928800 }}</ref> Laos,<ref>{{cite journal |author=Dance DAB |author2=Luangraj M |author3=Rattanavong S |author4=Sithivong N et al |title=Melioidosis in the Lao People's Democratic Republic |journal=Tropical Medicine and Infectious Disease |date=19 February 2018 |volume=3 |issue=1 |pages=21 |doi=10.3390/tropicalmed3010021 |pmid=30274419 |pmc=6136615 }}</ref> Singapore,<ref>{{cite journal |author=Sim SH |author2=Ong CEL |author3=Gan YH |author4=Wang D et al |title=Melioidosis in Singapore: Clinical, Veterinary, and Environmental Perspectives |journal=Tropical Medicine and Infectious Disease |date=12 March 2018 |volume=3 |issue=1 |pages=31 |doi=10.3390/tropicalmed3010031 |pmid=30274428 |pmc=6136607 }}</ref> Brunei,<ref>{{cite journal |author=Pande K |author2=Kadir KAA |author3=Asli R |author4=Chong VH |title=Melioidosis in Brunei Darussalam |journal=Tropical Medicine and Infectious Disease |date=19 February 2018 |volume=3 |issue=1 |pages=20 |doi=10.3390/tropicalmed3010020 |pmid=30274418 |pmc=6136610 }}</ref> Malaysia,<ref name="Nathan 2018">{{cite journal |author=Nathan S |author2=Chieng S |author3=Kingsley PV |author4=Mohan A |author5=Podin Y et al |title=Melioidosis in Malaysia: Incidence, Clinical Challenges, and Advances in Understanding Pathogenesis |journal=Tropical Medicine and Infectious Disease |date=27 February 2018 |volume=3 |issue=1 |pages=25 |doi=10.3390/tropicalmed3010025 |pmid=30274422 |pmc=6136604 }}</ref> Myanmar<ref>{{cite journal |author=Win MM |author2=Ashley EA |author3=Zin KN |author4=Aung MT |author5=Swee MMM et al |title=Melioidosis in Myanmar |journal=Tropical Medicine and Infectious Disease |date=1 March 2018 |volume=3 |issue=1 |pages=28 |doi=10.3390/tropicalmed3010028 |pmid=30274425 |pmc=6136617 }}</ref> and Vietnam<ref>{{cite journal |author=Trinh TT |author2=Nguyen LDN |author3=Nguyen TV |author4=Tran CX |author5=Le AV et al |title=Melioidosis in Vietnam: Recently Improved Recognition but still an Uncertain Disease Burden after Almost a Century of Reporting |journal=Tropical Medicine and Infectious Disease |date=9 April 2018 |volume=3 |issue=2 |pages=39 |doi=10.3390/tropicalmed3020039 |pmid=30274435 |pmc=6073866 }}</ref>), southern China,<ref>{{cite journal |author=Zheng X |author2=Xia Q |author3=Xia L |author4=Li W |title=Endemic Melioidosis in Southern China: Past and Present |journal=Tropical Medicine and Infectious Disease |date=25 February 2019 |volume=4 |issue=1 |pages=39 |doi=10.3390/tropicalmed4010039 |pmid=30823573 |pmc=6473618 }}</ref> Taiwan<ref>{{cite journal |author=Hsueh PT |author2=Huang WT |author3=Huseueh HK |author4=Chen YL |author5=Chen SY et al |title=Transmission Modes of Melioidosis in Taiwan |journal=Tropical Medicine and Infectious Disease |date=28 February 2018 |volume=3 |issue=1 |pages=26 |doi=10.3390/tropicalmed3010026 |pmid=30274423 |pmc=6136622 }}</ref> northern Australia.<ref name="Smith 2018">{{cite journal |author=Smith S |author2=Hanson J |author3=Currie BJ |title=Melioidosis: An Australian Perspective |journal=Tropical Medicine and Infectious Disease |date=1 March 2018 |volume=3 |issue=27 |pages=1 |doi=10.3390/tropicalmed3010027 |pmid=30274424 |pmc=6136632 }}</ref> India,<ref name="Mukhopadhyay 2018">{{cite journal |author=Mukhopadhyay C |author2=Shaw T |author3=Varghese GM |author4=Dance DAB |title=Melioidosis in South Asia (India, Nepal, Pakistan, Bhutan and Afghanistan) |journal=Tropical Medicine and Infectious Disease |date=22 May 2018 |volume=3 |issue=2 |pages=51 |doi=10.3390/tropicalmed3020051 |pmid=30274447 |pmc=6073985 }}</ref> and South America.<ref>{{cite journal |author=Rolim DB |author2=Lima RX |author3=Ribeiro AK |author4=Colares RM et al |title=Melioidosis in South America |journal=Tropical Medicine and Infectious Disease |date=5 June 2018 |volume=3 |issue=2 |pages=60 |doi=10.3390/tropicalmed3020060 |pmid=30274456 |pmc=6073846 }}</ref> Since 1991, a total of 583 cases were reported in India. Most Indian cases are located in [[w:Karnataka|Karnataka]] and [[w:Tamil Nadu|Tamil Nadu]].<ref name="Mukhopadhyay 2018"/> Fifty-one cases of melioidosis were reported in Bangladesh from 1961–2017. Nonetheless, lack of awareness and resources gives rise to underdiagnosis of the disease in the country.<ref>{{cite journal |author=Chowdhury FR |author2=Jilani MSA |author3=Barai L |author4=Rahman T et al |title=Melioidosis in Bangladesh: A Clinical and Epidemiological Analysis of Culture-Confirmed Cases |journal=Tropical Medicine and Infectious Disease |date=9 April 2018 |volume=3 |issue=2 |pages=40 |doi=10.3390/tropicalmed3020040 |pmid=30274436 |pmc=6073520 }}</ref> The true burden of melioidosis in Africa and Middle East remain unknown due to low amount of data. Several melioidosis cases were reported over the years. Although 24 African countries and three Middle Eastern countries predicted to be endemic with melioidosis, however not a single case was reported from these specific countries.<ref>{{cite journal |author=Steinmetz I |author2=Wagner GE |author3=Kanyala E et al |title=Melioidosis in Africa: Time to Uncover the True Disease Load |journal=Tropical Medicine and Infectious Disease |date=10 June 2018 |volume=3 |issue=2 |pages=62 |doi=10.3390/tropicalmed3020062 |pmid=30274458 |pmc=6073667 }}</ref> In the United States, two historical cases (1950 and 1971) and four recent cases (2010, 2011, 2013, 2020) have been reported amongst people that did not travel overseas.<ref name="Currie 2015"/><ref>{{cite journal |author=Cossaboom CM |author2=Marinova-Petkova A |author3=Strysko J et al |title=Melioidosis in a Resident of Texas with No Recent Travel History, United States |journal=Emerging Infectious Diseases |date=June 2020 |volume=26 |issue=6 |pages=1295–1299 |doi=10.3201/eid2606.190975 |pmid=32442394 |pmc=7258475 }}</ref> Despite extensive investigations, the source of melioidosis was never confirmed. One possible explanation is that importation of medicinal plant products or exotic reptiles could have resulted in the introduction of melioidosis in the United States.<ref name="Currie 2015"/> In 2021, there was a melioidosis outbreak in several states in the United States due to usage of contaminated aromatherapy spray imported from India.<ref>{{cite journal |author=Gee JE |author2=Bower WA |author3=Kunkel A et al |title=Multistate Outbreak of Melioidosis Associated with Imported Aromatherapy Spray |journal=New England Journal of Medicine |date=March 2022 |volume=386 |issue=9 |pages=861–868 |doi=10.1056/NEJMoa2116130 |pmid=35235727 }}</ref> There are also cases of infection through imported tropical fishes in home aquariums.<ref>{{cite journal |author=Dawson P |author2=Duwell MM |author3=Elrod MG et al |title=Human Melioidosis Caused by Novel Transmission of Burkholderia pseudomallei from Freshwater Home Aquarium, United States |journal=Emerging Infectious Diseases |date=December 2021 |volume=27 |issue=12 |pages=3030–3035 |doi=10.3201/eid2712.211756 |pmid=34570693 |pmc=8632198 }}</ref> In Europe, more than half of the melioidosis cases are imported from Thailand.<ref>{{cite journal |author=Tohic SL |author2=Montana M |author3=Koch L et al |title=A review of melioidosis cases imported into Europe |journal=European Journal of Clinical Microbiology & Infectious Diseases |date=25 March 2019 |volume=38 |issue=8 |pages=1395–1408 |doi=10.1007/s10096-019-03548-5 |pmid=30949898 }}</ref> Melioidosis is found in all age groups.<ref name="Joost 2018"/> For Australia and Thailand, the median age of infection is at 50 years; 5 to 10% of the patients are under 15 years.<ref name="Joost 2018"/> The single most important risk factor for developing melioidosis is [[w:diabetes mellitus|diabetes mellitus]], followed by hazardous alcohol use, chronic kidney disease, and chronic lung disease.<ref name="Suputtamongkol 1999">{{cite journal |author=Suputtamongkol Y |author2=Chaowagul W |author3=Chetchotisakd P |author4=Lertpatanasuwun N et al |year=1999 |title=Risk factors for melioidosis and bacteremic melioidosis |journal=Clinical Infectious Diseases |pmid=10476750 |volume=29 |issue=2 |pages=408–413 |doi=10.1086/520223 }}</ref> More than 50% of people with melioidosis have diabetes; diabetics have a 12-fold increased risk of contracting melioidosis. Diabetes decreases the ability of macrophages to fight the bacteria and reduced the [[w:T helper cell|T helper cell]] production. Excessive release of [[w:Tumor necrosis factor alpha|Tumor necrosis factor alpha]] and [[w:Interleukin 12|Interleukin 12]] by [[w:peripheral blood mononuclear cell|mononuclear cells]] increases the risk of septic shock.<ref name="Joost 2018"/> Other risk factors include [[w:thalassaemia|thalassaemia]], occupational exposure (e.g. [[w:rice paddy|rice paddy]] farmers),<ref name="Gassiep 2020"/> recreational exposure to soil, water, being male, age greater than 45 years, and prolonged steroid use/immunosuppression.<ref name="Joost 2018"/> However, 8% of children and 20% of adults with melioidosis have no risk factors.<ref name="Joost 2018"/> [[w:HIV|HIV]] infection does not appear to predispose to melioidosis, although several other co-infections have been reported.<ref name="Gassiep 2020"/> [[w:Infant|Infant]] cases have been reported possibly due to mother-to-child transmission, community-acquired infection, or healthcare-associated infection.<ref name="Joost 2018"/> Those who are well may also be infected with ''B. pseudomallei''. For example, 25% of children started [[w:seroconversion|producing antibodies]] against ''B. pseudomallei'' between 6 months to 4 years of staying in [[w:endemic (epidemiology)|endemic]] areas although they did not experience any melioidosis symptoms; suggesting they were exposed to it over this time. This means that many people without symptoms will test positive in serology tests in endemic areas.<ref name="Yi 2014"/> In Thailand, the seropositivity rate exceeds 50%, while in Australia the seropositivity rate is only 5%.<ref name="Currie 2015"/> The disease is clearly associated with increased rainfall, with the number of cases rising following increased precipitation. Severe rainfall increases the concentration of the bacteria in the topsoil, thus increasing thus of transmitting the bacteria through the air.<ref name="Gassiep 2020"/> A recent CDC Advisory indicated that the recent detection of the organism in the environment in Mississippi following the occurrence of two indigenous cases of melioidosis, confirms that parts of the southern USA should now be regarded as melioidosis-endemic.<ref>{{Cite web|url=https://www.emergency.cdc.gov/han/2022/pdf/CDC_HAN_470.pdf |title=Melioidosis Locally Endemic in Areas of the Mississippi Gulf Coast after Burkholderia pseudomallei Isolated in Soil and Water and Linked to Two Cases – Mississippi, 2020 and 2022|publisher=CDC|access-date=2022-08-05}}</ref> ==History== Pathologist [[w:Alfred Whitmore|Alfred Whitmore]] and his assistant Krishnaswami first reported melioidosis among beggars and morphine addicts at autopsy in Rangoon, present-day [[w:Burma|Myanmar]], in a report published in 1912.<ref name="Whitmore 1912">{{cite journal |author=Whitmore A |author2=Krishnaswami CS |title=A Hitherto Undescribed Infective Disease in Rangoon |journal=Indian Medical Gazette |date=1912 |volume=47 |issue=7 |pages=262–267 |pmid=29005374 |pmc=5168169 |doi= }}</ref> Whitmore was able to grow the organism in culture and its showed similarity with ''B. mallei'', another bacteria that was known causing glanders in animals. Therefore, he named the new organism ''Bacillus pseudomallei''. He did no further work on the organism.<ref name="Brightman 2020">{{cite journal |author=Brightman C |author2=Locum |title=Melioidosis, the Vietnamese time bomb |journal=Trends in Urology and Men's Health |year=4 June 2020 |volume=11 |issue=3 |pages=30–32 |pmid= |pmc= |doi=10.1002/tre.753 }}</ref> [[w:Arthur Conan Doyle|Arthur Conan Doyle]] may have read Whitmore's report before writing a short story that involved the fictitious tropical disease "Tapanuli fever" in a [[w:Sherlock Holmes|Sherlock Holmes]] story<ref name="pmid11823558">{{cite journal |author=Vora SK |title=Sherlock Holmes and a biological weapon |journal=Journal of the Royal Society of Medicine |volume=95 |issue=2 |pages=101–103 |date=February 2002 |pmid=11823558 |pmc=1279324 |doi=10.1258/jrsm.95.2.101 |url=http://www.jrsm.org/cgi/pmidlookup?view=long&pmid=11823558 }}</ref> titled “[[w:The Adventure of the Dying Detective|The Adventure of the Dying Detective]]” published in 1913.<ref name="Allen C 2005"/> In the same year, melioidosis outbreak occurred inside the Institute for Medical Research (IMR), [[w:Kuala Lumpur|Kuala Lumpur]], [[w:Federated Malay States|Malaya]] after its laboratory animals such as guinea pigs and rabbits were infected.<ref name="Nathan 2018"/> William Fletcher and Ambrose Thomas Stanton, doctors who worked at the IMR, were the next ones to study the organism. They were unable identify the organism that caused the outbreak. It was only in 1917, when Fletcher isolated an organism similar to Whitmore's bacillus from a Tamil rubber estate worker, the presence of the new species of bacteria was confirmed.<ref name="Brightman 2020"/> The term “melioidosis” was first coined in 1921. The name melioidosis is derived from the Greek ''melis'' (μηλις) meaning "a distemper of asses" with the suffixes -oid meaning "similar to" and -osis meaning "a condition", that is, a condition similar to glanders.<ref name="Stanton 1921">{{cite book |author=Stanton AT |author2=Fletcher W |chapter=Melioidosis, a new disease of the tropics |title=Far Eastern Association of Tropical Medicine: Transactions of the Fourth Congress |location=Batavia, Dutch East Indies |publisher=Javasche Boekhandel en Drukkerij |year=1921 }}</ref> ''B pseudomallei'' is similar in clinical presentation and genome make-up with ''B. mallei''<ref name="Godoy 2003">{{cite journal |author=Godoy D |author2=Randle G |author3=Simpson AJ |author4=Aanensen DM et al |title=Multilocus sequence typing and evolutionary relationships among the causative agents of melioidosis and glanders, Burkholderia pseudomallei and Burkholderia mallei |journal=Journal of Clinical Microbiology |year=May 2003 |volume=41 |issue=5 |pages=2068–2079 |pmid=12734250 |pmc=154742 |doi=10.1128/jcm.41.5.2068-2079.2003 }}</ref> but is distinguished from it due to epidemiological and [[w:zoonosis|zoonotic]] characteristics.<ref name="Yabuuci 1992"/> The first human case of melioidosis in South Asia was reported in Sri Lanka in 1927.<ref name="Mukhopadhyay 2018"/> In 1932, Thomas and Fletcher collected 83 cases of melioidosis from literature. In this [[w:cases series|case series]], there were only two survivals. Since then, more case series of melioidosis were reported.<ref name="Khaira 1959">{{cite journal |author=Khaira BS |author2=Young WB |author3=Hart PdeV |title=Melioidosis |journal=British Medical Journal |volume=1 |issue=5127 |pages=949–952 |date=11 April 1959 |pmid=13638596 |pmc=1993315 |doi=10.1136/bmj.1.5127.949 }}</ref> Thomas and Fletcher also pioneered the use of serological methods in diagnosing the disease. Thomas and Fletcher incorrectly believed that melioidosis infection came from human contact with [[w:rondent|rodents]]. However, observations on the disease noted that humans usually got it after exposure to mud or contaminated water. Besides, the organism was never grown from rats. This led to a search of the bacteria in the environment.<ref name="Ketheesan 2012">{{cite book |editor=Ketheesan N |chapter= Melioidosis in animals |title=Melioidosis: A Century of Observation and Research|author=Rush CM|author2=Thomas AD|isbn=978-0-444-53479-8 |location=Townsville, Australia |publisher=Elsevier B.V. |year=2012 |page=313 }}</ref> In 1936, the first animal (pig) case of melioidosis in Africa was reported in Madagascar.<ref name="Garin 2014">{{cite journal |author=Garin B |author2=Djaomazala I |author3=Dubois-Cauwelaert N et al |title=Autochthonous Melioidosis in Humans, Madagascar, 2012 and 2013 |journal=Emerging Infectious Diseases |volume=20 |issue=10 |pages=1739–1741 |date=October 2014 |pmid=25272365 |pmc=4193270 |doi=10.3201/eid2010.131524 }}</ref> In 1937, water was first identified as the habitat of ''B. pseudomallei''.<ref name="Limmathurotsakul review 2013 ">{{cite journal |author=Limmathurotsakul D |author2=Dance DAB |author3=Wuthiekanun V et al |title=Systematic Review and Consensus Guidelines for Environmental Sampling of Burkholderia pseudomallei |journal=PLOS Neglected Tropical Diseases |volume=7 |issue=3 |pages=e2105 |date=2013 |pmid=23556010 |pmc=3605150 |doi=10.1371/journal.pntd.0002105 }}</ref> The first case of Australian melioidosis was described in an outbreak in sheep in 1949 at [[w:North Queensland|North Queensland]]. This was followed by the first case of human melioidosis at [[w:Townsville|Townsville]] in 1950.<ref name="Currie 2000">{{cite journal |author=Currie BJ |author2=Fisher DA |author3=Howard DM et al |title=The epidemiology of melioidosis in Australia and Papua New Guinea |journal=Acta Tropica |volume=74 |issue=2 |pages=121–127 |date=February 2000 |pmid=10674639 |pmc= |doi=10.1016/s0001-706x(99)00060-1 }}</ref> Initially, the discovery of melioidosis in Australia had led to a debate on when and how the disease spread from Southeast Asia to a new distant environment.<ref name="Ketheesan 2012"/> However, this hypothesis was later disproved in 2017 when [[w:whole genome sequencing|whole genome sequencing]] of ''B. pseudomallei'' over 30 countries collected over 79 years suggested Australia as the early reservoir for melioidosis.<ref name="Chewapreecha 2017"/> In 1955, first case of local human melioidosis was reported in Thailand.<ref name="Hinjoy 2018"/> During the [[w:Vietnam War|Vietnam War]] from 1967 to 1973, 343 American soldiers were reported with melioidosis, with about 50 cases transmitted through inhalation.<ref name="Rubin 1963">{{cite journal |author=Rubin HL |author2=Alexander AD |author3=Yager RH |title=Melioidosis—a military medical problem? |journal=Military Medicine |volume=128 |issue= |pages=538–642 |date=1963 |pmid=13983001 |pmc= |doi=10.1093/milmed/128.6.538 }}</ref> An outbreak of melioidosis at the Paris Zoo in the 1970s (known as ''L'affaire du jardin des plantes'') was thought to have originated from an imported [[w:panda|panda]] or horses from Iran.<ref name="Allen C 2005"/><ref>{{cite journal |author=Mollaret HH |title='L'affaire du Jardin des plantes' ou comment le mélioïdose fit son apparition en France |journal=Médecine et Maladies Infectieuses |year=1988 |volume=18 |issue=Suppl 4 |pages=643–654 |doi=10.1016/S0399-077X(88)80175-6 }}</ref> It is unclear how imported melioidosis is able to persist in a completely new environment. Eventually, the outbreak terminated by itself after a period of time.<ref name="Ketheesan 2012"/> It was only during the 1980s, Infectious Disease Association of Thailand started took notice of this disease. First conference on melioidosis was held in 1985 in Thailand. It was during this meeting that collaboration between Sappasitprasong Hospital, Thailand, and Wellcome-Mahido-Oxford Tropical Medicine Research Programme was established. Such collaboration made Thailand a world leader in clinical and epidemiology research on melioidosis.<ref name="Ketheesan 2012"/> In 1989, several studies conducted in Thailand demonstrated ceftazidime as an effective antibiotic against melioidosis.<ref name="Ketheesan 2012"/> [[w:Ceftazidime|Ceftazidime]] had been shown to reduce the risk of death of melioidosis from 74% to 37%.<ref name="White 1989">{{cite journal |author=White NJ |author2=Dance DA |author3=Chaowagul W |author4=Wattanagoon Y et al |title=Halving of mortality of severe melioidosis by ceftazidime |journal=The Lancet |year=1989 |volume=23 |issue=2 |pages=697–701 |pmid=2570956 |pmc= |doi=10.1016/s0140-6736(89)90768-x }}</ref> In 1990, a non-virulent ‘arabinose-positive B. pseudomallei’ was found by Vanaporn Wuthiekanun. The organism was later reclassified into a new species called ''B. thailandensis''. This species has become a useful tool in the laboratory for the studies of the pathogenesis of ''B. pseudomallei''.<ref name="Ketheesan 2012"/> ''B. pseudomallei'' was previously classified as part of the genus ''[[w:Pseudomonas|Pseudomonas]]''. In 1992, the pathogen was formally named ''B. pseudomallei''.<ref name="Yabuuci 1992">{{cite journal |author=Yabuuci E |author2=Kosako Y |author3=Oyaizu H |author4=Yano I et al |title=Proposal of Burkholderia gen. nov. and transfer of seven species of the genus Pseudomonas homology group II to the new genus, with the type species Burkholderia cepacia (Palleroni and Holmes 1981) comb. nov |journal=Microbiology and Immunology |year=1993 |volume=36 |issue=12 |pages=1251–1275 |pmid=1283774 |pmc=154742 |doi=10.1111/j.1348-0421.1992.tb02129.x }}</ref> In 1994, First International Symposium on melioidosis was held in Kuala Lumpur where 80 delegates attended. Papers were presented and later published as a book.<ref name="Ketheesan 2012"/> Subsequent congresses were held in Thailand, Australia, and Singapore once every three years.<ref name="Ketheesan 2012"/> In 2002, ''B. pseudomallei'' was classified as a "Category B agent".<ref name="Rotz 2002">{{cite journal |author=Rotz LD |author2=Khan AS |author3=Lillibridge SR |author4=Ostroff SM |author5=Hughes JM |title=Public Health Assessment of Potential Biological Terrorism Agents |journal=Emerging Infectious Diseases |year=February 2002 |volume=8 |issue=2 |pages=225–230 |pmid=11897082 |pmc=2732458 |doi=10.3201/eid0802.010164 }}</ref> In 2004, the complete genome of ''B. pseudomallei'' was published.<ref name="Ketheesan 2012"/> In 2012, ''B pseudomallei'' was classified as a "Tier 1 select agent" by the U.S. Centers for Disease Control.<ref name="McRobb 2015">{{cite journal |author=McRobb E |author2=Sarovich DS |author3=Price EP |author4=Kaestli M et al |title=Tracing melioidosis back to the source: using whole-genome sequencing to investigate an outbreak originating from a contaminated domestic water supply |journal=Journal of Clinical Microbiology |year=April 2015 |volume=53 |issue=4 |pages=1144–1148 |pmid=25631791 |pmc=4365233 |doi=10.1128/JCM.03453-14 }}</ref> In 2014, [[w:co-trimoxazole|co-trimoxazole]] was established as the only oral eradication therapy rather than combination therapy of co-trimoxazole with doxycycline.<ref name="Chetchotisakd 2014">{{cite journal |author=Chetchotisakd P |author2=Chierakul W |author3=Chaowagul W |author4=Anunnatsiri S et al |title=Trimethoprim-sulfamethoxazole versus trimethoprim-sulfamethoxazole plus doxycycline as oral eradicative treatment for melioidosis (MERTH): a multicentre, double-blind, non-inferiority, randomised controlled trial |journal=The Lancet |year=1 March 2014 |volume=383 |issue=9919 |pages=807–814 |pmid=24284287 |pmc=3939931 |doi=10.1016/S0140-6736(13)61951-0 }}</ref> In 2016, a statistical model was developed to predict the occurrence of global melioidosis per year.<ref name="Limmathurotsakul 2016"/> ==Synonyms== *Pseudoglanders<ref>{{cite journal |author=Chai LYA |author2=Fisher D |title=Earth, wind, rain, and melioidosis |journal=The Lancet Planetary Health |date=1 August 2018 |volume=2 |issue=8 |pages=329–330 |doi=10.1016/S2542-5196(18)30165-7 |pmid=30082045 }}</ref> *Whitmore's disease (after Captain [[w:Alfred Whitmore|Alfred Whitmore]], who first described the disease)<ref name="Whitmore 1912"/> *Nightcliff gardener's disease ([[w:Nightcliff|Nightcliff]] is a suburb of [[w:Darwin, Northern Territory|Darwin, Australia]] where melioidosis is endemic)<ref>{{cite web |url=http://www.abc.net.au/pm/content/2005/s1285014.htm |title=Rise in melioidosis rates in NT |author=Barker A |date=19 June 2005 |accessdate=2007-06-24 |publisher=Australian Broadcasting Corporation }}</ref> *Paddy-field disease<ref>{{cite journal |author=Orellana C |title=Melioidosis strikes Singapore |journal=The Lancet Infectious Diseases |date=November 2004 |volume=4 |issue=11 |pages=655 |pmid=15534940 |doi=10.1016/S1473-3099(04)01190-9 }}</ref> *Morphia injector's septicaemia<ref name="Stevenson 1916">{{cite journal |author=Stevenson AC |title=Morphia injector's septicæmia (Whitmore's Disease) |journal=Transactions of The Royal Society of Tropical Medicine and Hygiene |date=June 1916 |volume=9 |issue=7 |pages=218–219 |pmid= |pmc= |doi=10.1016/S0035-9203(16)90035-X }}</ref> ==Biological warfare== Interest in melioidosis has been expressed because it has the potential to be developed as a [[w:biological warfare|biological weapon]]. Another similar bacterium, ''Burkholderia mallei'' was used by the Germans in [[w:World War I|World War I]] to infect livestock shipped to Allied countries.<ref name="Matthew 2018"/> Deliberate infection of human [[w:prisoners of war|prisoners of war]] and animals using ''B. mallei'' were carried out in China's [[w:Pingfang District|Pingfang District]] by the Japanese during World War II.<ref name="Allen C 2005"/> The [[w:Soviet Union|Soviet Union]] reportedly used ''B. mallei'' during the [[w:Soviet–Afghan War|Soviet–Afghan War]] in 1982 and 1984.<ref name="Matthew 2018"/> ''B. pseudomallei'', like ''B. mallei'', was studied by both the US<ref>{{cite book |editor=Withers MR |title=USAMRIID's Medical Management of Biological Casualties Handbook |edition=8th |publisher=U.S. Army Medical Institute of Infectious Diseases |location=Fort Detrick, Maryland|isbn=978-0-16-093126-0 |year=2014 |url=https://usamriid.health.mil/education/bluebookpdf/USAMRIID%20BlueBook%208th%20Edition%20-%20Sep%202014.pdf |pages=53 }}</ref> and Soviet Union as a potential biological warfare agent, but never weaponized.<ref name="Matthew 2018">{{cite journal |author=Nguyen HVN |author2=Smith ME |author3=Hayoun MA |title=Glanders and Melioidosis |date=27 October 2018 |pmid=28846298 |url=https://www.ncbi.nlm.nih.gov/books/NBK448110/ |accessdate=15 February 2019 |journal=StatPearls |publisher=StatPearls Publishing |id=NBK448110 }}</ref> Other countries such as Iran, Iraq, North Korea, and Syria may have investigated the properties of ''B. pseudomallei'' for biological weapons.<ref name="Yi 2014"/> The bacterium is readily available in the environment. It can also be aerosolized and transmitted via inhalation.<ref name="Yi 2014"/> However, the ''B. pseudomallei'' has never been used in biological warfare.<ref name="Yi 2014"/> The actual risk of the deliberate release of ''B. pseudomallei'' or ''B. mallei'' is unknown.<ref name="Gilad 1997">{{cite journal |author=Gilad J |author2=Harary I |author3=Dushnitsky T |author4=Schwartz D |author5=Amsalem Y |title=''Burkholderia mallei'' and ''Burkholderia pseudomallei'' as bioterrorism agents: national aspects of emergency preparedness |journal=The Israel Medical Association Journal |year=1997 |volume=9 |issue=7 |pages=499–503 |pmid=17710778 }}</ref> ==Additional information== ===Acknowledgements=== All authors contributed to the article. ===Conflict of interest=== No funding was received for the preparation of this article. No financial competing interests declared. ===Ethics statement=== No ethical permission required. ==References== {{Reflist}} dzlrb8gpczo2drcdaontsgjfkbs3lsq 0 263444 2416323 2415975 2022-08-18T22:24:38Z Scogdill 1331941 wikitext text/x-wiki == Overview == * Louise, Duchess of Devonshire was the subject of much comment and gossip in her life, arising in part from her prominence in society and in part from her conduct, which was not particularly controlled by middle-class notions of "respectability." * "Louise Alten, for her part, exemplifies another vanished feature [of Victorian England], the role of the political hostess. The part she played in Hartington's career even while she was his mistress gives a special twist to this function, and her more conventional ascendancy as Duchess of Devonshire, when she was said to be the most powerful person outside the government, shows the influence that could be wielded by women in a political system from which they were formally excluded."<ref name=":20">Vane, Henry. ''Affair of State: A Biography of the 8th Duke and Duchess of Devonshire''. Peter Owen, 2004.</ref>{{rp|12}} == Also Known As == *Louise, Duchess of Devonshire (15 January 1832 – 15 November 1911) *Louisa, Duchess of Manchester *Luise Friederike August Gräfin von Alten *Louisa Montagu *Louise Cavendish *The Double Duchess == Acquaintances, Friends and Enemies == === Friends === *[[Social Victorians/People/Albert Edward, Prince of Wales | Albert Edward, Prince of Wales]] (beginning about 1852) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], Lord Hartington (later 8th Duke of Devonshire) *Daisy, Lady Warwick *Lady Mayoress, Mrs. Benjamin Samuel Faudel-Phillips, 2nd Baronet,<ref>{{Cite journal|date=2020-08-25|title=Faudel-Phillips baronets|url=https://en.wikipedia.org/w/index.php?title=Faudel-Phillips_baronets&oldid=974879290|journal=Wikipedia|language=en}}</ref> presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4">"The Queen's Drawing Room" ''Morning Post'' 25 February 1897 Thursday: 5 [of 10], Col. 5a–7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970225/047/0005.</ref>{{rp|p. 5, Col. 6c}} *Mrs. J. E. Mellor, presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4" />{{rp|p. 5, Col. 6c}} === Enemies === * Consuelo, Duchess of Marlborough (at least, in 1901)<ref name=":1">Murphy, Sophia. ''The Duchess of Devonshire's Ball''. London: Sidgwick & Jackson, 1984.</ref>{{rp|pp. 31–32}} == Organizations == == Timeline == '''1852 July 22''', Luise Friederike Auguste Gräfin von Alten and William Drogo Montagu married.<ref name=":2">"Luise Friederike Auguste Gräfin von Alten." {{Cite web|url=http://www.thepeerage.com/p10947.htm#i109469|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1859''': "As for Louise, a glimpse of her unconventional ways is given by Lady Eleanor Stanley, describing a paper-chase at Kimbolton in 1859: 'The Duchess of Manchester, in getting too hastily over a stile, caught a hoop of her cage in it, and went head over heels, alighting on her feet with her cage and whole petticoats remaining above her head. they say there was never such a thing seen — and the other ladies hardly knew whether to be thankful or not that a part of her underclothing consisted of a pair of scarlet tartan knickerbockers (the things Charles shoots in) — which were revealed to all the world in general and the Duc de Malakoff in particular.' Lady Eleanor says that the other ladies hardly knew whether to be thankful because at this date a lady's normal wear under her petticoats was only a pair of leggings laced up to the waistband, an arrangement that could be unduly revealing in combination with a crinoline. ... But some of the more active ladies were beginning to adopt / Louise's expedient of donning a hidden pair of breeches for rambles in the country."<ref name=":20" />{{rp|25–26}} '''1863, early, or late 1862''', Louise and Spencer Compton Cavendish began a relationship.<ref name=":1" />{{rp|p. 26}} '''1873 December 10''', Mary Louise Elizabeth Montagu (daughter) and William Douglas-Hamilton married. '''1876 May 22''', Consuelo Iznaga y Clement and George Victor Drogo Montagu (son) married in Grace Church, New York City.<ref>{{Cite journal|date=2020-08-24|title=George Montagu, 8th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=George_Montagu,_8th_Duke_of_Manchester&oldid=974659520|journal=Wikipedia|language=en}}</ref><ref>{{Cite journal|date=2020-07-27|title=Consuelo Montagu, Duchess of Manchester|url=https://en.wikipedia.org/w/index.php?title=Consuelo_Montagu,_Duchess_of_Manchester&oldid=969888488|journal=Wikipedia|language=en}}</ref> '''1876 August 10''', Louisa Augusta Beatrice Montagu (daughter) and Archibald Acheson married. '''1889 January 5''', Alice Maude Olivia Montagu (daughter) and Edward Stanley married. '''1890 March 22''', William Drogo Montagu (7th Duke) died.<ref name=":3">"William Drogo Montagu, 7th Duke of Manchester." {{Cite web|url=http://www.thepeerage.com/p10128.htm#i101274|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1890 November 14''', William Angus Drogo Montagu (grandson) and Helena Zimmerman married secretly, in London.<ref>"Helena Zimmerman." {{Cite web|url=http://www.thepeerage.com/p34555.htm#i345545|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1892 August 16''', Louise Friederike Auguste Gräfin von Alten Montagu and Spencer Compton Cavendish, her second husband, married.<ref name=":2" /> '''1897 July 2, Friday''', Louise Cavendish (#18 on the list of attendees) hosted her famous [[Social Victorians/1897 Fancy Dress Ball| fancy-dress ball]] at Devonshire House in London. '''1897 July 20''', Mary Louise Elizabeth Montagu Douglas-Hamilton and Robert Carnaby Foster married. '''1900 November 14''', William Angus Drogo Montagu and Helena Zimmerman married.<ref>{{Cite journal|date=2020-07-17|title=Helena, Countess of Kintore|url=https://en.wikipedia.org/w/index.php?title=Helena,_Countess_of_Kintore&oldid=968067371|journal=Wikipedia|language=en}}</ref> '''1901 Spring''', Paris, Consuelo Spencer-Churchill, Duchess of Marlborough, describes a meeting with Louise Cavendish in the spring following Queen Victoria's death at the horse racetrack, Longchamps:<blockquote>A renowned character and virtually dictator of what was known as the fast set as opposed to the Victorian, Her Grace was a German aristocrat by birth. She had first been married to the impoverished Duke of Manchester, and when he died had improved her status by marriage to the rich Duke of Devonshire, who waged an undisputed influence in politics. Rumour had her beautiful, but when I knew her she was a raddled old woman, covering her wrinkles with paint and her pate with a brown wig. Her mouth was a red gash and from it, when she saw me, issued a stream of abuse. How could I, she complained, pointing to my white gloves, show so little respect to the memory of a great Queen? What a carefree world we must have lived in, that etiquette even in such small matters could assume so much importance?<ref>Balsan, Consuelo Vanderbilt. ''The Glitter and the Gold: The American Duchess — In Her Own Words''. New York: St. Martin's, 1953.</ref>{{rp|p. 115}}</blockquote> === Annual Events === Every year, as Duchess of Devonshire, Louise held a dance on the night after the Derby at Epsom Downs, which at this point was held on Wednesdays after Easter. == The Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == Louise, Duchess of Devonshire and Spencer Compton, 8th Duke of Devonshire hosted the famous [[Social Victorians/Terminology#Fancy-dress ball|fancy-dress ball]], which took place on Friday night. According to ''House Style: Five Centuries of Fashion at Chatsworth'',<blockquote>Such an event required extensive planning and attention to detail. The housekeeper aided by two secretaries was in charge of organisation inside the house with precise instructions from the Duchess on menus and all arrangements.<ref name=":18" />{{rp|137}}</blockquote> === The Devonshire House Staff at the Ball === The Dukes of Devonshire did not keep employment records of the staff working at their various properties. The Servants and Staff Database managed by Chatsworth House has a list of everyone ever named in any documents as a servant or staff member, and by 1891 census records included employment at places like the properties owned by the Duke of Devonshire.<ref name=":19" /> === Questions === # Who was housekeeper? Who were the secretaries? # Where did this info about the "precise instructions" come from? === The Historical Zenobia === Louise, Duchess of Devonshire attended the ball dressed as Zenobia, queen of Palmyra. Zenobia (240 – c. 274) was queen of the Syrian Palmyrene Empire, ruling as regent for her son after her husband's assassination.<ref>{{Cite journal|date=2022-05-03|title=Zenobia|url=https://en.wikipedia.org/w/index.php?title=Zenobia&oldid=1086005949|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Zenobia.</ref> She was the subject of much art in the 19th century, including literature, opera, sculpture, and paintings. Middle-eastern traveller Lady Hester Stanhope (1776–1839) discussed Zenobia in her memoirs, published in 1847.<ref>{{Cite journal|date=2022-03-07|title=Lady Hester Stanhope|url=https://en.wikipedia.org/w/index.php?title=Lady_Hester_Stanhope&oldid=1075838273|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lady_Hester_Stanhope.</ref> We do not know what sources, if any, were consulted by Louise, Duchess of Devonshire in preparing for personating Zenobia for the ball. The ''Encyclopædia Britannica'' offers a sense of what was generally available to those interested in that kind of research. The article on Zenobia in the 11th edition of the ''Encyclopædia Britannica'', published in 1911, written by George Albert Cooke<ref>Rev. George Albert Cooke (G.A.C.), Oriel Professor of the Interpretation of Holy Scripture in the University of Oxford, Fellow of Oriel College, Vol. XXVIII, p. viii. </ref>, is substantial, more than an entire column.<ref>"G.A.C." [George Albert Cooke]. "Zenobia." Encyclopaedia Britannica: An Dictionary of Arts, Sciences, and General Information. Ed. . Vol. XXVIII (Vol. 28): Vetch to Zymotic Diseases. ''Google Books''. Retrieved 11 August 2022. https://books.google.com/books?id=NEpAAQAAMAAJ.</ref>(p. 972, Col. 1a–2b) The prior edition<ref>The 10th edition of the ''Britannica'', 1902–1903, is the 9th edition plus 10 supplementary volumes.</ref>, the 9th edition, does not have an article on Zenobia, but she has an entry that is a cross reference to the Palmyra article, which is about 10 columns and includes a biographical sketch of her.<ref>By W.R.S., W. Robertson Smith, Lord Almoner's Professor of Arabic, University of Cambridge.</ref><ref name=":21">"W. R. S." [Smith, W. Robertson]. "Palmyra." ''The Encyclopaedia Britannica: A Dictionary of Arts, Sciences, and General Literature''. Ed., Thomas Spencer Baynes. 9th edition. Vol. XVIII (Vol. 18): Orne–Phthisis. 1885. Google Books. Retrieved 11 August 2022.https://books.google.com/books?id=0au0xN1PrAEC.</ref>(p. 1, Col. 1c, after p. 858) In a very long paragraph about her husband's father and then her husband, the ''Britannica''<nowiki/>'s discussion of Zenobia begins, <blockquote>Odænathus himself seems to have been engaged in almost constant warfare in the east and north against the Persians and perhaps the Scythians, but in his absence the reins of government were firmly held by his wife Zenobia, the most famous heroine of antiquity, to whom indeed Aurelian, in a letter preserved by Trebellius Pollio, ascribes the chief merit of all her husband's success. Septimia Zenobia was by firth a Palmyrene; her native name was Bath Zabbai ... [;] and Pollio's description of her dark beauty, black flashing eyes, and pearly teeth, together with her unusual physical endurance and the frank commanding manners which secured her authority in the camp and the desert, point emphatically to an Arabic rather than a Syrian descent. ... To the union of firmness and clemency, which is the most necessary quality of an Eastern sovereign, Zenobia added the rarer gifts of economy and organization, and an unusual range of intellectual culture. She spoke Coptic as well as Syriac, knew something of Latin, and had learned Greek from the famous Longinus, who remained at her court to the last, and paid the penalty of his life for his share in her counsels. She was also a diligent student of Eastern and Western history, and the statement that she enjoined her sons to speak Latin so that they had difficulty in using Greek implies a consistent and early adoption of the policy which made the sucess of Odænathus, and, taken in connexion with Aurelians' testimony, in a letter preserved by Pollio, that she had the chief merit of her husband's exploits, seems to justify the conclusion that it was her educated political insight that created the fortunes of the short-lived dynasty. ... In the zenith of his fame Odænathus was cut off by assassination along with his eldest son Herod, and it is generally assumed that the murder took place under Gallienus. ... [p. 201, Col. 2c – p. 202, Col. 1a] The fact seems to be that, while Odaenathus was busy at the other end of his kingdom, Zenobia administered the government at Palmyra and directed the conquest of Egypt, still nominally acting under the emperor at Rome, whose authority on the Nile was disputed by one or more pretenders. ... It still seems ... strange and yet an undoubted fact that Zenobia, who not only enjoyed the real authority behind her beardless son, but placed her name before his on public inscriptions, ... struck no coins till the second year of Aurelian, when the breach with Rome took place, and she suddenly appears as an empress (Σεβαστή, Augusta) of five years' standing. Up to that date the royal pair probably did not venture to coin in open defiance to Rome, and yet were unwilling to circulate an acknowledgment of vassalship in all the bazaars of the East. ... ... Zenobia, supported by her two generals, kinsmen of her husband, was now face to face with a Roman invasion. She held Egypt, Syria, Mesopotamia, and Asia Minor as far as Ancyra; and Bithynia was ready to join her party had not the army of Aurelian appeared just in time from Byzantium. She could count too on the Armenians and the Arabs, but the loyalty of Syria was doubtful: the towns disliked a rule which was essentially "barbarian," and in Antioch at least the patroness of the Monarchian bishop Paul of Samosata could to be popular with the large Christian party by whom he was bitterly hated. There were many Romans [p. 202, Col. 1c – p. 202, Col. 2a] in Zenobia's force, and it was they who bore the brunt of the two great battles at Antioch and Emesa, which followed Aurelian's rapid advance through Asia Minor. But Zenobia made light of these defeats, — "I have suffered no great loss" was her message to Aurelian, "for almost all who have fallen are Romans." ... But the empire of Palmyra came four centuries too soon. Rome was not yet exhausted, and Zenobia had neither the religious discipline of Islam to hold the Arabs together nor the spoil of the treasuries of Persia to keep their enthusiasm always fresh. Aurelian's military skill was strained to the uttermost by the prudence and energy of Zenobia, but he succeeded in forming and maintaining the siege of Palmyra in spite of its bulwark of desert, and his gold corrupted the Arab and Armenian auxiliaries. Zenobia attempted to flee and throw herself on the Persians, but she was pursued and taken, and then the Palmyrene lost heart and capitulated. Aurelian seized the wealth of the city, but spared the inhabitants, and to Zenobia he granted her life while he put her advisors to death. She figured in his splendid triumph, and by the most probable account accepted her fall with dignity, and closed her days at Tibur, where she lived with her sons the life of a Roman matron. The fall of Zenobia may be placed in the spring of 272.<ref name=":21" /> (p. 201, Col. 2a – p. 202, Col. 2b) </blockquote> == The Duchess of Devonshire's Costume == [[File:Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra.jpg|thumb|Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra|alt=Louise, Duchess of Devonshire in costume as Zenobia, Queen of Palmyra]] At their fancy-dress ball, Louisa, Duchess of Devonshire sat at Table 1 during the first seating for supper, escorted in to the table by the Prince of Wales.<ref name=":7">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref>{{rp|p. 7, Col. 4c}} Her costume was designed by M. Comelli (Attillo Giuseppe Comelli, 1858–1925, artist and costumier for opera, ballet and theatre in London as well as Europe and the U.S.<ref name=":13">{{Cite book|url=https://books.google.com/books?id=SZh2DwAAQBAJ&pg=PT207&lpg=PT207&dq=Attilio+Comelli&source=bl&ots=lFB0If7CwV&sig=ACfU3U1_Ost_lhmMvzMMs6NvuhK5SlRhJw&hl=en&sa=X&ved=2ahUKEwjKlsTw2sH3AhXYAp0JHVIxDWA4KBDoAXoECBAQAw#v=onepage&q=Attilio%20Comelli&f=false|title=Forgotten Designers Costume Designers of American Broadway Revues and Musicals From 1900-1930|last=Unruh|first=Delbert|date=2018-11-06|publisher=Page Publishing Inc|isbn=978-1-64082-758-5|language=en}} N.P.</ref>)<ref name=":5">“The Devonshire House Ball.” The ''Man of Ross'' 10 July 1897, Saturday: 2 [of 8], Col. 4b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0001463/18970710/033/0002.</ref> <ref name=":8">"The Duchess of Devonshire's Fancy Dress Ball. Special Telegram." ''Belfast News-Letter'' Saturday 03 July 1897: 5 [of 8], Col. 9 [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000038/18970703/015/0005.</ref>{{rp|p. 5, Col. 9a}} <ref name=":9">"By One Who Was There." “The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.</ref> and constructed by the House of Worth. Comelli seems to have designed [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess and Her Entourage|the costumes of her retinue as well]]. According to Russell Harris,<blockquote>For her costume, the Duchess commissioned Monsieur Comelli (1858-1925), a well-known designer of opera costumes for the London theatre and opera stage, and then had the design made up by Worth of Paris. ''Munsey’s Magazine'' noted “it is safe to say that the Queen of Palmyra never owned such a sumptuous costume in her lifetime.”<ref>Harris, Russell. {{Cite web|url=http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html|title=Louise, Duchess of Devonshire, née Countess von Alten of Hanover (1832-1911), as Zenobia, Queen of Palmyra|website=www.rvondeh.dircon.co.uk|access-date=2022-05-05}} ''Narrated in Calm Prose: Photographs from the V&A's Lafayette Archive of Guests in Costume at the Duchess of Devonshire's Diamond Jubilee Ball, July 1897''. http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html.</ref></blockquote>Lafayette's portrait of "Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester)" in costume is photogravure #5 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref>"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.</ref> The printing on the portrait says, "The Duchess of Devonshire as Zenobia Queen of Palmyra," with a Long S in ''Duchess''.<ref>"Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra." Devonshire House Fancy Dress Ball Album. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158357/Louise-Frederica-Augusta-Cavendish-ne-von-Alten-Duchess-of-Devonshire-formerly-Duchess-of-Manchester-as-Zenobia-Queen-of-Palmyra.</ref> Often, the V&A Lafayette Archive contains more than one portrait of a sitter for this ball, but the uncropped portrait (above right), which shows the unfinished end of the balustrade in front of the Duchess and the edge of the painted drop behind it, seems to have been the only portrait taken by Lafayette of the Duchess in costume. The copy owned by the National Portrait Gallery in London and the copy included in the album are cropped so that those unfinished edges do not show, but they appear to be from the same photograph. That is, there is one source for our information about this portrait: this single photograph. === Newspaper Descriptions of the Duchess's Costume === Newspaper articles about the Duchess's presence at the ball focused on her hosting, her costume, [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Jewelry|her jewelry]], and [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Entourage|her entourage]], often in the same story. These almost exactly identical descriptions suggest [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]] or a shared primary source: * The London ''Evening Standard'', ''Morning Post'' and ''Times'' published almost identical descriptions, varying only by the occasional punctuation mark or short phrase ("supplied by Worth, of Paris"). The fullest version is this one: "The Duchess of Devonshire, as Zenobia, Queen of Palmyra, wore a magnificent costume, supplied by Worth, of Paris. The skirt of gold tissue was embroidered all over in a star-like design in emeralds, sapphires, diamonds, and other jewels, outlined with gold, the corners where it opened in front being elaborately wrought in the same jewels and gold to represent peacocks' outspread tails. This opened to show an under-dress of cream crêpe de chine, delicately embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was attached to the shoulders by two slender points, and was fastened at the waist with a large diamond ornament. It was of green velvet of a lovely shade, and was superbly embroidered in Oriental designs, introducing the lotus flower in rubies, sapphires, amethysts, emeralds, and diamonds, in four borderings on contrasting grounds, separated with gold cord. The train was lined with turquoise satin. The bodice was composed of gold tissue to match the skirt, and the front was of crêpe de chine, hidden with a [[Social Victorians/Terminology#Stomacher|stomacher]] of real diamonds, rubies, and emeralds, and there was a jewelled belt."<ref name=":16">“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.</ref>{{rp|p. 3, Col. 2b}} <ref name=":7" />{{rp|p. 7, Col. 7a}} <ref name=":6">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1A–4C ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref>{{rp|p. 12, Col. 3b}} *"The Duchess of Devonshire was a dazzling vision, dressed as 'Zenobia,' in a glistening gold gauze gown, elaborately ornamented with suns and discs, wrought in purple and green gems outlined with gold, and having a large diamond as centre. The space between was fluted with fine silver spangles. This robe was open in front over an under dress of white crépe de chine, delicately worked in crystals, and at each side of the opening on the gold robe were large fan-shaped groups of peacock feathers, worked in multicoloured jewels. The [[Social Victorians/Terminology#Corsage|corsage]] was to correspond, and had a magnificent [[Social Victorians/Terminology#Girdle|girdle]] of jewels, the train of bright green velvet, hung like a fan, without folds, being fastened at each side of the shoulders by diamond brooches, and caught at the waist with a similar ornament. It was a mass of gorgeous embroidery, carried out in heliotrope velvet, lotus flowers studded with tinted gems, and other devices in terra-cotta and electric blue velvet — all enriched with gold, diamond, and jewelled embroidery — and lined with pale blue satin. ... Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail."<ref name=":15">"Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses." London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.</ref>{{rp|p. 5, Col. 6a}} * According to the article in ''The Graphic'' written by [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]], though this caption to the Lafayette photograph may not be hers, the Duchess of Devonshire wore a "Skirt of gold tissue, embroidered all over with emeralds, sapphires, diamonds, and other jewels outlined with gold. This opened to show an underdress of crème crêpe de chine, embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was green velvet, superbly embroidered in Oriental designs. The bodice was composed of gold tissue, and the front was of crêpe de chine hidden with a stomacher of diamonds, rubies, and emeralds."<ref name=":10">Greville, Violet, Lady. "Devonshire House Ball." The ''Graphic'' Saturday 10 July 1897: 15 [of 24]: Col. 1a–16, Col. 1c. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000057/18970710/019/0015.</ref>{{rp|p. 15, Col. 3b}} Because it is in black and white, the photograph of the Duchess taken by Lafayette in her costume (above right) can tell us almost nothing about the colors of the costume. Fortunately, this costume has survived and is held in Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth). ''House Style: Five Centuries of Fashion at Chatsworth'', the volume about an exhibition mounted 25 March to 22 October 2017, has beautiful photographs from several perspective of the costume, including the train, and accessories. Published photographs of it '''show X'''.<ref name=":18">Burlington, Julia, and Hamish Bowles. ''House Style: Five Centuries of Fashion at Chatsworth''. Skira Rizzoli, 2017. Based on an exhibition 25 March to 22 October 2017.</ref> It has been worn by '''Deborah, 11th Duchess of Devonshire when the Dowager Duchess at some point for some reason'''. Some accounts of her costume suggest that the reporter thought that all the jewels sewn onto the dress were actual precious or semiprecious stones, but the London ''Daily News'' says some of them were crystals.<ref name=":15" />(p. 5, Col. 6a) According to ''House Style: Five Centuries of Fashion at Chatsworth'',<blockquote>The skirt of gold gauze, appliquéd with tinsel medallions and peacock plumes worked in bright foils, wire coils and spangled with sequins, was worn over an ivory satin underskirt wrought over with silver thread and diamonds. Attached to the shoulders was a long graduated train in the most vivid emerald-green velvet, appliquéd with velvet and gold work in an eastern design and studied with jewels.<ref name=":18" />(128)</blockquote> === Observations on the Duchess's Costume === In the photograph of her in costume, the Duchess is standing in a three-quarter front pose. That static position in a two-dimensional image makes it difficult to understand exactly how the costume and headdress worked. The terms we use are these: stomacher, train, bodice (what's on the torso, no matter how many layers), overskirt, underskirt, belt (really an accessory, like the jewelry), sleeves. The 1897 newspaper reports use different terminology for the parts of the dress than we would now. Some of the terms that appear in the newspaper reports include overdress, under-dress, bodice, stomacher robe, train, corsage, girdle. The Duchess's Body how the fabric draped, how tight the bodice was, what the stomacher looked like, what pearls she is wearing. Is her hair down or up? ==== Fit ==== *The recent exhibitions of the dress show a different fit than what is shown in the Lafayette photograph. Louise, the 8th Duchess's bodice is draped loosely over the stomacher, but photographs of Deborah, 11th Duchess or of the mannequin in the 2017 exhibition show a bodice fitted to the stomacher. * Also, the costume itself may have been altered or adjusted to make the waist more defined and the line more Elizabethan (the wide, square neckline; the A-line, the split skirt with the petticoat in the middle). * The dress seems to have been designed to flatter the 8th Duchess's statuesque figure, especially her waist: the belt falls down to the hips, the A-line; the way what we can see of the under-dress is narrow at the top and widens toward the bottom. * The drape of the skirt is straighter on the 8th Duchess as well as the Dowager Duchess than on the mannequins. ==== Inconsistencies in the Descriptions and Images ==== * The newspaper accounts of the Duchess's costume do not agree on several particulars, including the color of the train and its lining. ==== Changing Victorian Design Elements ==== The costume as Louise, the 8th Duchess wore it is vaguely Middle Eastern and exotic, consistent with her '''leading''' the "Oriental" procession. The sleeve treatment, the drape of the bodice and the skirt are consistent with the line of something from the Middle East. This would have been what Comelli designed. Early clothing history books existed by the middle of the century, and it would be logical that Comelli would be familiar with them. The versions of the costume shown in recent exhibitions make it seem more Victorian than what we see in Lafayette's portrait. The tightness of the fit of the bodice indicates a Victorian corset under the costume. The line of the bodice and skirt, the bare arms in the sleeve treatment, the train, the belt, The versions of the dress the the tightness of the dress against the stomacher all look Victorian. A-line drape of the skirt was done for the exhibition to show the artistry and complexity of the trim. * The sleeve treatment in the 1897 photograph is caught up at the shoulder. On the Dowager Duchess and the mannequins, the sleeves are released. ==== Trim ==== * The sequins attached to the costume would have been metal. They appear to be silver that has tarnished, which is why they are dark now. Currently, the tarnish on the sequins and embroidery thread cannot be removed without damage to the fabric. * The gold thread in the embroidery has not tarnished, but the silver thread, which would have sparkled in 1897, has. ==== Accessories ==== * Belt * Headdress * Do the horns on [[Social Victorians/People/Louisa Montagu Cavendish#Zenobia's Crown|her crown]] go toward the back from the apex that rises from the jewel-encrusted band or do they stick straight out to the sides? Does the headdress include the traditional three white plumes? * Jewelry * Fan * Can't tell how many strands of pearls are around Louise's neck in the Lafayette photo, but the photographs of the recreation of the crown and of the costume from the 2017 exhibit show [[Social Victorians/People/Louisa Montagu Cavendish#Goldsmith, Pearl & Diamond Merchant, & Silversmith|five strands]]. ==== The Train ==== * The reporter's description of the Duchess's train in the London ''Daily News'' is confusing: it says the train is "bright green velvet" and "It was a mass of gorgeous embroidery, carried out in heliotrope velvet." Embroidery is not "carried out" in velvet. Heliotrope is not green, it's pinkish purple. * Furthermore, heliotrope appears to have been a new word for a color, introduced perhaps in 1882 in A. Maerz and M. Rea Paul ''A Dictionary of Colour'', New York: '''1930 McGraw-Hill Page 196'''; Color Sample of Heliotrope: Page 131 Plate 54 Colour Sample C10 ("Heliotrope," Wikipedia [[wikipedia:Heliotrope_(color)#cite_note-2|https://en.wikipedia.org/wiki/Heliotrope_(color)#]]<nowiki/>[[wikipedia:Heliotrope_(color)#cite_note-2|c]][[wikipedia:Heliotrope_(color)#cite_note-2|ite_note-2]]). The ''OED'' lists 3 newspaper sources for the 1880s, The ''World'' (1882), ''Truth'' (1886), and the ''Daily News'' (1887) ("heliotrope" 1.d. Retrieved 6 August 2022). The ''Daily News'' describes it well: "that peculiar mauve known as heliotrope." The word itself has existed in English for many centuries, <nowiki/>both for the flower (which turns to follow the sun) as well as for instruments that reflect or measure sunlight and also for a variety of quartz (also called bloodstone). * Color of green velvet on the train: the colors differ, depending on the photograph. One photograph shows a very vivid green, and the rest show a lighter green. We're not sure what's up with that. == The Duchess's Jewelry == The Duchess was known fo<nowiki/>r her jewelry, and as she aged and after her marriage to the 8th Duke in 1892, the jewelry she wore only increased in scale and notability. It is not surprising that her jewelry would make up copy for the newspapers. Newspaper reports before the ball gossiped about the jewelry associated with the costumes being developed. For example, according to the Edinburgh ''Evening News'' on 21 June 1897, less than two weeks before the party, "The ball being a fancy dress one, men as well as women will be able in certain characters to wear jewels. The Duchess of Devonshire, who is to appear as Zenobia, is getting her jewels reset after the antique style."<ref>“The Duchess of Devonshire’s Ball.” Edinburgh ''Evening News'' 21 June 1897, Monday: 4 [of 6], Col. 5c [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18970621/079/0004.</ref> While almost all descriptions of her mention her jewels because they were so much a topic of conversation and because they were sewn onto the costume itself, these emphasize her jewelry and most are very similar to each other: * "The Duchess was attired with great Oriental magnificence as Zenobia. Her dress was a tissue of silver, embroidered with gold and jewels, an overmantle of cloth of gold embroidered in the same manner hung from the shoulders, and she wore a bandeau of gold studded with gems, and surrounded by hanging chains of pearls over her elaborate headdress; strings and ropes of jewels and pearls were worn round the neck, and hung down almost to the knees."<ref>“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032. </ref>{{rp|p. 32, Cols. 1c–2a}} <ref name=":17">"Duchess of Devonshire's Fancy-Dress Ball. Brilliant Spectacle." The [Guernsey] ''Star'' 6 July 1897, Tuesday: 1 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000184/18970706/003/0001.</ref>{{rp|p. 1, Col. 2a}} * "A wonderfully beautiful dress was that which was worn by the Duchess of Devonshire as Zenobia, Queen of Palmyra. It was of golden tissue, sewn with silver paillettes, and jewelled with diamonds and other precious stones. In front there were silk embroideries, in many vivid shades of colour, and here the golden draperies opened to show a petticoat of white crêpe de chine, embroidered with pearls and gold. The short train was of brilliant green velvet, exquisitely embroidered. One of the Duchess of Devonshire’s beautiful diamond and emerald tiaras had been taken to pieces to form a stomacher, the effect of which was dazzling in its brilliancy. Long chains of pearls and other wonderful jewels were worn with this beautiful dress."<ref>“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a [of 3]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.</ref>{{rp|p. 7, Col. 2b}} * In the article about the ball in the ''Graphic'', [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]] says, "The Ducal hostess herself elected to appear as Zenobia, Queen of Palmyra, with lavish magnificence, and wearing a corruscation of jewels which must have eclipsed the state of even the all-subduing majesty the Duchess impersonated."<ref name=":10" />{{rp|p. 16, Col. 1a}} *The Duchess was dressed "as Zenobia, Queen of Palmyra, her dress a marvel of soft tissues and exquisite ornament, and her tiara a still greater marvel of the jeweller's art."<ref name=":6" />{{rp|p. 12, Col. 2a}} <ref>"The Duchess of Devonshire’s Historic Ball. Some of the Fancy Costumes." Supplement. The ''Leicester Chronicle and Leicestershire Mercury'' 10 July 1897, Saturday: 11 [of 12], Cols. 4a–b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000173/18970710/141/0011.</ref>{{rp|p. 11, 4a}} As with the colors, Lafayette's photograph of the Duchess in costume does not show the jewels very clearly. We cannot see the stomacher or the "long chains of pearls and other wonderful jewels" or the pearls that "hung down almost to the knees" (although, of course, any newspaper account can be wrong because reporters were not present at the ball). === Zenobia's Crown<ref name=":10" /> === The crown that the Duchess wore as Zenobia is difficult to see clearly in the Lafayette photograph (above right), and no other images of the crown exist. It was lavish, "encrusted" with jewels and featuring pearls: * London ''Daily News'' says, "The crown worn with this was high, and of filigree gold, surmounted with two horns, each tipped with a large diamond. It was encrusted with large diamonds, rubies, and emeralds, and long chains of pearls fell under the chin and about the head — one magnificent pear-shaped pearl resting on the forehead."<ref name=":15" />{{rp|p. 5, Col. 6a}} * These descriptions are all identical except for the addition or subtraction of an occasional comma: The London ''Evening Standard'' says, "A gold crown encrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end, and two upstanding white ostrich feathers in the centre, and round the front were festoons of pearls, with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":16" />{{rp|p. 3, Col. 2b}} <ref name=":7" />{{rp|p. 7, Col. 7a}} <ref name=":6" />{{rp|p. 12, Col. 3b}} <ref name=":10" />{{rp|p. 15, Col. 3b}} * The Guernsey ''Star'' says, "Her Grace wore a bandeau of gold round her head, studded with diamonds, turquoise, and emeralds, and surrounded by hanging chains of superb pearls."<ref name=":17" />{{rp|p. 1, Col. 2a}} This crown was recreated from the single Lafayette photograph of the Duchess in costume for the ''House Style: Five Centuries of Fashion at Chatsworth'', the exhibition mounted in 2017 and exhibited on the mannequin wearing the Duchess's costume. A photograph of this recreation separate from the costume, but with the pearls and the white plumes, can be seen on [https://www.sothebys.com/en/articles/how-c-w-sellors-transformed-five-centuries-of-fashion-into-modern-jewellery this Sotheby's page] about C. W Sellor's, the jewelry firm that did the recreation as well as a number of other recreations for the exhibit.<ref>Sotheby's. "How C. W. Sellors Transformed Five Centuries of Fashion Into Modern Jewellery." ''Sotheby's'' 9 May 2017. Retrieved 11 August 2022. https://www.sothebys.com/en/articles/how-c-w-sellors-transformed-five-centuries-of-fashion-into-modern-jewellery.</ref> Although Lafayette's photograph is not conclusive, in Sellor's recreation the horns on the crown point toward back of the head rather than out to the sides. === Goldsmith, Pearl & Diamond Merchant, & Silversmith === The Duchess's pearls, which were an important feature of her costume, occasioned a great deal of direct commentary in the newspaper accounts. The Duchess's jewelry occasioned a great deal of reportage in the articles about the ball. '''It was reported that she had her jewels restrung to be used in the costume. stomacher and review of jewelry in more general articles''' An invoice and receipt in the Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) is from a concern whose preprinted stationery has a crown in the upper-left corner, suggesting that they had a royal warrant, and no name other than Goldsmith, Pearl & Diamond Merchant, & Silversmith. This document offers a unique view into the evolution of one necklace, at least, over the years. It lists what are apparently three restringing of some pearls of Louise, Duchess of Devonshire. The three restringings appear to be dated: The first necklace is a "Pearl Necklet in original 4 rows." The invoice is dated 20 October 1892 (but the stationery was printed to assume the invoice would be used in the 1880s, so the 9 is written over the second 8, and the 2 has been added).<ref name=":14">Invoice and receipt. Goldsmith, Pearl & Diamond Merchant & Silversmith. Date of itemized invoices for restringing pearls: 20 October 1892, 1 March 1897, 1909. The Devonshire Collections, Chatsworth, Reference number FIS/4/1/2.</ref>(p. 1) The necklet she brought in to be restrung contained a "Total [of] Total 224 large pearls": # 1st [row] 51 large pearls # 2nd 53 large pearls # 3rd 57 large pearls # 4th 63 large pearls The second necklace is a "Necklet as re-strung on October 15th 1892, with addition of small pearls supplied, now consists of 5 rows, containing" a total of "224 large pearls & 227 small"<ref name=":14" />(p. 1): # 1st 41 large pearls & 40 small # 2nd 42 large pearls & 42 small # 3rd 44 large pearls & 45 small # 4th 47 large pearls & 48 small # 5th 50 large pearls & 51 small The third necklace is a "Pearl Necklet as again re-strung with additional pearls supplied 1 March 1897, now consisting of 5 Rows containing" a total of "262 Large Pearls & 267 Small"<ref name=":14" />(p. 2): # 1st Row 45 Large Pearls & 44 Small # 2nd Row 48 large Pearls & 49 Small # 3rd Row 51 Large Pearls & 52 Small # 4th Row 56 Large Pearls & 65 small Possibly these pearls may have been restrung in 1909 into a coronet, but the handwriting is not clear enough to read.<ref name=":14" />(p. 2) If the Duchess wore one of these stringings of her pearls for the ball, then it must have been the second necklet, strung in 1892, a 5-strand necklace. None of the newspaper accounts refer to a 5-strand pearl necklace, although her pearls are often mentioned. == The Duchess's Entourage == Besides the Duke of Devonshire, the retinue of Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra, included her grandson, [[Social Victorians/People/William Angus Drogo Montagu|William Angus Drago Montagu, 9th Duke of Manchester]], dressed as a Georgian courtier. Four newspapers say that the Duchess's entourage included three groups, all in costume: children, trumpeters and fan-bearers. The ''Westminster Gazette'' and the ''Man of Ross'' list the groups but do not mention the number of members of the groups. According to two sources, probably in error, the London ''Daily News''<ref name=":15" />{{rp|p. 5, Col. 6a}} and the Belfast ''News-Letter,''<ref name=":8" />{{rp|p. 5, Col. 9a}} these groups each had four members. The London ''Daily News'' is likely the source (because it was published at 5:00 a.m.<ref>Mitchell, Charles. ''Newspaper Press Directory'', 1895. [Hathi Trust via U Wisconsin Madison.] http://hdl.handle.net/2027/mdp.39015085486150.</ref>{{rp|55}}) for the Belfast ''News-Letter'', which took part in [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]], like so many other newspapers of the 19th century. ['''check this: which one was published earlier in the day, and on which day?'''] These four sources describe the Duchess's retinue and how the people in it were dressed: *"The Duchess of Devonshire was dazzingly [sic] magnificent as 'Zenobia,' arrayed in the glistening fabrics and massive jewels in which artists have delighted to depict the Warrior Queen, the costume in this case being specially designed by the clever French artist, M. Comelli, who was also responsible for the splendid attire of the Queen's suite. This was composed of four children in white Assyrian robes, draped with pink shawls; four trumpeters in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel; and four fan-bearers attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold — a gloriously magnificent pageant."<ref name=":8" />{{rp|p. 5, Col. 9a}} *"Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail."<ref name=":15" />{{rp|p. 5, Col. 6a}} *"The duchess was dressed as Zenobia, in gold cloth, gorgeously embroidered in gold, brilliants, and coloured stones, and opening over an under dress of white crêpe de Chine, worked finely in brilliants. The train of light green velvet was lined with blue, and sumptuously embroidered in jewels and gold, the colouring being particularly artistic. With this dress were worn splendid jewels, and a large horn crown, encrusted with diamonds, emeralds, and rubies. The duchess was attended by a suite of children, trumpeters, and fan-bearers, all picturesquely attired in Assyian [sic] costumes — the whole group being specially designed by M. Comelli."<ref name=":5" /> *"The host was dressed as Charles V. of Germany, in black velvet, satin, and fur; and the Duchess made the most gorgeous of Zenobias, in a gown of gold gauze, and a green velvet train — both a mass of exquisite oriental embroidery. The crown and hanging ropes of pearls, the jewelled girdle, and the train of children, fan-bearers, and trumpeters — all in Babylonish garb — as designed by M. Comelli, made a gloriously imposing and picturesque group."<ref name=":9" /> === Details of the Costumes in the Entourage === The Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) has "receipts" or invoices that functioned as receipts for several commercial concerns that were involved in making costumes or accessories for costumes for this ball. They are the following: * [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|M. (Attillo Giuseppe) Comelli]] * [[Social Victorians/People/Louisa Montagu Cavendish#B. Burnet & Co.|B. Burnet & Co.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Arthur Millward, Theatrical Jeweller|Arthur Millward, Theatrical Jeweller]] * [[Social Victorians/People/Louisa Montagu Cavendish#Liberty & Co., Ltd.|Liberty & Co., Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Lafayette, Ltd.|Lafayette, Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Goldsmith, Pearl & Diamond Merchant, & Silversmith|Goldsmith, Pearl & Diamond Merchant, & Silversmith]] This list of commercial concerns almost certainly cannot be the complete list of all concerns that contributed to the costumes. These are the only receipts or invoices about expenses for the ball, however, that the Chatsworth Archive contains; similar documents were likely not even kept or were destroyed with other papers not retained at some point in time. The business concerns listed above were specialized and likely used for different elements of the costumes. As a theatrical designer, Comelli would have depended on the suppliers he knew and arranged with them for the construction of these costumes. The Chatsworth Archive calls these documents ''receipts'', which indeed they are because they were returned to Devonshire House as receipts for payment. From our perspective, though, they are invoices that contain specifics about what was used to make the costumes. The analysis of these invoices has led to an understanding of what the people who attended the Duchess in her entourage wore and a clearer sense, perhaps, of how many people walked in that entourage. This analysis is based on the items listed on the invoices and their pricing, most of which is included in the section for each invoice, below. While the Belfast ''News-Letter'' says that each group contained four members,<ref name=":8" />{{rp|p. 5, Col. 9a}} the invoices and receipts suggest that the newspaper, the single source for this information, was wrong about the number of people in each group. It is theoretically possible, of course, that suppliers other than the ones in the Chatsworth Archive made some of these costumes and that other invoices and receipts must have existed at that time. But the [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|Comelli memo, below]], seems definitive: he designed and seems to have overseen the construction of the costumes, which numbered six rather than twelve. '''[Collier discussion?]''' Besides providing welcome detail about the costumes of the people in the Duchess's entourage, which is available nowhere else, these invoices also raise at least as many questions as they answer. ==== M. (Attillo Giuseppe) Comelli ==== Attillo Giuseppe Comelli was a designer for opera, ballet and theatre in Europe, the UK and North America.<ref name=":13" /> The receipt in the Chatsworth Archive was sent from Covent Garden. The invoice lists £4 for "Making six costumes," 3''s'' for "Extras" and 12s for "Cab fares for men paid by the request of M. Comelli."<ref>M. Comelli, Covent Garden, to Duchess of Devonshire. Date of invoice 13 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Three other names are on this invoice and receipt: * L. L[?] Collier [?], written under and perhaps as part of the direction to the Duchess of Devonshire * Mr. Strong ("Forwarded to Mr Strong. [sic] by the instructions of M. Comelli," written in the same hand as wrote the majority of the memo) * Floyd [?] Collier [??] ("Received with Thanks," presumably thanking for the payment, in a different hand) ==== B. Burnet & Co. ==== An invoice and receipt from B. Burnet & Co., held in the Archives of the Duke of Devonshire, has specific information about some of the fabrics, trims and accessories purchased for the costumes of the Duchess's retinue.<ref name=":11">B. Burnet & Co. to Louise, Duchess of Devonshire. Date of invoice 2 and 6 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Besides itemizing some costume or accessory elements that seem clearly to be for the groups, the invoice also lists items not easy to associate with particular costumes, like the following: * 12 yards of White silk fringe 8in deep<ref name=":11" />{{rp|back left}} * 12 1/2 yards of "wht cloth"<ref name=":11" />{{rp|back left}} * 9 yards of "[[Social Victorians/Terminology#Selesia|Selesia]]"<ref name=":11" />{{rp|back left}} * 2 yards of Canvas<ref name=":11" />{{rp|back right}} * 4 Tan Wool Tights<ref name=":11" />{{rp|back right}} * 2 Tan Boys Tights<ref name=":11" />{{rp|back right}} At this time, we are not sure which costumes these elements were used for. Possibly the white silk fringe and the white cloth would have been used to construct the robes for the children and trumpeters in the entourage. The number of tights suggests that the six costumes on this invoice all included tights. With other elements of the trumpeters' costumes, the Burnet invoice also lists "6 prs Assyrian Buskins." Probably, to a late Victorian, buskins would have been "defensive leggings"<ref>Demmin, Auguste. An illustrated History of Arms and Armour: From the Earliest Period to the Present Time. George Bell, 1894. Google Books https://books.google.com/books?id=ArRCAAAAYAAJ: 106.</ref> laced together and covering the lower leg and often feet of a soldier. To a clothing and military historian, buskins (or greaves) were worn by people in a number of cultures over millennia and varied widely in style and construction. Buskins appear in Assyrian art held at the time by the British Museum. Listing six pairs of buskins suggests that every costume in the Duchess's entrourage included buskins, possibly worn over the tan tights. The Burnet invoice lists "4 Broad Belts," which may have held "4 Skins Fleshers."<ref name=":11" />{{rp|p. 1, front of invoice}} (A skin flesher is a kind of knife used to separate the skin from the meat in animals.) If each group included only two members, then perhaps the belts and fleshers were worn not only by the trumpeters but also by the fan-bearers. The Millward invoice (specifics in the section on the Millward invoice below) lists "8 Doz 'Plaques' for Belts'" with a drawing of an upright rectangle with a circle in the middle, which might have been a jewel. Double lines around the rectangle suggest that the plaques were not flat or the metal was not thin. The drawing does not give any ideas about how the plaques were attached to the belts, if they were. It is impossible to tell if the plaques were attached to the "4 Broad Belts" (likely for the trumpeters and fan-bearers), but unless they were quite tiny, "8 Doz 'Plaques'" would be far too many for the belts of only the two children. A different hand, probably "[L.??] L. Collier," wrote the following sentence at the end of the invoice and receipt, above the postmark:<blockquote>All the above named articles were used for the six [?] dresses made for the Devonshire Ball.<ref name=":11" />{{rp|back right}}</blockquote> This same hand, signing what is possibly "Floyd Collier," also signed the postmark of the Comelli invoice and receipt. On the Burnet document, this writer, possibly an assistant or employee of the Duchess of Devonshire, says that "six dresses" were made (if in fact, that word is ''six''). (No "Collier" is listed among the staff or servants of the Duke of Devonshire at the end of the 19th century.<ref name=":19">"Servants A-H." ''Historic Servants and Staff. Servants and Staff Database''. Retrieved 18 July 2022 https://www.chatsworth.org/media/11528/servants-a-h.pdf.</ref> The invoice appears to itemize materials used for six costumes: two children, two trumpeters and two fan-bearers. ==== Arthur Millward, Theatrical Jeweller ==== An invoice and request for payment from Arthur Millward, Theatrical Jeweller, held in the Archives of Chatsworth House, has more specifics about some of the fabrics, trims and accessories for the costumes of the Duchess's retinue.<ref name=":12">Memorandum. Arthur Millward, Theatrical Jeweller, to Louise, Duchess of Devonshire. Date of itemized invoice 1 July 1897; date of request for payment(?) 27 August 1897. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> This invoice lists the following, which could have been used in any of the costumes for the entourage: * 8 Doz 'Plaques' for Belts [discussed with the belts in the section on the Burnet invoice, above] * 4 Large Armlets * 4 Bracelets * 8 Armlets<ref name=":12" />{{rp|p. 2, back}} Because Millward was a Theatrical Jeweller, it seems likely that most (if not all) of the items listed on the invoice were made of metal and the jewels mentioned were artificial, made of glass or paste. Other items on the invoice seem to belong to the costumes of the trumpeters, which the Belfast ''News-Letter'' says included helmets: * 2 Helmets * 2 Centre pieces The Millward invoice shows tiny line drawings next to the words ''2 Helmets'' and ''2 Centre pieces''. These drawings suggest that the Centre pieces were attached to the helmets rather than being anything that would have been put on a table as decoration. Other items seem to belong to the costumes of the fan-bearers: * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers The "Pearl & Gold Headdresses" were likely the "jewelled diadems" mentioned in the Belfast ''News-Letter''. The "Fan properties with Feathers" are likely to have been the "long-handled fans of white feathers, mounted in blue and gold" mentioned in the newspaper report.<ref name=":8" />{{rp|p. 5, Col. 9a}} At the end of the Millward invoice, a "reduction as agreed with M [Mr?] Commelli [sic]" of £1 10''s'' is subtracted from a total of £22 3''s''. No reason for this reduction is given.<ref name=":12" />{{rp|p. 2, back}} ==== Liberty & Co., Ltd. ==== One invoice and receipt from the Chatsworth Archive, dated 12 July, to the Duchess of Devonshire, lists "13 yds S&W Satin[?]," 7 yards of blue and 6 of purple.<ref>Invoice and receipt. Liberty & Co. Ltd. To Her Grace, The Duchess of Devonshire. Date of itemized invoice 12 July [1897]. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Because the fabric is satin and from Liberty, it is possible that it was not used in the costumes of the people in the entourage but perhaps for the costume of the Duchess herself??? ==== Lafayette, Ltd. ==== The invoice and receipt from Lafayette, Ltd., the photographer that set up a temporary studio in the garden to take portraits of people at the ball in their costumes, may not be related to the ball at all.<ref>Invoice and receipt. Lafayette, Ltd. To His Grace The Duke of Devonshire. Date of itemized invoice 12 April 1897; addressed to the Duke, 18 February 1898; receipt and thanks for payment, 7 April 1898. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Three dates are written on the preprinted stationery: # 18/2/98 (18 February 1898), under the direction to "His Grace The Duke of Devonshire" # 4/12/97 (4 December 1897), next to the single item on the invoice for which a charge is listed: "6 [??] £1.10.0" # 7/4/98 (7 April 1898), in a different hand, with "Recd by cheque 7/4/98 Lafayette Ltd pp[?] [??] thanks" At the bottom of the page, in the hand that wrote all of the invoice except the receipt and thanks, is "With Lafayette Ltds Compliments." ==== Details for the Children in the Entourage ==== According to the ''Belfast News-Letter'', four children were "in white Assyrian robes, draped with pink shawls."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for "White Cloth Dresses":<ref name=":11" />{{rp|p. 2, back left of invoice}} * "2 Terra Gown draperies with Stars 200 in all" * "2 Cloth fronts embroidered with Square Medallions down centre" * "2 do do [ditto ditto, that is, cloth fronts] embroidered double border down front each side and collar" * "4 Sleeves embroidered Small Medallions" The Burnet & Co. invoice lists 6 yards of "Terra" Silk Fringe, which perhaps was used to trim the "terra draperies," or shawls, made from 3 1/4 yards of "Light Terra Satinette" worn by the children? ==== Details for the Trumpeters in the Entourage ==== According to the ''Belfast News-Letter'', four trumpeters were "in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel."<ref name=":8" />{{rp|p. 5, Col. 9a}} The trumpeters appear to have been dressed as soldiers or military men. According to the B. Burnet invoice, the following was purchased for the trumpeters' costumes:<ref name=":11" />{{rp|p. 1, front of invoice}} * 7 '''units (yards?)''' of purple silk [probably used for shawls?] * "2 skirt fronts with border alround [sic]" * "2 sets of Leather Cuarasses [sic] Embroidered front & back" * "4 Sleeves embroidered loop stitch" The Millward invoice lists * 2 Helmets * 2 Centre Pieces [probably for helmets rather than table decorations] ==== Details for the Fan-bearers in the Entourage ==== According to the ''Belfast News-Letter'', four fan-bearers were "attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for the fan bearers's costumes:<ref name=":11" />{{rp|pp. 1–2, front and left-back of invoice}} * "Embroidering 2 Crimson draperies with Stars 334 in all" * "2 Top [?] fronts embroidered & round necks" * "4 Sleeves embroidered long stitch"The Millward invoice lists * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers<ref name=":12" />{{rp|p. 2, back}} The Burnet & Co. invoice lists 12 yards of "Red Silk Fringe," which perhaps was used to trim the "crimson shawls" or "Crimson draperies," which may have been made from the 5 yards of "Red Satinette." Again, this list suggests two rather than four costumes. == Demographics == *Nationality: born in Hanover, in what is now Germany<ref name=":0">{{Cite journal|date=2020-07-27|title=Louisa Cavendish, Duchess of Devonshire|url=https://en.wikipedia.org/w/index.php?title=Louisa_Cavendish,_Duchess_of_Devonshire&oldid=969824214|journal=Wikipedia|language=en}}</ref> === Residences === ==== As Duchess of Manchester ==== *Kimbolton Castle, Huntingdonshire *Manchester House, London ==== As Duchess of Devonshire ==== *Devonshire House, London (mid-April until mid-July, for the Season) *Compton Place, Eastbourne (mid-July until 12 August<ref name=":1" />{{rp|p. 32}}) *Bolton Abbey, Yorkshire (12 August until the middle of September<ref name=":1" />{{rp|p. 32}}) *Chatsworth, Derbyshire (middle of September until early Spring<ref name=":1" />{{rp|p. 32}}) *Lismore Castle, County Waterford (early Spring until the middle of April<ref name=":1" />{{rp|p. 32}}) == Family == *Louisa (or Luise) Friederike Auguste Gräfin von Alten Montagu Cavendish (15 January 1832 – 15 November 1911)<ref name=":2" /><ref name=":0" /> *William Drogo Montagu, 7th Duke of Manchester (15 October 1823 – 22 March 1890)<ref name=":3" /><ref>{{Cite journal|date=2020-09-07|title=William Montagu, 7th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=William_Montagu,_7th_Duke_of_Manchester&oldid=977197445|journal=Wikipedia|language=en}}</ref> #George Victor Drogo Montagu, 8th Duke of Manchester (17 June 1853 – 18 August 1892) #Mary Louise [Louisa?] Elizabeth Montagu Douglas-Hamilton Forster (27 December 1854 – 10 February 1934) #Louisa Augusta Beatrice Montagu Acheson (c. 1856 – 3 March 1944) #Charles William Augustus Montagu (23 November 1860 – 10 November 1939) #Alice Maude Olivia Montagu Stanley (15 August 1862 – 23 July 1957) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], 8th Duke of Devonshire (23 July 1833 – 24 March 1908) == Notes and Questions == #As Duchess of Manchester Luise was not invited to the wedding between Bertie and Alix, Victoria's punishment for Luise's having gotten the Duke of Derby to promise her the position of Mistress of the Robes (and then exacting that promise).<ref>Leslie, Anita. ''The Marlborough House Set''. New York: Doubleday, 1973.</ref>{{rp|pp. 47–48}} #"As a young woman she was extremely beautiful; Princess Catherine Radziwill saw her at a reception given by the Empress of Germany and recalls on being introduced to her 'how she struck me as the loveliest creature I had ever set eyes upon. Indeed I have only met three women in my whole existence who could be compared to her.'"<ref name=":1" />{{rp|p. 21}} == Footnotes == {{reflist}} pkmu3oqykhr076u0no3gm0s78m5ujrk 2416324 2416323 2022-08-18T22:27:08Z Scogdill 1331941 /* The Historical Zenobia */ wikitext text/x-wiki == Overview == * Louise, Duchess of Devonshire was the subject of much comment and gossip in her life, arising in part from her prominence in society and in part from her conduct, which was not particularly controlled by middle-class notions of "respectability." * "Louise Alten, for her part, exemplifies another vanished feature [of Victorian England], the role of the political hostess. The part she played in Hartington's career even while she was his mistress gives a special twist to this function, and her more conventional ascendancy as Duchess of Devonshire, when she was said to be the most powerful person outside the government, shows the influence that could be wielded by women in a political system from which they were formally excluded."<ref name=":20">Vane, Henry. ''Affair of State: A Biography of the 8th Duke and Duchess of Devonshire''. Peter Owen, 2004.</ref>{{rp|12}} == Also Known As == *Louise, Duchess of Devonshire (15 January 1832 – 15 November 1911) *Louisa, Duchess of Manchester *Luise Friederike August Gräfin von Alten *Louisa Montagu *Louise Cavendish *The Double Duchess == Acquaintances, Friends and Enemies == === Friends === *[[Social Victorians/People/Albert Edward, Prince of Wales | Albert Edward, Prince of Wales]] (beginning about 1852) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], Lord Hartington (later 8th Duke of Devonshire) *Daisy, Lady Warwick *Lady Mayoress, Mrs. Benjamin Samuel Faudel-Phillips, 2nd Baronet,<ref>{{Cite journal|date=2020-08-25|title=Faudel-Phillips baronets|url=https://en.wikipedia.org/w/index.php?title=Faudel-Phillips_baronets&oldid=974879290|journal=Wikipedia|language=en}}</ref> presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4">"The Queen's Drawing Room" ''Morning Post'' 25 February 1897 Thursday: 5 [of 10], Col. 5a–7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970225/047/0005.</ref>{{rp|p. 5, Col. 6c}} *Mrs. J. E. Mellor, presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4" />{{rp|p. 5, Col. 6c}} === Enemies === * Consuelo, Duchess of Marlborough (at least, in 1901)<ref name=":1">Murphy, Sophia. ''The Duchess of Devonshire's Ball''. London: Sidgwick & Jackson, 1984.</ref>{{rp|pp. 31–32}} == Organizations == == Timeline == '''1852 July 22''', Luise Friederike Auguste Gräfin von Alten and William Drogo Montagu married.<ref name=":2">"Luise Friederike Auguste Gräfin von Alten." {{Cite web|url=http://www.thepeerage.com/p10947.htm#i109469|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1859''': "As for Louise, a glimpse of her unconventional ways is given by Lady Eleanor Stanley, describing a paper-chase at Kimbolton in 1859: 'The Duchess of Manchester, in getting too hastily over a stile, caught a hoop of her cage in it, and went head over heels, alighting on her feet with her cage and whole petticoats remaining above her head. they say there was never such a thing seen — and the other ladies hardly knew whether to be thankful or not that a part of her underclothing consisted of a pair of scarlet tartan knickerbockers (the things Charles shoots in) — which were revealed to all the world in general and the Duc de Malakoff in particular.' Lady Eleanor says that the other ladies hardly knew whether to be thankful because at this date a lady's normal wear under her petticoats was only a pair of leggings laced up to the waistband, an arrangement that could be unduly revealing in combination with a crinoline. ... But some of the more active ladies were beginning to adopt / Louise's expedient of donning a hidden pair of breeches for rambles in the country."<ref name=":20" />{{rp|25–26}} '''1863, early, or late 1862''', Louise and Spencer Compton Cavendish began a relationship.<ref name=":1" />{{rp|p. 26}} '''1873 December 10''', Mary Louise Elizabeth Montagu (daughter) and William Douglas-Hamilton married. '''1876 May 22''', Consuelo Iznaga y Clement and George Victor Drogo Montagu (son) married in Grace Church, New York City.<ref>{{Cite journal|date=2020-08-24|title=George Montagu, 8th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=George_Montagu,_8th_Duke_of_Manchester&oldid=974659520|journal=Wikipedia|language=en}}</ref><ref>{{Cite journal|date=2020-07-27|title=Consuelo Montagu, Duchess of Manchester|url=https://en.wikipedia.org/w/index.php?title=Consuelo_Montagu,_Duchess_of_Manchester&oldid=969888488|journal=Wikipedia|language=en}}</ref> '''1876 August 10''', Louisa Augusta Beatrice Montagu (daughter) and Archibald Acheson married. '''1889 January 5''', Alice Maude Olivia Montagu (daughter) and Edward Stanley married. '''1890 March 22''', William Drogo Montagu (7th Duke) died.<ref name=":3">"William Drogo Montagu, 7th Duke of Manchester." {{Cite web|url=http://www.thepeerage.com/p10128.htm#i101274|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1890 November 14''', William Angus Drogo Montagu (grandson) and Helena Zimmerman married secretly, in London.<ref>"Helena Zimmerman." {{Cite web|url=http://www.thepeerage.com/p34555.htm#i345545|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1892 August 16''', Louise Friederike Auguste Gräfin von Alten Montagu and Spencer Compton Cavendish, her second husband, married.<ref name=":2" /> '''1897 July 2, Friday''', Louise Cavendish (#18 on the list of attendees) hosted her famous [[Social Victorians/1897 Fancy Dress Ball| fancy-dress ball]] at Devonshire House in London. '''1897 July 20''', Mary Louise Elizabeth Montagu Douglas-Hamilton and Robert Carnaby Foster married. '''1900 November 14''', William Angus Drogo Montagu and Helena Zimmerman married.<ref>{{Cite journal|date=2020-07-17|title=Helena, Countess of Kintore|url=https://en.wikipedia.org/w/index.php?title=Helena,_Countess_of_Kintore&oldid=968067371|journal=Wikipedia|language=en}}</ref> '''1901 Spring''', Paris, Consuelo Spencer-Churchill, Duchess of Marlborough, describes a meeting with Louise Cavendish in the spring following Queen Victoria's death at the horse racetrack, Longchamps:<blockquote>A renowned character and virtually dictator of what was known as the fast set as opposed to the Victorian, Her Grace was a German aristocrat by birth. She had first been married to the impoverished Duke of Manchester, and when he died had improved her status by marriage to the rich Duke of Devonshire, who waged an undisputed influence in politics. Rumour had her beautiful, but when I knew her she was a raddled old woman, covering her wrinkles with paint and her pate with a brown wig. Her mouth was a red gash and from it, when she saw me, issued a stream of abuse. How could I, she complained, pointing to my white gloves, show so little respect to the memory of a great Queen? What a carefree world we must have lived in, that etiquette even in such small matters could assume so much importance?<ref>Balsan, Consuelo Vanderbilt. ''The Glitter and the Gold: The American Duchess — In Her Own Words''. New York: St. Martin's, 1953.</ref>{{rp|p. 115}}</blockquote> === Annual Events === Every year, as Duchess of Devonshire, Louise held a dance on the night after the Derby at Epsom Downs, which at this point was held on Wednesdays after Easter. == The Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == Louise, Duchess of Devonshire and Spencer Compton, 8th Duke of Devonshire hosted the famous [[Social Victorians/Terminology#Fancy-dress ball|fancy-dress ball]], which took place on Friday night. According to ''House Style: Five Centuries of Fashion at Chatsworth'',<blockquote>Such an event required extensive planning and attention to detail. The housekeeper aided by two secretaries was in charge of organisation inside the house with precise instructions from the Duchess on menus and all arrangements.<ref name=":18" />{{rp|137}}</blockquote> === The Devonshire House Staff at the Ball === The Dukes of Devonshire did not keep employment records of the staff working at their various properties. The Servants and Staff Database managed by Chatsworth House has a list of everyone ever named in any documents as a servant or staff member, and by 1891 census records included employment at places like the properties owned by the Duke of Devonshire.<ref name=":19" /> === Questions === # Who was housekeeper? Who were the secretaries? # Where did this info about the "precise instructions" come from? === The Historical Zenobia === Louise, Duchess of Devonshire attended the ball dressed as Zenobia, queen of Palmyra. Zenobia (240 – c. 274) was queen of the Syrian Palmyrene Empire, ruling as regent for her son after her husband's assassination.<ref>{{Cite journal|date=2022-05-03|title=Zenobia|url=https://en.wikipedia.org/w/index.php?title=Zenobia&oldid=1086005949|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Zenobia.</ref> She was the subject of much art in the 19th century, including literature, opera, sculpture, and paintings. Middle-eastern traveller Lady Hester Stanhope (1776–1839) discussed Zenobia in her memoirs, published in 1847.<ref>{{Cite journal|date=2022-03-07|title=Lady Hester Stanhope|url=https://en.wikipedia.org/w/index.php?title=Lady_Hester_Stanhope&oldid=1075838273|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lady_Hester_Stanhope.</ref> We do not know what sources, if any, were consulted by Louise, Duchess of Devonshire in preparing for personating Zenobia for the ball. The ''Encyclopædia Britannica'' offers a sense of what was generally available to those interested in that kind of research. The article on Zenobia in the 11th edition of the ''Encyclopædia Britannica'', published in 1911, written by George Albert Cooke<ref>Rev. George Albert Cooke (G.A.C.), Oriel Professor of the Interpretation of Holy Scripture in the University of Oxford, Fellow of Oriel College, Vol. XXVIII, p. viii. </ref>, is substantial, more than an entire column.<ref>"G.A.C." [George Albert Cooke]. "Zenobia." Encyclopaedia Britannica: An Dictionary of Arts, Sciences, and General Information. Ed. . Vol. XXVIII (Vol. 28): Vetch to Zymotic Diseases. ''Google Books''. Retrieved 11 August 2022. https://books.google.com/books?id=NEpAAQAAMAAJ.</ref>{{rp|p. 972, Col. 1a–2b}} The prior edition<ref>The 10th edition of the ''Britannica'', 1902–1903, is the 9th edition plus 10 supplementary volumes.</ref>, the 9th edition, does not have an article on Zenobia, but she has an entry that is a cross reference to the Palmyra article, which is about 10 columns and includes a biographical sketch of her.<ref>By W.R.S., W. Robertson Smith, Lord Almoner's Professor of Arabic, University of Cambridge.</ref><ref name=":21">"W. R. S." [Smith, W. Robertson]. "Palmyra." ''The Encyclopaedia Britannica: A Dictionary of Arts, Sciences, and General Literature''. Ed., Thomas Spencer Baynes. 9th edition. Vol. XVIII (Vol. 18): Orne–Phthisis. 1885. Google Books. Retrieved 11 August 2022.https://books.google.com/books?id=0au0xN1PrAEC.</ref>{{rp|p. 1, Col. 1c, after p. 858}} In a very long paragraph about her husband's father and then her husband, the ''Britannica''<nowiki/>'s discussion of Zenobia begins, <blockquote>Odænathus himself seems to have been engaged in almost constant warfare in the east and north against the Persians and perhaps the Scythians, but in his absence the reins of government were firmly held by his wife Zenobia, the most famous heroine of antiquity, to whom indeed Aurelian, in a letter preserved by Trebellius Pollio, ascribes the chief merit of all her husband's success. Septimia Zenobia was by firth a Palmyrene; her native name was Bath Zabbai ... [;] and Pollio's description of her dark beauty, black flashing eyes, and pearly teeth, together with her unusual physical endurance and the frank commanding manners which secured her authority in the camp and the desert, point emphatically to an Arabic rather than a Syrian descent. ... To the union of firmness and clemency, which is the most necessary quality of an Eastern sovereign, Zenobia added the rarer gifts of economy and organization, and an unusual range of intellectual culture. She spoke Coptic as well as Syriac, knew something of Latin, and had learned Greek from the famous Longinus, who remained at her court to the last, and paid the penalty of his life for his share in her counsels. She was also a diligent student of Eastern and Western history, and the statement that she enjoined her sons to speak Latin so that they had difficulty in using Greek implies a consistent and early adoption of the policy which made the sucess of Odænathus, and, taken in connexion with Aurelians' testimony, in a letter preserved by Pollio, that she had the chief merit of her husband's exploits, seems to justify the conclusion that it was her educated political insight that created the fortunes of the short-lived dynasty. ... In the zenith of his fame Odænathus was cut off by assassination along with his eldest son Herod, and it is generally assumed that the murder took place under Gallienus. ... [p. 201, Col. 2c – p. 202, Col. 1a] The fact seems to be that, while Odaenathus was busy at the other end of his kingdom, Zenobia administered the government at Palmyra and directed the conquest of Egypt, still nominally acting under the emperor at Rome, whose authority on the Nile was disputed by one or more pretenders. ... It still seems ... strange and yet an undoubted fact that Zenobia, who not only enjoyed the real authority behind her beardless son, but placed her name before his on public inscriptions, ... struck no coins till the second year of Aurelian, when the breach with Rome took place, and she suddenly appears as an empress (Σεβαστή, Augusta) of five years' standing. Up to that date the royal pair probably did not venture to coin in open defiance to Rome, and yet were unwilling to circulate an acknowledgment of vassalship in all the bazaars of the East. ... ... Zenobia, supported by her two generals, kinsmen of her husband, was now face to face with a Roman invasion. She held Egypt, Syria, Mesopotamia, and Asia Minor as far as Ancyra; and Bithynia was ready to join her party had not the army of Aurelian appeared just in time from Byzantium. She could count too on the Armenians and the Arabs, but the loyalty of Syria was doubtful: the towns disliked a rule which was essentially "barbarian," and in Antioch at least the patroness of the Monarchian bishop Paul of Samosata could to be popular with the large Christian party by whom he was bitterly hated. There were many Romans [p. 202, Col. 1c – p. 202, Col. 2a] in Zenobia's force, and it was they who bore the brunt of the two great battles at Antioch and Emesa, which followed Aurelian's rapid advance through Asia Minor. But Zenobia made light of these defeats, — "I have suffered no great loss" was her message to Aurelian, "for almost all who have fallen are Romans." ... But the empire of Palmyra came four centuries too soon. Rome was not yet exhausted, and Zenobia had neither the religious discipline of Islam to hold the Arabs together nor the spoil of the treasuries of Persia to keep their enthusiasm always fresh. Aurelian's military skill was strained to the uttermost by the prudence and energy of Zenobia, but he succeeded in forming and maintaining the siege of Palmyra in spite of its bulwark of desert, and his gold corrupted the Arab and Armenian auxiliaries. Zenobia attempted to flee and throw herself on the Persians, but she was pursued and taken, and then the Palmyrene lost heart and capitulated. Aurelian seized the wealth of the city, but spared the inhabitants, and to Zenobia he granted her life while he put her advisors to death. She figured in his splendid triumph, and by the most probable account accepted her fall with dignity, and closed her days at Tibur, where she lived with her sons the life of a Roman matron. The fall of Zenobia may be placed in the spring of 272.<ref name=":21" /> (p. 201, Col. 2a – p. 202, Col. 2b) </blockquote> == The Duchess of Devonshire's Costume == [[File:Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra.jpg|thumb|Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra|alt=Louise, Duchess of Devonshire in costume as Zenobia, Queen of Palmyra]] At their fancy-dress ball, Louisa, Duchess of Devonshire sat at Table 1 during the first seating for supper, escorted in to the table by the Prince of Wales.<ref name=":7">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref>{{rp|p. 7, Col. 4c}} Her costume was designed by M. Comelli (Attillo Giuseppe Comelli, 1858–1925, artist and costumier for opera, ballet and theatre in London as well as Europe and the U.S.<ref name=":13">{{Cite book|url=https://books.google.com/books?id=SZh2DwAAQBAJ&pg=PT207&lpg=PT207&dq=Attilio+Comelli&source=bl&ots=lFB0If7CwV&sig=ACfU3U1_Ost_lhmMvzMMs6NvuhK5SlRhJw&hl=en&sa=X&ved=2ahUKEwjKlsTw2sH3AhXYAp0JHVIxDWA4KBDoAXoECBAQAw#v=onepage&q=Attilio%20Comelli&f=false|title=Forgotten Designers Costume Designers of American Broadway Revues and Musicals From 1900-1930|last=Unruh|first=Delbert|date=2018-11-06|publisher=Page Publishing Inc|isbn=978-1-64082-758-5|language=en}} N.P.</ref>)<ref name=":5">“The Devonshire House Ball.” The ''Man of Ross'' 10 July 1897, Saturday: 2 [of 8], Col. 4b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0001463/18970710/033/0002.</ref> <ref name=":8">"The Duchess of Devonshire's Fancy Dress Ball. Special Telegram." ''Belfast News-Letter'' Saturday 03 July 1897: 5 [of 8], Col. 9 [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000038/18970703/015/0005.</ref>{{rp|p. 5, Col. 9a}} <ref name=":9">"By One Who Was There." “The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.</ref> and constructed by the House of Worth. Comelli seems to have designed [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess and Her Entourage|the costumes of her retinue as well]]. According to Russell Harris,<blockquote>For her costume, the Duchess commissioned Monsieur Comelli (1858-1925), a well-known designer of opera costumes for the London theatre and opera stage, and then had the design made up by Worth of Paris. ''Munsey’s Magazine'' noted “it is safe to say that the Queen of Palmyra never owned such a sumptuous costume in her lifetime.”<ref>Harris, Russell. {{Cite web|url=http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html|title=Louise, Duchess of Devonshire, née Countess von Alten of Hanover (1832-1911), as Zenobia, Queen of Palmyra|website=www.rvondeh.dircon.co.uk|access-date=2022-05-05}} ''Narrated in Calm Prose: Photographs from the V&A's Lafayette Archive of Guests in Costume at the Duchess of Devonshire's Diamond Jubilee Ball, July 1897''. http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html.</ref></blockquote>Lafayette's portrait of "Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester)" in costume is photogravure #5 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref>"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.</ref> The printing on the portrait says, "The Duchess of Devonshire as Zenobia Queen of Palmyra," with a Long S in ''Duchess''.<ref>"Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra." Devonshire House Fancy Dress Ball Album. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158357/Louise-Frederica-Augusta-Cavendish-ne-von-Alten-Duchess-of-Devonshire-formerly-Duchess-of-Manchester-as-Zenobia-Queen-of-Palmyra.</ref> Often, the V&A Lafayette Archive contains more than one portrait of a sitter for this ball, but the uncropped portrait (above right), which shows the unfinished end of the balustrade in front of the Duchess and the edge of the painted drop behind it, seems to have been the only portrait taken by Lafayette of the Duchess in costume. The copy owned by the National Portrait Gallery in London and the copy included in the album are cropped so that those unfinished edges do not show, but they appear to be from the same photograph. That is, there is one source for our information about this portrait: this single photograph. === Newspaper Descriptions of the Duchess's Costume === Newspaper articles about the Duchess's presence at the ball focused on her hosting, her costume, [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Jewelry|her jewelry]], and [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Entourage|her entourage]], often in the same story. These almost exactly identical descriptions suggest [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]] or a shared primary source: * The London ''Evening Standard'', ''Morning Post'' and ''Times'' published almost identical descriptions, varying only by the occasional punctuation mark or short phrase ("supplied by Worth, of Paris"). The fullest version is this one: "The Duchess of Devonshire, as Zenobia, Queen of Palmyra, wore a magnificent costume, supplied by Worth, of Paris. The skirt of gold tissue was embroidered all over in a star-like design in emeralds, sapphires, diamonds, and other jewels, outlined with gold, the corners where it opened in front being elaborately wrought in the same jewels and gold to represent peacocks' outspread tails. This opened to show an under-dress of cream crêpe de chine, delicately embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was attached to the shoulders by two slender points, and was fastened at the waist with a large diamond ornament. It was of green velvet of a lovely shade, and was superbly embroidered in Oriental designs, introducing the lotus flower in rubies, sapphires, amethysts, emeralds, and diamonds, in four borderings on contrasting grounds, separated with gold cord. The train was lined with turquoise satin. The bodice was composed of gold tissue to match the skirt, and the front was of crêpe de chine, hidden with a [[Social Victorians/Terminology#Stomacher|stomacher]] of real diamonds, rubies, and emeralds, and there was a jewelled belt."<ref name=":16">“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.</ref>{{rp|p. 3, Col. 2b}} <ref name=":7" />{{rp|p. 7, Col. 7a}} <ref name=":6">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1A–4C ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref>{{rp|p. 12, Col. 3b}} *"The Duchess of Devonshire was a dazzling vision, dressed as 'Zenobia,' in a glistening gold gauze gown, elaborately ornamented with suns and discs, wrought in purple and green gems outlined with gold, and having a large diamond as centre. The space between was fluted with fine silver spangles. This robe was open in front over an under dress of white crépe de chine, delicately worked in crystals, and at each side of the opening on the gold robe were large fan-shaped groups of peacock feathers, worked in multicoloured jewels. The [[Social Victorians/Terminology#Corsage|corsage]] was to correspond, and had a magnificent [[Social Victorians/Terminology#Girdle|girdle]] of jewels, the train of bright green velvet, hung like a fan, without folds, being fastened at each side of the shoulders by diamond brooches, and caught at the waist with a similar ornament. It was a mass of gorgeous embroidery, carried out in heliotrope velvet, lotus flowers studded with tinted gems, and other devices in terra-cotta and electric blue velvet — all enriched with gold, diamond, and jewelled embroidery — and lined with pale blue satin. ... Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail."<ref name=":15">"Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses." London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.</ref>{{rp|p. 5, Col. 6a}} * According to the article in ''The Graphic'' written by [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]], though this caption to the Lafayette photograph may not be hers, the Duchess of Devonshire wore a "Skirt of gold tissue, embroidered all over with emeralds, sapphires, diamonds, and other jewels outlined with gold. This opened to show an underdress of crème crêpe de chine, embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was green velvet, superbly embroidered in Oriental designs. The bodice was composed of gold tissue, and the front was of crêpe de chine hidden with a stomacher of diamonds, rubies, and emeralds."<ref name=":10">Greville, Violet, Lady. "Devonshire House Ball." The ''Graphic'' Saturday 10 July 1897: 15 [of 24]: Col. 1a–16, Col. 1c. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000057/18970710/019/0015.</ref>{{rp|p. 15, Col. 3b}} Because it is in black and white, the photograph of the Duchess taken by Lafayette in her costume (above right) can tell us almost nothing about the colors of the costume. Fortunately, this costume has survived and is held in Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth). ''House Style: Five Centuries of Fashion at Chatsworth'', the volume about an exhibition mounted 25 March to 22 October 2017, has beautiful photographs from several perspective of the costume, including the train, and accessories. Published photographs of it '''show X'''.<ref name=":18">Burlington, Julia, and Hamish Bowles. ''House Style: Five Centuries of Fashion at Chatsworth''. Skira Rizzoli, 2017. Based on an exhibition 25 March to 22 October 2017.</ref> It has been worn by '''Deborah, 11th Duchess of Devonshire when the Dowager Duchess at some point for some reason'''. Some accounts of her costume suggest that the reporter thought that all the jewels sewn onto the dress were actual precious or semiprecious stones, but the London ''Daily News'' says some of them were crystals.<ref name=":15" />(p. 5, Col. 6a) According to ''House Style: Five Centuries of Fashion at Chatsworth'',<blockquote>The skirt of gold gauze, appliquéd with tinsel medallions and peacock plumes worked in bright foils, wire coils and spangled with sequins, was worn over an ivory satin underskirt wrought over with silver thread and diamonds. Attached to the shoulders was a long graduated train in the most vivid emerald-green velvet, appliquéd with velvet and gold work in an eastern design and studied with jewels.<ref name=":18" />(128)</blockquote> === Observations on the Duchess's Costume === In the photograph of her in costume, the Duchess is standing in a three-quarter front pose. That static position in a two-dimensional image makes it difficult to understand exactly how the costume and headdress worked. The terms we use are these: stomacher, train, bodice (what's on the torso, no matter how many layers), overskirt, underskirt, belt (really an accessory, like the jewelry), sleeves. The 1897 newspaper reports use different terminology for the parts of the dress than we would now. Some of the terms that appear in the newspaper reports include overdress, under-dress, bodice, stomacher robe, train, corsage, girdle. The Duchess's Body how the fabric draped, how tight the bodice was, what the stomacher looked like, what pearls she is wearing. Is her hair down or up? ==== Fit ==== *The recent exhibitions of the dress show a different fit than what is shown in the Lafayette photograph. Louise, the 8th Duchess's bodice is draped loosely over the stomacher, but photographs of Deborah, 11th Duchess or of the mannequin in the 2017 exhibition show a bodice fitted to the stomacher. * Also, the costume itself may have been altered or adjusted to make the waist more defined and the line more Elizabethan (the wide, square neckline; the A-line, the split skirt with the petticoat in the middle). * The dress seems to have been designed to flatter the 8th Duchess's statuesque figure, especially her waist: the belt falls down to the hips, the A-line; the way what we can see of the under-dress is narrow at the top and widens toward the bottom. * The drape of the skirt is straighter on the 8th Duchess as well as the Dowager Duchess than on the mannequins. ==== Inconsistencies in the Descriptions and Images ==== * The newspaper accounts of the Duchess's costume do not agree on several particulars, including the color of the train and its lining. ==== Changing Victorian Design Elements ==== The costume as Louise, the 8th Duchess wore it is vaguely Middle Eastern and exotic, consistent with her '''leading''' the "Oriental" procession. The sleeve treatment, the drape of the bodice and the skirt are consistent with the line of something from the Middle East. This would have been what Comelli designed. Early clothing history books existed by the middle of the century, and it would be logical that Comelli would be familiar with them. The versions of the costume shown in recent exhibitions make it seem more Victorian than what we see in Lafayette's portrait. The tightness of the fit of the bodice indicates a Victorian corset under the costume. The line of the bodice and skirt, the bare arms in the sleeve treatment, the train, the belt, The versions of the dress the the tightness of the dress against the stomacher all look Victorian. A-line drape of the skirt was done for the exhibition to show the artistry and complexity of the trim. * The sleeve treatment in the 1897 photograph is caught up at the shoulder. On the Dowager Duchess and the mannequins, the sleeves are released. ==== Trim ==== * The sequins attached to the costume would have been metal. They appear to be silver that has tarnished, which is why they are dark now. Currently, the tarnish on the sequins and embroidery thread cannot be removed without damage to the fabric. * The gold thread in the embroidery has not tarnished, but the silver thread, which would have sparkled in 1897, has. ==== Accessories ==== * Belt * Headdress * Do the horns on [[Social Victorians/People/Louisa Montagu Cavendish#Zenobia's Crown|her crown]] go toward the back from the apex that rises from the jewel-encrusted band or do they stick straight out to the sides? Does the headdress include the traditional three white plumes? * Jewelry * Fan * Can't tell how many strands of pearls are around Louise's neck in the Lafayette photo, but the photographs of the recreation of the crown and of the costume from the 2017 exhibit show [[Social Victorians/People/Louisa Montagu Cavendish#Goldsmith, Pearl & Diamond Merchant, & Silversmith|five strands]]. ==== The Train ==== * The reporter's description of the Duchess's train in the London ''Daily News'' is confusing: it says the train is "bright green velvet" and "It was a mass of gorgeous embroidery, carried out in heliotrope velvet." Embroidery is not "carried out" in velvet. Heliotrope is not green, it's pinkish purple. * Furthermore, heliotrope appears to have been a new word for a color, introduced perhaps in 1882 in A. Maerz and M. Rea Paul ''A Dictionary of Colour'', New York: '''1930 McGraw-Hill Page 196'''; Color Sample of Heliotrope: Page 131 Plate 54 Colour Sample C10 ("Heliotrope," Wikipedia [[wikipedia:Heliotrope_(color)#cite_note-2|https://en.wikipedia.org/wiki/Heliotrope_(color)#]]<nowiki/>[[wikipedia:Heliotrope_(color)#cite_note-2|c]][[wikipedia:Heliotrope_(color)#cite_note-2|ite_note-2]]). The ''OED'' lists 3 newspaper sources for the 1880s, The ''World'' (1882), ''Truth'' (1886), and the ''Daily News'' (1887) ("heliotrope" 1.d. Retrieved 6 August 2022). The ''Daily News'' describes it well: "that peculiar mauve known as heliotrope." The word itself has existed in English for many centuries, <nowiki/>both for the flower (which turns to follow the sun) as well as for instruments that reflect or measure sunlight and also for a variety of quartz (also called bloodstone). * Color of green velvet on the train: the colors differ, depending on the photograph. One photograph shows a very vivid green, and the rest show a lighter green. We're not sure what's up with that. == The Duchess's Jewelry == The Duchess was known fo<nowiki/>r her jewelry, and as she aged and after her marriage to the 8th Duke in 1892, the jewelry she wore only increased in scale and notability. It is not surprising that her jewelry would make up copy for the newspapers. Newspaper reports before the ball gossiped about the jewelry associated with the costumes being developed. For example, according to the Edinburgh ''Evening News'' on 21 June 1897, less than two weeks before the party, "The ball being a fancy dress one, men as well as women will be able in certain characters to wear jewels. The Duchess of Devonshire, who is to appear as Zenobia, is getting her jewels reset after the antique style."<ref>“The Duchess of Devonshire’s Ball.” Edinburgh ''Evening News'' 21 June 1897, Monday: 4 [of 6], Col. 5c [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18970621/079/0004.</ref> While almost all descriptions of her mention her jewels because they were so much a topic of conversation and because they were sewn onto the costume itself, these emphasize her jewelry and most are very similar to each other: * "The Duchess was attired with great Oriental magnificence as Zenobia. Her dress was a tissue of silver, embroidered with gold and jewels, an overmantle of cloth of gold embroidered in the same manner hung from the shoulders, and she wore a bandeau of gold studded with gems, and surrounded by hanging chains of pearls over her elaborate headdress; strings and ropes of jewels and pearls were worn round the neck, and hung down almost to the knees."<ref>“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032. </ref>{{rp|p. 32, Cols. 1c–2a}} <ref name=":17">"Duchess of Devonshire's Fancy-Dress Ball. Brilliant Spectacle." The [Guernsey] ''Star'' 6 July 1897, Tuesday: 1 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000184/18970706/003/0001.</ref>{{rp|p. 1, Col. 2a}} * "A wonderfully beautiful dress was that which was worn by the Duchess of Devonshire as Zenobia, Queen of Palmyra. It was of golden tissue, sewn with silver paillettes, and jewelled with diamonds and other precious stones. In front there were silk embroideries, in many vivid shades of colour, and here the golden draperies opened to show a petticoat of white crêpe de chine, embroidered with pearls and gold. The short train was of brilliant green velvet, exquisitely embroidered. One of the Duchess of Devonshire’s beautiful diamond and emerald tiaras had been taken to pieces to form a stomacher, the effect of which was dazzling in its brilliancy. Long chains of pearls and other wonderful jewels were worn with this beautiful dress."<ref>“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a [of 3]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.</ref>{{rp|p. 7, Col. 2b}} * In the article about the ball in the ''Graphic'', [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]] says, "The Ducal hostess herself elected to appear as Zenobia, Queen of Palmyra, with lavish magnificence, and wearing a corruscation of jewels which must have eclipsed the state of even the all-subduing majesty the Duchess impersonated."<ref name=":10" />{{rp|p. 16, Col. 1a}} *The Duchess was dressed "as Zenobia, Queen of Palmyra, her dress a marvel of soft tissues and exquisite ornament, and her tiara a still greater marvel of the jeweller's art."<ref name=":6" />{{rp|p. 12, Col. 2a}} <ref>"The Duchess of Devonshire’s Historic Ball. Some of the Fancy Costumes." Supplement. The ''Leicester Chronicle and Leicestershire Mercury'' 10 July 1897, Saturday: 11 [of 12], Cols. 4a–b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000173/18970710/141/0011.</ref>{{rp|p. 11, 4a}} As with the colors, Lafayette's photograph of the Duchess in costume does not show the jewels very clearly. We cannot see the stomacher or the "long chains of pearls and other wonderful jewels" or the pearls that "hung down almost to the knees" (although, of course, any newspaper account can be wrong because reporters were not present at the ball). === Zenobia's Crown<ref name=":10" /> === The crown that the Duchess wore as Zenobia is difficult to see clearly in the Lafayette photograph (above right), and no other images of the crown exist. It was lavish, "encrusted" with jewels and featuring pearls: * London ''Daily News'' says, "The crown worn with this was high, and of filigree gold, surmounted with two horns, each tipped with a large diamond. It was encrusted with large diamonds, rubies, and emeralds, and long chains of pearls fell under the chin and about the head — one magnificent pear-shaped pearl resting on the forehead."<ref name=":15" />{{rp|p. 5, Col. 6a}} * These descriptions are all identical except for the addition or subtraction of an occasional comma: The London ''Evening Standard'' says, "A gold crown encrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end, and two upstanding white ostrich feathers in the centre, and round the front were festoons of pearls, with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":16" />{{rp|p. 3, Col. 2b}} <ref name=":7" />{{rp|p. 7, Col. 7a}} <ref name=":6" />{{rp|p. 12, Col. 3b}} <ref name=":10" />{{rp|p. 15, Col. 3b}} * The Guernsey ''Star'' says, "Her Grace wore a bandeau of gold round her head, studded with diamonds, turquoise, and emeralds, and surrounded by hanging chains of superb pearls."<ref name=":17" />{{rp|p. 1, Col. 2a}} This crown was recreated from the single Lafayette photograph of the Duchess in costume for the ''House Style: Five Centuries of Fashion at Chatsworth'', the exhibition mounted in 2017 and exhibited on the mannequin wearing the Duchess's costume. A photograph of this recreation separate from the costume, but with the pearls and the white plumes, can be seen on [https://www.sothebys.com/en/articles/how-c-w-sellors-transformed-five-centuries-of-fashion-into-modern-jewellery this Sotheby's page] about C. W Sellor's, the jewelry firm that did the recreation as well as a number of other recreations for the exhibit.<ref>Sotheby's. "How C. W. Sellors Transformed Five Centuries of Fashion Into Modern Jewellery." ''Sotheby's'' 9 May 2017. Retrieved 11 August 2022. https://www.sothebys.com/en/articles/how-c-w-sellors-transformed-five-centuries-of-fashion-into-modern-jewellery.</ref> Although Lafayette's photograph is not conclusive, in Sellor's recreation the horns on the crown point toward back of the head rather than out to the sides. === Goldsmith, Pearl & Diamond Merchant, & Silversmith === The Duchess's pearls, which were an important feature of her costume, occasioned a great deal of direct commentary in the newspaper accounts. The Duchess's jewelry occasioned a great deal of reportage in the articles about the ball. '''It was reported that she had her jewels restrung to be used in the costume. stomacher and review of jewelry in more general articles''' An invoice and receipt in the Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) is from a concern whose preprinted stationery has a crown in the upper-left corner, suggesting that they had a royal warrant, and no name other than Goldsmith, Pearl & Diamond Merchant, & Silversmith. This document offers a unique view into the evolution of one necklace, at least, over the years. It lists what are apparently three restringing of some pearls of Louise, Duchess of Devonshire. The three restringings appear to be dated: The first necklace is a "Pearl Necklet in original 4 rows." The invoice is dated 20 October 1892 (but the stationery was printed to assume the invoice would be used in the 1880s, so the 9 is written over the second 8, and the 2 has been added).<ref name=":14">Invoice and receipt. Goldsmith, Pearl & Diamond Merchant & Silversmith. Date of itemized invoices for restringing pearls: 20 October 1892, 1 March 1897, 1909. The Devonshire Collections, Chatsworth, Reference number FIS/4/1/2.</ref>(p. 1) The necklet she brought in to be restrung contained a "Total [of] Total 224 large pearls": # 1st [row] 51 large pearls # 2nd 53 large pearls # 3rd 57 large pearls # 4th 63 large pearls The second necklace is a "Necklet as re-strung on October 15th 1892, with addition of small pearls supplied, now consists of 5 rows, containing" a total of "224 large pearls & 227 small"<ref name=":14" />(p. 1): # 1st 41 large pearls & 40 small # 2nd 42 large pearls & 42 small # 3rd 44 large pearls & 45 small # 4th 47 large pearls & 48 small # 5th 50 large pearls & 51 small The third necklace is a "Pearl Necklet as again re-strung with additional pearls supplied 1 March 1897, now consisting of 5 Rows containing" a total of "262 Large Pearls & 267 Small"<ref name=":14" />(p. 2): # 1st Row 45 Large Pearls & 44 Small # 2nd Row 48 large Pearls & 49 Small # 3rd Row 51 Large Pearls & 52 Small # 4th Row 56 Large Pearls & 65 small Possibly these pearls may have been restrung in 1909 into a coronet, but the handwriting is not clear enough to read.<ref name=":14" />(p. 2) If the Duchess wore one of these stringings of her pearls for the ball, then it must have been the second necklet, strung in 1892, a 5-strand necklace. None of the newspaper accounts refer to a 5-strand pearl necklace, although her pearls are often mentioned. == The Duchess's Entourage == Besides the Duke of Devonshire, the retinue of Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra, included her grandson, [[Social Victorians/People/William Angus Drogo Montagu|William Angus Drago Montagu, 9th Duke of Manchester]], dressed as a Georgian courtier. Four newspapers say that the Duchess's entourage included three groups, all in costume: children, trumpeters and fan-bearers. The ''Westminster Gazette'' and the ''Man of Ross'' list the groups but do not mention the number of members of the groups. According to two sources, probably in error, the London ''Daily News''<ref name=":15" />{{rp|p. 5, Col. 6a}} and the Belfast ''News-Letter,''<ref name=":8" />{{rp|p. 5, Col. 9a}} these groups each had four members. The London ''Daily News'' is likely the source (because it was published at 5:00 a.m.<ref>Mitchell, Charles. ''Newspaper Press Directory'', 1895. [Hathi Trust via U Wisconsin Madison.] http://hdl.handle.net/2027/mdp.39015085486150.</ref>{{rp|55}}) for the Belfast ''News-Letter'', which took part in [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]], like so many other newspapers of the 19th century. ['''check this: which one was published earlier in the day, and on which day?'''] These four sources describe the Duchess's retinue and how the people in it were dressed: *"The Duchess of Devonshire was dazzingly [sic] magnificent as 'Zenobia,' arrayed in the glistening fabrics and massive jewels in which artists have delighted to depict the Warrior Queen, the costume in this case being specially designed by the clever French artist, M. Comelli, who was also responsible for the splendid attire of the Queen's suite. This was composed of four children in white Assyrian robes, draped with pink shawls; four trumpeters in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel; and four fan-bearers attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold — a gloriously magnificent pageant."<ref name=":8" />{{rp|p. 5, Col. 9a}} *"Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail."<ref name=":15" />{{rp|p. 5, Col. 6a}} *"The duchess was dressed as Zenobia, in gold cloth, gorgeously embroidered in gold, brilliants, and coloured stones, and opening over an under dress of white crêpe de Chine, worked finely in brilliants. The train of light green velvet was lined with blue, and sumptuously embroidered in jewels and gold, the colouring being particularly artistic. With this dress were worn splendid jewels, and a large horn crown, encrusted with diamonds, emeralds, and rubies. The duchess was attended by a suite of children, trumpeters, and fan-bearers, all picturesquely attired in Assyian [sic] costumes — the whole group being specially designed by M. Comelli."<ref name=":5" /> *"The host was dressed as Charles V. of Germany, in black velvet, satin, and fur; and the Duchess made the most gorgeous of Zenobias, in a gown of gold gauze, and a green velvet train — both a mass of exquisite oriental embroidery. The crown and hanging ropes of pearls, the jewelled girdle, and the train of children, fan-bearers, and trumpeters — all in Babylonish garb — as designed by M. Comelli, made a gloriously imposing and picturesque group."<ref name=":9" /> === Details of the Costumes in the Entourage === The Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) has "receipts" or invoices that functioned as receipts for several commercial concerns that were involved in making costumes or accessories for costumes for this ball. They are the following: * [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|M. (Attillo Giuseppe) Comelli]] * [[Social Victorians/People/Louisa Montagu Cavendish#B. Burnet & Co.|B. Burnet & Co.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Arthur Millward, Theatrical Jeweller|Arthur Millward, Theatrical Jeweller]] * [[Social Victorians/People/Louisa Montagu Cavendish#Liberty & Co., Ltd.|Liberty & Co., Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Lafayette, Ltd.|Lafayette, Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Goldsmith, Pearl & Diamond Merchant, & Silversmith|Goldsmith, Pearl & Diamond Merchant, & Silversmith]] This list of commercial concerns almost certainly cannot be the complete list of all concerns that contributed to the costumes. These are the only receipts or invoices about expenses for the ball, however, that the Chatsworth Archive contains; similar documents were likely not even kept or were destroyed with other papers not retained at some point in time. The business concerns listed above were specialized and likely used for different elements of the costumes. As a theatrical designer, Comelli would have depended on the suppliers he knew and arranged with them for the construction of these costumes. The Chatsworth Archive calls these documents ''receipts'', which indeed they are because they were returned to Devonshire House as receipts for payment. From our perspective, though, they are invoices that contain specifics about what was used to make the costumes. The analysis of these invoices has led to an understanding of what the people who attended the Duchess in her entourage wore and a clearer sense, perhaps, of how many people walked in that entourage. This analysis is based on the items listed on the invoices and their pricing, most of which is included in the section for each invoice, below. While the Belfast ''News-Letter'' says that each group contained four members,<ref name=":8" />{{rp|p. 5, Col. 9a}} the invoices and receipts suggest that the newspaper, the single source for this information, was wrong about the number of people in each group. It is theoretically possible, of course, that suppliers other than the ones in the Chatsworth Archive made some of these costumes and that other invoices and receipts must have existed at that time. But the [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|Comelli memo, below]], seems definitive: he designed and seems to have overseen the construction of the costumes, which numbered six rather than twelve. '''[Collier discussion?]''' Besides providing welcome detail about the costumes of the people in the Duchess's entourage, which is available nowhere else, these invoices also raise at least as many questions as they answer. ==== M. (Attillo Giuseppe) Comelli ==== Attillo Giuseppe Comelli was a designer for opera, ballet and theatre in Europe, the UK and North America.<ref name=":13" /> The receipt in the Chatsworth Archive was sent from Covent Garden. The invoice lists £4 for "Making six costumes," 3''s'' for "Extras" and 12s for "Cab fares for men paid by the request of M. Comelli."<ref>M. Comelli, Covent Garden, to Duchess of Devonshire. Date of invoice 13 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Three other names are on this invoice and receipt: * L. L[?] Collier [?], written under and perhaps as part of the direction to the Duchess of Devonshire * Mr. Strong ("Forwarded to Mr Strong. [sic] by the instructions of M. Comelli," written in the same hand as wrote the majority of the memo) * Floyd [?] Collier [??] ("Received with Thanks," presumably thanking for the payment, in a different hand) ==== B. Burnet & Co. ==== An invoice and receipt from B. Burnet & Co., held in the Archives of the Duke of Devonshire, has specific information about some of the fabrics, trims and accessories purchased for the costumes of the Duchess's retinue.<ref name=":11">B. Burnet & Co. to Louise, Duchess of Devonshire. Date of invoice 2 and 6 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Besides itemizing some costume or accessory elements that seem clearly to be for the groups, the invoice also lists items not easy to associate with particular costumes, like the following: * 12 yards of White silk fringe 8in deep<ref name=":11" />{{rp|back left}} * 12 1/2 yards of "wht cloth"<ref name=":11" />{{rp|back left}} * 9 yards of "[[Social Victorians/Terminology#Selesia|Selesia]]"<ref name=":11" />{{rp|back left}} * 2 yards of Canvas<ref name=":11" />{{rp|back right}} * 4 Tan Wool Tights<ref name=":11" />{{rp|back right}} * 2 Tan Boys Tights<ref name=":11" />{{rp|back right}} At this time, we are not sure which costumes these elements were used for. Possibly the white silk fringe and the white cloth would have been used to construct the robes for the children and trumpeters in the entourage. The number of tights suggests that the six costumes on this invoice all included tights. With other elements of the trumpeters' costumes, the Burnet invoice also lists "6 prs Assyrian Buskins." Probably, to a late Victorian, buskins would have been "defensive leggings"<ref>Demmin, Auguste. An illustrated History of Arms and Armour: From the Earliest Period to the Present Time. George Bell, 1894. Google Books https://books.google.com/books?id=ArRCAAAAYAAJ: 106.</ref> laced together and covering the lower leg and often feet of a soldier. To a clothing and military historian, buskins (or greaves) were worn by people in a number of cultures over millennia and varied widely in style and construction. Buskins appear in Assyrian art held at the time by the British Museum. Listing six pairs of buskins suggests that every costume in the Duchess's entrourage included buskins, possibly worn over the tan tights. The Burnet invoice lists "4 Broad Belts," which may have held "4 Skins Fleshers."<ref name=":11" />{{rp|p. 1, front of invoice}} (A skin flesher is a kind of knife used to separate the skin from the meat in animals.) If each group included only two members, then perhaps the belts and fleshers were worn not only by the trumpeters but also by the fan-bearers. The Millward invoice (specifics in the section on the Millward invoice below) lists "8 Doz 'Plaques' for Belts'" with a drawing of an upright rectangle with a circle in the middle, which might have been a jewel. Double lines around the rectangle suggest that the plaques were not flat or the metal was not thin. The drawing does not give any ideas about how the plaques were attached to the belts, if they were. It is impossible to tell if the plaques were attached to the "4 Broad Belts" (likely for the trumpeters and fan-bearers), but unless they were quite tiny, "8 Doz 'Plaques'" would be far too many for the belts of only the two children. A different hand, probably "[L.??] L. Collier," wrote the following sentence at the end of the invoice and receipt, above the postmark:<blockquote>All the above named articles were used for the six [?] dresses made for the Devonshire Ball.<ref name=":11" />{{rp|back right}}</blockquote> This same hand, signing what is possibly "Floyd Collier," also signed the postmark of the Comelli invoice and receipt. On the Burnet document, this writer, possibly an assistant or employee of the Duchess of Devonshire, says that "six dresses" were made (if in fact, that word is ''six''). (No "Collier" is listed among the staff or servants of the Duke of Devonshire at the end of the 19th century.<ref name=":19">"Servants A-H." ''Historic Servants and Staff. Servants and Staff Database''. Retrieved 18 July 2022 https://www.chatsworth.org/media/11528/servants-a-h.pdf.</ref> The invoice appears to itemize materials used for six costumes: two children, two trumpeters and two fan-bearers. ==== Arthur Millward, Theatrical Jeweller ==== An invoice and request for payment from Arthur Millward, Theatrical Jeweller, held in the Archives of Chatsworth House, has more specifics about some of the fabrics, trims and accessories for the costumes of the Duchess's retinue.<ref name=":12">Memorandum. Arthur Millward, Theatrical Jeweller, to Louise, Duchess of Devonshire. Date of itemized invoice 1 July 1897; date of request for payment(?) 27 August 1897. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> This invoice lists the following, which could have been used in any of the costumes for the entourage: * 8 Doz 'Plaques' for Belts [discussed with the belts in the section on the Burnet invoice, above] * 4 Large Armlets * 4 Bracelets * 8 Armlets<ref name=":12" />{{rp|p. 2, back}} Because Millward was a Theatrical Jeweller, it seems likely that most (if not all) of the items listed on the invoice were made of metal and the jewels mentioned were artificial, made of glass or paste. Other items on the invoice seem to belong to the costumes of the trumpeters, which the Belfast ''News-Letter'' says included helmets: * 2 Helmets * 2 Centre pieces The Millward invoice shows tiny line drawings next to the words ''2 Helmets'' and ''2 Centre pieces''. These drawings suggest that the Centre pieces were attached to the helmets rather than being anything that would have been put on a table as decoration. Other items seem to belong to the costumes of the fan-bearers: * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers The "Pearl & Gold Headdresses" were likely the "jewelled diadems" mentioned in the Belfast ''News-Letter''. The "Fan properties with Feathers" are likely to have been the "long-handled fans of white feathers, mounted in blue and gold" mentioned in the newspaper report.<ref name=":8" />{{rp|p. 5, Col. 9a}} At the end of the Millward invoice, a "reduction as agreed with M [Mr?] Commelli [sic]" of £1 10''s'' is subtracted from a total of £22 3''s''. No reason for this reduction is given.<ref name=":12" />{{rp|p. 2, back}} ==== Liberty & Co., Ltd. ==== One invoice and receipt from the Chatsworth Archive, dated 12 July, to the Duchess of Devonshire, lists "13 yds S&W Satin[?]," 7 yards of blue and 6 of purple.<ref>Invoice and receipt. Liberty & Co. Ltd. To Her Grace, The Duchess of Devonshire. Date of itemized invoice 12 July [1897]. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Because the fabric is satin and from Liberty, it is possible that it was not used in the costumes of the people in the entourage but perhaps for the costume of the Duchess herself??? ==== Lafayette, Ltd. ==== The invoice and receipt from Lafayette, Ltd., the photographer that set up a temporary studio in the garden to take portraits of people at the ball in their costumes, may not be related to the ball at all.<ref>Invoice and receipt. Lafayette, Ltd. To His Grace The Duke of Devonshire. Date of itemized invoice 12 April 1897; addressed to the Duke, 18 February 1898; receipt and thanks for payment, 7 April 1898. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Three dates are written on the preprinted stationery: # 18/2/98 (18 February 1898), under the direction to "His Grace The Duke of Devonshire" # 4/12/97 (4 December 1897), next to the single item on the invoice for which a charge is listed: "6 [??] £1.10.0" # 7/4/98 (7 April 1898), in a different hand, with "Recd by cheque 7/4/98 Lafayette Ltd pp[?] [??] thanks" At the bottom of the page, in the hand that wrote all of the invoice except the receipt and thanks, is "With Lafayette Ltds Compliments." ==== Details for the Children in the Entourage ==== According to the ''Belfast News-Letter'', four children were "in white Assyrian robes, draped with pink shawls."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for "White Cloth Dresses":<ref name=":11" />{{rp|p. 2, back left of invoice}} * "2 Terra Gown draperies with Stars 200 in all" * "2 Cloth fronts embroidered with Square Medallions down centre" * "2 do do [ditto ditto, that is, cloth fronts] embroidered double border down front each side and collar" * "4 Sleeves embroidered Small Medallions" The Burnet & Co. invoice lists 6 yards of "Terra" Silk Fringe, which perhaps was used to trim the "terra draperies," or shawls, made from 3 1/4 yards of "Light Terra Satinette" worn by the children? ==== Details for the Trumpeters in the Entourage ==== According to the ''Belfast News-Letter'', four trumpeters were "in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel."<ref name=":8" />{{rp|p. 5, Col. 9a}} The trumpeters appear to have been dressed as soldiers or military men. According to the B. Burnet invoice, the following was purchased for the trumpeters' costumes:<ref name=":11" />{{rp|p. 1, front of invoice}} * 7 '''units (yards?)''' of purple silk [probably used for shawls?] * "2 skirt fronts with border alround [sic]" * "2 sets of Leather Cuarasses [sic] Embroidered front & back" * "4 Sleeves embroidered loop stitch" The Millward invoice lists * 2 Helmets * 2 Centre Pieces [probably for helmets rather than table decorations] ==== Details for the Fan-bearers in the Entourage ==== According to the ''Belfast News-Letter'', four fan-bearers were "attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for the fan bearers's costumes:<ref name=":11" />{{rp|pp. 1–2, front and left-back of invoice}} * "Embroidering 2 Crimson draperies with Stars 334 in all" * "2 Top [?] fronts embroidered & round necks" * "4 Sleeves embroidered long stitch"The Millward invoice lists * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers<ref name=":12" />{{rp|p. 2, back}} The Burnet & Co. invoice lists 12 yards of "Red Silk Fringe," which perhaps was used to trim the "crimson shawls" or "Crimson draperies," which may have been made from the 5 yards of "Red Satinette." Again, this list suggests two rather than four costumes. == Demographics == *Nationality: born in Hanover, in what is now Germany<ref name=":0">{{Cite journal|date=2020-07-27|title=Louisa Cavendish, Duchess of Devonshire|url=https://en.wikipedia.org/w/index.php?title=Louisa_Cavendish,_Duchess_of_Devonshire&oldid=969824214|journal=Wikipedia|language=en}}</ref> === Residences === ==== As Duchess of Manchester ==== *Kimbolton Castle, Huntingdonshire *Manchester House, London ==== As Duchess of Devonshire ==== *Devonshire House, London (mid-April until mid-July, for the Season) *Compton Place, Eastbourne (mid-July until 12 August<ref name=":1" />{{rp|p. 32}}) *Bolton Abbey, Yorkshire (12 August until the middle of September<ref name=":1" />{{rp|p. 32}}) *Chatsworth, Derbyshire (middle of September until early Spring<ref name=":1" />{{rp|p. 32}}) *Lismore Castle, County Waterford (early Spring until the middle of April<ref name=":1" />{{rp|p. 32}}) == Family == *Louisa (or Luise) Friederike Auguste Gräfin von Alten Montagu Cavendish (15 January 1832 – 15 November 1911)<ref name=":2" /><ref name=":0" /> *William Drogo Montagu, 7th Duke of Manchester (15 October 1823 – 22 March 1890)<ref name=":3" /><ref>{{Cite journal|date=2020-09-07|title=William Montagu, 7th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=William_Montagu,_7th_Duke_of_Manchester&oldid=977197445|journal=Wikipedia|language=en}}</ref> #George Victor Drogo Montagu, 8th Duke of Manchester (17 June 1853 – 18 August 1892) #Mary Louise [Louisa?] Elizabeth Montagu Douglas-Hamilton Forster (27 December 1854 – 10 February 1934) #Louisa Augusta Beatrice Montagu Acheson (c. 1856 – 3 March 1944) #Charles William Augustus Montagu (23 November 1860 – 10 November 1939) #Alice Maude Olivia Montagu Stanley (15 August 1862 – 23 July 1957) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], 8th Duke of Devonshire (23 July 1833 – 24 March 1908) == Notes and Questions == #As Duchess of Manchester Luise was not invited to the wedding between Bertie and Alix, Victoria's punishment for Luise's having gotten the Duke of Derby to promise her the position of Mistress of the Robes (and then exacting that promise).<ref>Leslie, Anita. ''The Marlborough House Set''. New York: Doubleday, 1973.</ref>{{rp|pp. 47–48}} #"As a young woman she was extremely beautiful; Princess Catherine Radziwill saw her at a reception given by the Empress of Germany and recalls on being introduced to her 'how she struck me as the loveliest creature I had ever set eyes upon. Indeed I have only met three women in my whole existence who could be compared to her.'"<ref name=":1" />{{rp|p. 21}} == Footnotes == {{reflist}} smh4ww89unuv5i6ayszm827eshs8qxj 2416325 2416324 2022-08-18T22:27:48Z Scogdill 1331941 /* The Historical Zenobia */ wikitext text/x-wiki == Overview == * Louise, Duchess of Devonshire was the subject of much comment and gossip in her life, arising in part from her prominence in society and in part from her conduct, which was not particularly controlled by middle-class notions of "respectability." * "Louise Alten, for her part, exemplifies another vanished feature [of Victorian England], the role of the political hostess. The part she played in Hartington's career even while she was his mistress gives a special twist to this function, and her more conventional ascendancy as Duchess of Devonshire, when she was said to be the most powerful person outside the government, shows the influence that could be wielded by women in a political system from which they were formally excluded."<ref name=":20">Vane, Henry. ''Affair of State: A Biography of the 8th Duke and Duchess of Devonshire''. Peter Owen, 2004.</ref>{{rp|12}} == Also Known As == *Louise, Duchess of Devonshire (15 January 1832 – 15 November 1911) *Louisa, Duchess of Manchester *Luise Friederike August Gräfin von Alten *Louisa Montagu *Louise Cavendish *The Double Duchess == Acquaintances, Friends and Enemies == === Friends === *[[Social Victorians/People/Albert Edward, Prince of Wales | Albert Edward, Prince of Wales]] (beginning about 1852) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], Lord Hartington (later 8th Duke of Devonshire) *Daisy, Lady Warwick *Lady Mayoress, Mrs. Benjamin Samuel Faudel-Phillips, 2nd Baronet,<ref>{{Cite journal|date=2020-08-25|title=Faudel-Phillips baronets|url=https://en.wikipedia.org/w/index.php?title=Faudel-Phillips_baronets&oldid=974879290|journal=Wikipedia|language=en}}</ref> presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4">"The Queen's Drawing Room" ''Morning Post'' 25 February 1897 Thursday: 5 [of 10], Col. 5a–7b [of 8]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970225/047/0005.</ref>{{rp|p. 5, Col. 6c}} *Mrs. J. E. Mellor, presented to Victoria by Louisa Cavendish at a Queen's Drawing-room on Wednesday, 24 February 1897 at Buckingham Palace.<ref name=":4" />{{rp|p. 5, Col. 6c}} === Enemies === * Consuelo, Duchess of Marlborough (at least, in 1901)<ref name=":1">Murphy, Sophia. ''The Duchess of Devonshire's Ball''. London: Sidgwick & Jackson, 1984.</ref>{{rp|pp. 31–32}} == Organizations == == Timeline == '''1852 July 22''', Luise Friederike Auguste Gräfin von Alten and William Drogo Montagu married.<ref name=":2">"Luise Friederike Auguste Gräfin von Alten." {{Cite web|url=http://www.thepeerage.com/p10947.htm#i109469|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1859''': "As for Louise, a glimpse of her unconventional ways is given by Lady Eleanor Stanley, describing a paper-chase at Kimbolton in 1859: 'The Duchess of Manchester, in getting too hastily over a stile, caught a hoop of her cage in it, and went head over heels, alighting on her feet with her cage and whole petticoats remaining above her head. they say there was never such a thing seen — and the other ladies hardly knew whether to be thankful or not that a part of her underclothing consisted of a pair of scarlet tartan knickerbockers (the things Charles shoots in) — which were revealed to all the world in general and the Duc de Malakoff in particular.' Lady Eleanor says that the other ladies hardly knew whether to be thankful because at this date a lady's normal wear under her petticoats was only a pair of leggings laced up to the waistband, an arrangement that could be unduly revealing in combination with a crinoline. ... But some of the more active ladies were beginning to adopt / Louise's expedient of donning a hidden pair of breeches for rambles in the country."<ref name=":20" />{{rp|25–26}} '''1863, early, or late 1862''', Louise and Spencer Compton Cavendish began a relationship.<ref name=":1" />{{rp|p. 26}} '''1873 December 10''', Mary Louise Elizabeth Montagu (daughter) and William Douglas-Hamilton married. '''1876 May 22''', Consuelo Iznaga y Clement and George Victor Drogo Montagu (son) married in Grace Church, New York City.<ref>{{Cite journal|date=2020-08-24|title=George Montagu, 8th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=George_Montagu,_8th_Duke_of_Manchester&oldid=974659520|journal=Wikipedia|language=en}}</ref><ref>{{Cite journal|date=2020-07-27|title=Consuelo Montagu, Duchess of Manchester|url=https://en.wikipedia.org/w/index.php?title=Consuelo_Montagu,_Duchess_of_Manchester&oldid=969888488|journal=Wikipedia|language=en}}</ref> '''1876 August 10''', Louisa Augusta Beatrice Montagu (daughter) and Archibald Acheson married. '''1889 January 5''', Alice Maude Olivia Montagu (daughter) and Edward Stanley married. '''1890 March 22''', William Drogo Montagu (7th Duke) died.<ref name=":3">"William Drogo Montagu, 7th Duke of Manchester." {{Cite web|url=http://www.thepeerage.com/p10128.htm#i101274|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1890 November 14''', William Angus Drogo Montagu (grandson) and Helena Zimmerman married secretly, in London.<ref>"Helena Zimmerman." {{Cite web|url=http://www.thepeerage.com/p34555.htm#i345545|title=Person Page|website=www.thepeerage.com|access-date=2020-09-25}}</ref> '''1892 August 16''', Louise Friederike Auguste Gräfin von Alten Montagu and Spencer Compton Cavendish, her second husband, married.<ref name=":2" /> '''1897 July 2, Friday''', Louise Cavendish (#18 on the list of attendees) hosted her famous [[Social Victorians/1897 Fancy Dress Ball| fancy-dress ball]] at Devonshire House in London. '''1897 July 20''', Mary Louise Elizabeth Montagu Douglas-Hamilton and Robert Carnaby Foster married. '''1900 November 14''', William Angus Drogo Montagu and Helena Zimmerman married.<ref>{{Cite journal|date=2020-07-17|title=Helena, Countess of Kintore|url=https://en.wikipedia.org/w/index.php?title=Helena,_Countess_of_Kintore&oldid=968067371|journal=Wikipedia|language=en}}</ref> '''1901 Spring''', Paris, Consuelo Spencer-Churchill, Duchess of Marlborough, describes a meeting with Louise Cavendish in the spring following Queen Victoria's death at the horse racetrack, Longchamps:<blockquote>A renowned character and virtually dictator of what was known as the fast set as opposed to the Victorian, Her Grace was a German aristocrat by birth. She had first been married to the impoverished Duke of Manchester, and when he died had improved her status by marriage to the rich Duke of Devonshire, who waged an undisputed influence in politics. Rumour had her beautiful, but when I knew her she was a raddled old woman, covering her wrinkles with paint and her pate with a brown wig. Her mouth was a red gash and from it, when she saw me, issued a stream of abuse. How could I, she complained, pointing to my white gloves, show so little respect to the memory of a great Queen? What a carefree world we must have lived in, that etiquette even in such small matters could assume so much importance?<ref>Balsan, Consuelo Vanderbilt. ''The Glitter and the Gold: The American Duchess — In Her Own Words''. New York: St. Martin's, 1953.</ref>{{rp|p. 115}}</blockquote> === Annual Events === Every year, as Duchess of Devonshire, Louise held a dance on the night after the Derby at Epsom Downs, which at this point was held on Wednesdays after Easter. == The Duchess of Devonshire's 2 July 1897 Fancy-dress Ball == Louise, Duchess of Devonshire and Spencer Compton, 8th Duke of Devonshire hosted the famous [[Social Victorians/Terminology#Fancy-dress ball|fancy-dress ball]], which took place on Friday night. According to ''House Style: Five Centuries of Fashion at Chatsworth'',<blockquote>Such an event required extensive planning and attention to detail. The housekeeper aided by two secretaries was in charge of organisation inside the house with precise instructions from the Duchess on menus and all arrangements.<ref name=":18" />{{rp|137}}</blockquote> === The Devonshire House Staff at the Ball === The Dukes of Devonshire did not keep employment records of the staff working at their various properties. The Servants and Staff Database managed by Chatsworth House has a list of everyone ever named in any documents as a servant or staff member, and by 1891 census records included employment at places like the properties owned by the Duke of Devonshire.<ref name=":19" /> === Questions === # Who was housekeeper? Who were the secretaries? # Where did this info about the "precise instructions" come from? === The Historical Zenobia === Louise, Duchess of Devonshire attended the ball dressed as Zenobia, queen of Palmyra. Zenobia (240 – c. 274) was queen of the Syrian Palmyrene Empire, ruling as regent for her son after her husband's assassination.<ref>{{Cite journal|date=2022-05-03|title=Zenobia|url=https://en.wikipedia.org/w/index.php?title=Zenobia&oldid=1086005949|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Zenobia.</ref> She was the subject of much art in the 19th century, including literature, opera, sculpture, and paintings. Middle-eastern traveller Lady Hester Stanhope (1776–1839) discussed Zenobia in her memoirs, published in 1847.<ref>{{Cite journal|date=2022-03-07|title=Lady Hester Stanhope|url=https://en.wikipedia.org/w/index.php?title=Lady_Hester_Stanhope&oldid=1075838273|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lady_Hester_Stanhope.</ref> We do not know what sources, if any, were consulted by Louise, Duchess of Devonshire in preparing for personating Zenobia for the ball. The ''Encyclopædia Britannica'' offers a sense of what was generally available to those interested in that kind of research. The article on Zenobia in the 11th edition of the ''Encyclopædia Britannica'', published in 1911, written by George Albert Cooke<ref>Rev. George Albert Cooke (G.A.C.), Oriel Professor of the Interpretation of Holy Scripture in the University of Oxford, Fellow of Oriel College, Vol. XXVIII, p. viii. </ref>, is substantial, more than an entire column.<ref>"G.A.C." [George Albert Cooke]. "Zenobia." Encyclopaedia Britannica: An Dictionary of Arts, Sciences, and General Information. Ed. . Vol. XXVIII (Vol. 28): Vetch to Zymotic Diseases. ''Google Books''. Retrieved 11 August 2022. https://books.google.com/books?id=NEpAAQAAMAAJ.</ref>{{rp|p. 972, Col. 1a–2b}} The prior edition<ref>The 10th edition of the ''Britannica'', 1902–1903, is the 9th edition plus 10 supplementary volumes.</ref>, the 9th edition, does not have an article on Zenobia, but she has an entry that is a cross reference to the Palmyra article, which is about 10 columns and includes a biographical sketch of her.<ref>By W.R.S., W. Robertson Smith, Lord Almoner's Professor of Arabic, University of Cambridge.</ref><ref name=":21">"W. R. S." [Smith, W. Robertson]. "Palmyra." ''The Encyclopaedia Britannica: A Dictionary of Arts, Sciences, and General Literature''. Ed., Thomas Spencer Baynes. 9th edition. Vol. XVIII (Vol. 18): Orne–Phthisis. 1885. Google Books. Retrieved 11 August 2022.https://books.google.com/books?id=0au0xN1PrAEC.</ref>{{rp|p. 1, Col. 1c, after p. 858}} In a very long paragraph about her husband's father and then her husband, the ''Britannica''<nowiki/>'s discussion of Zenobia begins, <blockquote>Odænathus himself seems to have been engaged in almost constant warfare in the east and north against the Persians and perhaps the Scythians, but in his absence the reins of government were firmly held by his wife Zenobia, the most famous heroine of antiquity, to whom indeed Aurelian, in a letter preserved by Trebellius Pollio, ascribes the chief merit of all her husband's success. Septimia Zenobia was by firth a Palmyrene; her native name was Bath Zabbai ... [;] and Pollio's description of her dark beauty, black flashing eyes, and pearly teeth, together with her unusual physical endurance and the frank commanding manners which secured her authority in the camp and the desert, point emphatically to an Arabic rather than a Syrian descent. ... To the union of firmness and clemency, which is the most necessary quality of an Eastern sovereign, Zenobia added the rarer gifts of economy and organization, and an unusual range of intellectual culture. She spoke Coptic as well as Syriac, knew something of Latin, and had learned Greek from the famous Longinus, who remained at her court to the last, and paid the penalty of his life for his share in her counsels. She was also a diligent student of Eastern and Western history, and the statement that she enjoined her sons to speak Latin so that they had difficulty in using Greek implies a consistent and early adoption of the policy which made the sucess of Odænathus, and, taken in connexion with Aurelians' testimony, in a letter preserved by Pollio, that she had the chief merit of her husband's exploits, seems to justify the conclusion that it was her educated political insight that created the fortunes of the short-lived dynasty. ... In the zenith of his fame Odænathus was cut off by assassination along with his eldest son Herod, and it is generally assumed that the murder took place under Gallienus. ... [p. 201, Col. 2c – p. 202, Col. 1a] The fact seems to be that, while Odaenathus was busy at the other end of his kingdom, Zenobia administered the government at Palmyra and directed the conquest of Egypt, still nominally acting under the emperor at Rome, whose authority on the Nile was disputed by one or more pretenders. ... It still seems ... strange and yet an undoubted fact that Zenobia, who not only enjoyed the real authority behind her beardless son, but placed her name before his on public inscriptions, ... struck no coins till the second year of Aurelian, when the breach with Rome took place, and she suddenly appears as an empress (Σεβαστή, Augusta) of five years' standing. Up to that date the royal pair probably did not venture to coin in open defiance to Rome, and yet were unwilling to circulate an acknowledgment of vassalship in all the bazaars of the East. ... ... Zenobia, supported by her two generals, kinsmen of her husband, was now face to face with a Roman invasion. She held Egypt, Syria, Mesopotamia, and Asia Minor as far as Ancyra; and Bithynia was ready to join her party had not the army of Aurelian appeared just in time from Byzantium. She could count too on the Armenians and the Arabs, but the loyalty of Syria was doubtful: the towns disliked a rule which was essentially "barbarian," and in Antioch at least the patroness of the Monarchian bishop Paul of Samosata could to be popular with the large Christian party by whom he was bitterly hated. There were many Romans [p. 202, Col. 1c – p. 202, Col. 2a] in Zenobia's force, and it was they who bore the brunt of the two great battles at Antioch and Emesa, which followed Aurelian's rapid advance through Asia Minor. But Zenobia made light of these defeats, — "I have suffered no great loss" was her message to Aurelian, "for almost all who have fallen are Romans." ... But the empire of Palmyra came four centuries too soon. Rome was not yet exhausted, and Zenobia had neither the religious discipline of Islam to hold the Arabs together nor the spoil of the treasuries of Persia to keep their enthusiasm always fresh. Aurelian's military skill was strained to the uttermost by the prudence and energy of Zenobia, but he succeeded in forming and maintaining the siege of Palmyra in spite of its bulwark of desert, and his gold corrupted the Arab and Armenian auxiliaries. Zenobia attempted to flee and throw herself on the Persians, but she was pursued and taken, and then the Palmyrene lost heart and capitulated. Aurelian seized the wealth of the city, but spared the inhabitants, and to Zenobia he granted her life while he put her advisors to death. She figured in his splendid triumph, and by the most probable account accepted her fall with dignity, and closed her days at Tibur, where she lived with her sons the life of a Roman matron. The fall of Zenobia may be placed in the spring of 272.<ref name=":21" />{{rp|p. 201, Col. 2a – p. 202, Col. 2b}} </blockquote> == The Duchess of Devonshire's Costume == [[File:Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra.jpg|thumb|Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra|alt=Louise, Duchess of Devonshire in costume as Zenobia, Queen of Palmyra]] At their fancy-dress ball, Louisa, Duchess of Devonshire sat at Table 1 during the first seating for supper, escorted in to the table by the Prince of Wales.<ref name=":7">"Fancy Dress Ball at Devonshire House." ''Morning Post'' Saturday 3 July 1897: 7 [of 12], Col. 4a–8 Col. 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970703/054/0007.</ref>{{rp|p. 7, Col. 4c}} Her costume was designed by M. Comelli (Attillo Giuseppe Comelli, 1858–1925, artist and costumier for opera, ballet and theatre in London as well as Europe and the U.S.<ref name=":13">{{Cite book|url=https://books.google.com/books?id=SZh2DwAAQBAJ&pg=PT207&lpg=PT207&dq=Attilio+Comelli&source=bl&ots=lFB0If7CwV&sig=ACfU3U1_Ost_lhmMvzMMs6NvuhK5SlRhJw&hl=en&sa=X&ved=2ahUKEwjKlsTw2sH3AhXYAp0JHVIxDWA4KBDoAXoECBAQAw#v=onepage&q=Attilio%20Comelli&f=false|title=Forgotten Designers Costume Designers of American Broadway Revues and Musicals From 1900-1930|last=Unruh|first=Delbert|date=2018-11-06|publisher=Page Publishing Inc|isbn=978-1-64082-758-5|language=en}} N.P.</ref>)<ref name=":5">“The Devonshire House Ball.” The ''Man of Ross'' 10 July 1897, Saturday: 2 [of 8], Col. 4b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0001463/18970710/033/0002.</ref> <ref name=":8">"The Duchess of Devonshire's Fancy Dress Ball. Special Telegram." ''Belfast News-Letter'' Saturday 03 July 1897: 5 [of 8], Col. 9 [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/BL/0000038/18970703/015/0005.</ref>{{rp|p. 5, Col. 9a}} <ref name=":9">"By One Who Was There." “The Duchess’s Costume Ball.” ''Westminster Gazette'' 03 July 1897 Saturday: 5 [of 8], Cols. 1a–3b [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002947/18970703/035/0005.</ref> and constructed by the House of Worth. Comelli seems to have designed [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess and Her Entourage|the costumes of her retinue as well]]. According to Russell Harris,<blockquote>For her costume, the Duchess commissioned Monsieur Comelli (1858-1925), a well-known designer of opera costumes for the London theatre and opera stage, and then had the design made up by Worth of Paris. ''Munsey’s Magazine'' noted “it is safe to say that the Queen of Palmyra never owned such a sumptuous costume in her lifetime.”<ref>Harris, Russell. {{Cite web|url=http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html|title=Louise, Duchess of Devonshire, née Countess von Alten of Hanover (1832-1911), as Zenobia, Queen of Palmyra|website=www.rvondeh.dircon.co.uk|access-date=2022-05-05}} ''Narrated in Calm Prose: Photographs from the V&A's Lafayette Archive of Guests in Costume at the Duchess of Devonshire's Diamond Jubilee Ball, July 1897''. http://www.rvondeh.dircon.co.uk/incalmprose/devonshiredss.html.</ref></blockquote>Lafayette's portrait of "Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester)" in costume is photogravure #5 in the album presented to the Duchess of Devonshire and now in the National Portrait Gallery.<ref>"Devonshire House Fancy Dress Ball (1897): photogravures by Walker & Boutall after various photographers." 1899. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait-list.php?set=515.</ref> The printing on the portrait says, "The Duchess of Devonshire as Zenobia Queen of Palmyra," with a Long S in ''Duchess''.<ref>"Louise Frederica Augusta Cavendish (née von Alten), Duchess of Devonshire (formerly Duchess of Manchester) as Zenobia, Queen of Palmyra." Devonshire House Fancy Dress Ball Album. National Portrait Gallery https://www.npg.org.uk/collections/search/portrait/mw158357/Louise-Frederica-Augusta-Cavendish-ne-von-Alten-Duchess-of-Devonshire-formerly-Duchess-of-Manchester-as-Zenobia-Queen-of-Palmyra.</ref> Often, the V&A Lafayette Archive contains more than one portrait of a sitter for this ball, but the uncropped portrait (above right), which shows the unfinished end of the balustrade in front of the Duchess and the edge of the painted drop behind it, seems to have been the only portrait taken by Lafayette of the Duchess in costume. The copy owned by the National Portrait Gallery in London and the copy included in the album are cropped so that those unfinished edges do not show, but they appear to be from the same photograph. That is, there is one source for our information about this portrait: this single photograph. === Newspaper Descriptions of the Duchess's Costume === Newspaper articles about the Duchess's presence at the ball focused on her hosting, her costume, [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Jewelry|her jewelry]], and [[Social Victorians/People/Louisa Montagu Cavendish#The Duchess's Entourage|her entourage]], often in the same story. These almost exactly identical descriptions suggest [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]] or a shared primary source: * The London ''Evening Standard'', ''Morning Post'' and ''Times'' published almost identical descriptions, varying only by the occasional punctuation mark or short phrase ("supplied by Worth, of Paris"). The fullest version is this one: "The Duchess of Devonshire, as Zenobia, Queen of Palmyra, wore a magnificent costume, supplied by Worth, of Paris. The skirt of gold tissue was embroidered all over in a star-like design in emeralds, sapphires, diamonds, and other jewels, outlined with gold, the corners where it opened in front being elaborately wrought in the same jewels and gold to represent peacocks' outspread tails. This opened to show an under-dress of cream crêpe de chine, delicately embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was attached to the shoulders by two slender points, and was fastened at the waist with a large diamond ornament. It was of green velvet of a lovely shade, and was superbly embroidered in Oriental designs, introducing the lotus flower in rubies, sapphires, amethysts, emeralds, and diamonds, in four borderings on contrasting grounds, separated with gold cord. The train was lined with turquoise satin. The bodice was composed of gold tissue to match the skirt, and the front was of crêpe de chine, hidden with a [[Social Victorians/Terminology#Stomacher|stomacher]] of real diamonds, rubies, and emeralds, and there was a jewelled belt."<ref name=":16">“The Ball at Devonshire House. Magnificent Spectacle. Description of the Dresses.” London ''Evening Standard'' 3 July 1897 Saturday: 3 [of 12], Cols. 1a–5b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18970703/015/0004.</ref>{{rp|p. 3, Col. 2b}} <ref name=":7" />{{rp|p. 7, Col. 7a}} <ref name=":6">"Ball at Devonshire House." The ''Times'' Saturday 3 July 1897: 12, Cols. 1A–4C ''The Times Digital Archive''. Web. 28 Nov. 2015.</ref>{{rp|p. 12, Col. 3b}} *"The Duchess of Devonshire was a dazzling vision, dressed as 'Zenobia,' in a glistening gold gauze gown, elaborately ornamented with suns and discs, wrought in purple and green gems outlined with gold, and having a large diamond as centre. The space between was fluted with fine silver spangles. This robe was open in front over an under dress of white crépe de chine, delicately worked in crystals, and at each side of the opening on the gold robe were large fan-shaped groups of peacock feathers, worked in multicoloured jewels. The [[Social Victorians/Terminology#Corsage|corsage]] was to correspond, and had a magnificent [[Social Victorians/Terminology#Girdle|girdle]] of jewels, the train of bright green velvet, hung like a fan, without folds, being fastened at each side of the shoulders by diamond brooches, and caught at the waist with a similar ornament. It was a mass of gorgeous embroidery, carried out in heliotrope velvet, lotus flowers studded with tinted gems, and other devices in terra-cotta and electric blue velvet — all enriched with gold, diamond, and jewelled embroidery — and lined with pale blue satin. ... Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail."<ref name=":15">"Duchess of Devonshire's Fancy Ball. A Brilliant Spectacle. Some of the Dresses." London ''Daily News'' Saturday 3 July 1897: 5 [of 10], Col. 6a–6, Col. 1b. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970703/024/0005 and http://www.britishnewspaperarchive.co.uk/viewer/BL/0000051/18970703/024/0006.</ref>{{rp|p. 5, Col. 6a}} * According to the article in ''The Graphic'' written by [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]], though this caption to the Lafayette photograph may not be hers, the Duchess of Devonshire wore a "Skirt of gold tissue, embroidered all over with emeralds, sapphires, diamonds, and other jewels outlined with gold. This opened to show an underdress of crème crêpe de chine, embroidered in silver, gold, and pearls, and sprinkled all over with diamonds. The train was green velvet, superbly embroidered in Oriental designs. The bodice was composed of gold tissue, and the front was of crêpe de chine hidden with a stomacher of diamonds, rubies, and emeralds."<ref name=":10">Greville, Violet, Lady. "Devonshire House Ball." The ''Graphic'' Saturday 10 July 1897: 15 [of 24]: Col. 1a–16, Col. 1c. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000057/18970710/019/0015.</ref>{{rp|p. 15, Col. 3b}} Because it is in black and white, the photograph of the Duchess taken by Lafayette in her costume (above right) can tell us almost nothing about the colors of the costume. Fortunately, this costume has survived and is held in Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth). ''House Style: Five Centuries of Fashion at Chatsworth'', the volume about an exhibition mounted 25 March to 22 October 2017, has beautiful photographs from several perspective of the costume, including the train, and accessories. Published photographs of it '''show X'''.<ref name=":18">Burlington, Julia, and Hamish Bowles. ''House Style: Five Centuries of Fashion at Chatsworth''. Skira Rizzoli, 2017. Based on an exhibition 25 March to 22 October 2017.</ref> It has been worn by '''Deborah, 11th Duchess of Devonshire when the Dowager Duchess at some point for some reason'''. Some accounts of her costume suggest that the reporter thought that all the jewels sewn onto the dress were actual precious or semiprecious stones, but the London ''Daily News'' says some of them were crystals.<ref name=":15" />(p. 5, Col. 6a) According to ''House Style: Five Centuries of Fashion at Chatsworth'',<blockquote>The skirt of gold gauze, appliquéd with tinsel medallions and peacock plumes worked in bright foils, wire coils and spangled with sequins, was worn over an ivory satin underskirt wrought over with silver thread and diamonds. Attached to the shoulders was a long graduated train in the most vivid emerald-green velvet, appliquéd with velvet and gold work in an eastern design and studied with jewels.<ref name=":18" />(128)</blockquote> === Observations on the Duchess's Costume === In the photograph of her in costume, the Duchess is standing in a three-quarter front pose. That static position in a two-dimensional image makes it difficult to understand exactly how the costume and headdress worked. The terms we use are these: stomacher, train, bodice (what's on the torso, no matter how many layers), overskirt, underskirt, belt (really an accessory, like the jewelry), sleeves. The 1897 newspaper reports use different terminology for the parts of the dress than we would now. Some of the terms that appear in the newspaper reports include overdress, under-dress, bodice, stomacher robe, train, corsage, girdle. The Duchess's Body how the fabric draped, how tight the bodice was, what the stomacher looked like, what pearls she is wearing. Is her hair down or up? ==== Fit ==== *The recent exhibitions of the dress show a different fit than what is shown in the Lafayette photograph. Louise, the 8th Duchess's bodice is draped loosely over the stomacher, but photographs of Deborah, 11th Duchess or of the mannequin in the 2017 exhibition show a bodice fitted to the stomacher. * Also, the costume itself may have been altered or adjusted to make the waist more defined and the line more Elizabethan (the wide, square neckline; the A-line, the split skirt with the petticoat in the middle). * The dress seems to have been designed to flatter the 8th Duchess's statuesque figure, especially her waist: the belt falls down to the hips, the A-line; the way what we can see of the under-dress is narrow at the top and widens toward the bottom. * The drape of the skirt is straighter on the 8th Duchess as well as the Dowager Duchess than on the mannequins. ==== Inconsistencies in the Descriptions and Images ==== * The newspaper accounts of the Duchess's costume do not agree on several particulars, including the color of the train and its lining. ==== Changing Victorian Design Elements ==== The costume as Louise, the 8th Duchess wore it is vaguely Middle Eastern and exotic, consistent with her '''leading''' the "Oriental" procession. The sleeve treatment, the drape of the bodice and the skirt are consistent with the line of something from the Middle East. This would have been what Comelli designed. Early clothing history books existed by the middle of the century, and it would be logical that Comelli would be familiar with them. The versions of the costume shown in recent exhibitions make it seem more Victorian than what we see in Lafayette's portrait. The tightness of the fit of the bodice indicates a Victorian corset under the costume. The line of the bodice and skirt, the bare arms in the sleeve treatment, the train, the belt, The versions of the dress the the tightness of the dress against the stomacher all look Victorian. A-line drape of the skirt was done for the exhibition to show the artistry and complexity of the trim. * The sleeve treatment in the 1897 photograph is caught up at the shoulder. On the Dowager Duchess and the mannequins, the sleeves are released. ==== Trim ==== * The sequins attached to the costume would have been metal. They appear to be silver that has tarnished, which is why they are dark now. Currently, the tarnish on the sequins and embroidery thread cannot be removed without damage to the fabric. * The gold thread in the embroidery has not tarnished, but the silver thread, which would have sparkled in 1897, has. ==== Accessories ==== * Belt * Headdress * Do the horns on [[Social Victorians/People/Louisa Montagu Cavendish#Zenobia's Crown|her crown]] go toward the back from the apex that rises from the jewel-encrusted band or do they stick straight out to the sides? Does the headdress include the traditional three white plumes? * Jewelry * Fan * Can't tell how many strands of pearls are around Louise's neck in the Lafayette photo, but the photographs of the recreation of the crown and of the costume from the 2017 exhibit show [[Social Victorians/People/Louisa Montagu Cavendish#Goldsmith, Pearl & Diamond Merchant, & Silversmith|five strands]]. ==== The Train ==== * The reporter's description of the Duchess's train in the London ''Daily News'' is confusing: it says the train is "bright green velvet" and "It was a mass of gorgeous embroidery, carried out in heliotrope velvet." Embroidery is not "carried out" in velvet. Heliotrope is not green, it's pinkish purple. * Furthermore, heliotrope appears to have been a new word for a color, introduced perhaps in 1882 in A. Maerz and M. Rea Paul ''A Dictionary of Colour'', New York: '''1930 McGraw-Hill Page 196'''; Color Sample of Heliotrope: Page 131 Plate 54 Colour Sample C10 ("Heliotrope," Wikipedia [[wikipedia:Heliotrope_(color)#cite_note-2|https://en.wikipedia.org/wiki/Heliotrope_(color)#]]<nowiki/>[[wikipedia:Heliotrope_(color)#cite_note-2|c]][[wikipedia:Heliotrope_(color)#cite_note-2|ite_note-2]]). The ''OED'' lists 3 newspaper sources for the 1880s, The ''World'' (1882), ''Truth'' (1886), and the ''Daily News'' (1887) ("heliotrope" 1.d. Retrieved 6 August 2022). The ''Daily News'' describes it well: "that peculiar mauve known as heliotrope." The word itself has existed in English for many centuries, <nowiki/>both for the flower (which turns to follow the sun) as well as for instruments that reflect or measure sunlight and also for a variety of quartz (also called bloodstone). * Color of green velvet on the train: the colors differ, depending on the photograph. One photograph shows a very vivid green, and the rest show a lighter green. We're not sure what's up with that. == The Duchess's Jewelry == The Duchess was known fo<nowiki/>r her jewelry, and as she aged and after her marriage to the 8th Duke in 1892, the jewelry she wore only increased in scale and notability. It is not surprising that her jewelry would make up copy for the newspapers. Newspaper reports before the ball gossiped about the jewelry associated with the costumes being developed. For example, according to the Edinburgh ''Evening News'' on 21 June 1897, less than two weeks before the party, "The ball being a fancy dress one, men as well as women will be able in certain characters to wear jewels. The Duchess of Devonshire, who is to appear as Zenobia, is getting her jewels reset after the antique style."<ref>“The Duchess of Devonshire’s Ball.” Edinburgh ''Evening News'' 21 June 1897, Monday: 4 [of 6], Col. 5c [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000452/18970621/079/0004.</ref> While almost all descriptions of her mention her jewels because they were so much a topic of conversation and because they were sewn onto the costume itself, these emphasize her jewelry and most are very similar to each other: * "The Duchess was attired with great Oriental magnificence as Zenobia. Her dress was a tissue of silver, embroidered with gold and jewels, an overmantle of cloth of gold embroidered in the same manner hung from the shoulders, and she wore a bandeau of gold studded with gems, and surrounded by hanging chains of pearls over her elaborate headdress; strings and ropes of jewels and pearls were worn round the neck, and hung down almost to the knees."<ref>“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032. </ref>{{rp|p. 32, Cols. 1c–2a}} <ref name=":17">"Duchess of Devonshire's Fancy-Dress Ball. Brilliant Spectacle." The [Guernsey] ''Star'' 6 July 1897, Tuesday: 1 [of 4], Col. 1a–2b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000184/18970706/003/0001.</ref>{{rp|p. 1, Col. 2a}} * "A wonderfully beautiful dress was that which was worn by the Duchess of Devonshire as Zenobia, Queen of Palmyra. It was of golden tissue, sewn with silver paillettes, and jewelled with diamonds and other precious stones. In front there were silk embroideries, in many vivid shades of colour, and here the golden draperies opened to show a petticoat of white crêpe de chine, embroidered with pearls and gold. The short train was of brilliant green velvet, exquisitely embroidered. One of the Duchess of Devonshire’s beautiful diamond and emerald tiaras had been taken to pieces to form a stomacher, the effect of which was dazzling in its brilliancy. Long chains of pearls and other wonderful jewels were worn with this beautiful dress."<ref>“The Devonshire House Ball. A Brilliant Gathering.” The ''Pall Mall Gazette'' 3 July 1897, Saturday: 7 [of 10], Col. 2a–3a [of 3]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000098/18970703/019/0007.</ref>{{rp|p. 7, Col. 2b}} * In the article about the ball in the ''Graphic'', [[Social Victorians/People/Lady Violet Greville|Lady Violet Greville]] says, "The Ducal hostess herself elected to appear as Zenobia, Queen of Palmyra, with lavish magnificence, and wearing a corruscation of jewels which must have eclipsed the state of even the all-subduing majesty the Duchess impersonated."<ref name=":10" />{{rp|p. 16, Col. 1a}} *The Duchess was dressed "as Zenobia, Queen of Palmyra, her dress a marvel of soft tissues and exquisite ornament, and her tiara a still greater marvel of the jeweller's art."<ref name=":6" />{{rp|p. 12, Col. 2a}} <ref>"The Duchess of Devonshire’s Historic Ball. Some of the Fancy Costumes." Supplement. The ''Leicester Chronicle and Leicestershire Mercury'' 10 July 1897, Saturday: 11 [of 12], Cols. 4a–b [of 7]. ''British Newspaper Archive'' http://www.britishnewspaperarchive.co.uk/viewer/bl/0000173/18970710/141/0011.</ref>{{rp|p. 11, 4a}} As with the colors, Lafayette's photograph of the Duchess in costume does not show the jewels very clearly. We cannot see the stomacher or the "long chains of pearls and other wonderful jewels" or the pearls that "hung down almost to the knees" (although, of course, any newspaper account can be wrong because reporters were not present at the ball). === Zenobia's Crown<ref name=":10" /> === The crown that the Duchess wore as Zenobia is difficult to see clearly in the Lafayette photograph (above right), and no other images of the crown exist. It was lavish, "encrusted" with jewels and featuring pearls: * London ''Daily News'' says, "The crown worn with this was high, and of filigree gold, surmounted with two horns, each tipped with a large diamond. It was encrusted with large diamonds, rubies, and emeralds, and long chains of pearls fell under the chin and about the head — one magnificent pear-shaped pearl resting on the forehead."<ref name=":15" />{{rp|p. 5, Col. 6a}} * These descriptions are all identical except for the addition or subtraction of an occasional comma: The London ''Evening Standard'' says, "A gold crown encrusted with emeralds, diamonds, and rubies, with a diamond drop at each curved end, and two upstanding white ostrich feathers in the centre, and round the front were festoons of pearls, with a large pear-shaped pearl in the centre falling on the forehead."<ref name=":16" />{{rp|p. 3, Col. 2b}} <ref name=":7" />{{rp|p. 7, Col. 7a}} <ref name=":6" />{{rp|p. 12, Col. 3b}} <ref name=":10" />{{rp|p. 15, Col. 3b}} * The Guernsey ''Star'' says, "Her Grace wore a bandeau of gold round her head, studded with diamonds, turquoise, and emeralds, and surrounded by hanging chains of superb pearls."<ref name=":17" />{{rp|p. 1, Col. 2a}} This crown was recreated from the single Lafayette photograph of the Duchess in costume for the ''House Style: Five Centuries of Fashion at Chatsworth'', the exhibition mounted in 2017 and exhibited on the mannequin wearing the Duchess's costume. A photograph of this recreation separate from the costume, but with the pearls and the white plumes, can be seen on [https://www.sothebys.com/en/articles/how-c-w-sellors-transformed-five-centuries-of-fashion-into-modern-jewellery this Sotheby's page] about C. W Sellor's, the jewelry firm that did the recreation as well as a number of other recreations for the exhibit.<ref>Sotheby's. "How C. W. Sellors Transformed Five Centuries of Fashion Into Modern Jewellery." ''Sotheby's'' 9 May 2017. Retrieved 11 August 2022. https://www.sothebys.com/en/articles/how-c-w-sellors-transformed-five-centuries-of-fashion-into-modern-jewellery.</ref> Although Lafayette's photograph is not conclusive, in Sellor's recreation the horns on the crown point toward back of the head rather than out to the sides. === Goldsmith, Pearl & Diamond Merchant, & Silversmith === The Duchess's pearls, which were an important feature of her costume, occasioned a great deal of direct commentary in the newspaper accounts. The Duchess's jewelry occasioned a great deal of reportage in the articles about the ball. '''It was reported that she had her jewels restrung to be used in the costume. stomacher and review of jewelry in more general articles''' An invoice and receipt in the Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) is from a concern whose preprinted stationery has a crown in the upper-left corner, suggesting that they had a royal warrant, and no name other than Goldsmith, Pearl & Diamond Merchant, & Silversmith. This document offers a unique view into the evolution of one necklace, at least, over the years. It lists what are apparently three restringing of some pearls of Louise, Duchess of Devonshire. The three restringings appear to be dated: The first necklace is a "Pearl Necklet in original 4 rows." The invoice is dated 20 October 1892 (but the stationery was printed to assume the invoice would be used in the 1880s, so the 9 is written over the second 8, and the 2 has been added).<ref name=":14">Invoice and receipt. Goldsmith, Pearl & Diamond Merchant & Silversmith. Date of itemized invoices for restringing pearls: 20 October 1892, 1 March 1897, 1909. The Devonshire Collections, Chatsworth, Reference number FIS/4/1/2.</ref>(p. 1) The necklet she brought in to be restrung contained a "Total [of] Total 224 large pearls": # 1st [row] 51 large pearls # 2nd 53 large pearls # 3rd 57 large pearls # 4th 63 large pearls The second necklace is a "Necklet as re-strung on October 15th 1892, with addition of small pearls supplied, now consists of 5 rows, containing" a total of "224 large pearls & 227 small"<ref name=":14" />(p. 1): # 1st 41 large pearls & 40 small # 2nd 42 large pearls & 42 small # 3rd 44 large pearls & 45 small # 4th 47 large pearls & 48 small # 5th 50 large pearls & 51 small The third necklace is a "Pearl Necklet as again re-strung with additional pearls supplied 1 March 1897, now consisting of 5 Rows containing" a total of "262 Large Pearls & 267 Small"<ref name=":14" />(p. 2): # 1st Row 45 Large Pearls & 44 Small # 2nd Row 48 large Pearls & 49 Small # 3rd Row 51 Large Pearls & 52 Small # 4th Row 56 Large Pearls & 65 small Possibly these pearls may have been restrung in 1909 into a coronet, but the handwriting is not clear enough to read.<ref name=":14" />(p. 2) If the Duchess wore one of these stringings of her pearls for the ball, then it must have been the second necklet, strung in 1892, a 5-strand necklace. None of the newspaper accounts refer to a 5-strand pearl necklace, although her pearls are often mentioned. == The Duchess's Entourage == Besides the Duke of Devonshire, the retinue of Louise, Duchess of Devonshire as Zenobia, Queen of Palmyra, included her grandson, [[Social Victorians/People/William Angus Drogo Montagu|William Angus Drago Montagu, 9th Duke of Manchester]], dressed as a Georgian courtier. Four newspapers say that the Duchess's entourage included three groups, all in costume: children, trumpeters and fan-bearers. The ''Westminster Gazette'' and the ''Man of Ross'' list the groups but do not mention the number of members of the groups. According to two sources, probably in error, the London ''Daily News''<ref name=":15" />{{rp|p. 5, Col. 6a}} and the Belfast ''News-Letter,''<ref name=":8" />{{rp|p. 5, Col. 9a}} these groups each had four members. The London ''Daily News'' is likely the source (because it was published at 5:00 a.m.<ref>Mitchell, Charles. ''Newspaper Press Directory'', 1895. [Hathi Trust via U Wisconsin Madison.] http://hdl.handle.net/2027/mdp.39015085486150.</ref>{{rp|55}}) for the Belfast ''News-Letter'', which took part in [[Social Victorians/1897 Fancy Dress Ball/anthology#Scissors-and-Paste Journalism|scissors-and-paste journalism]], like so many other newspapers of the 19th century. ['''check this: which one was published earlier in the day, and on which day?'''] These four sources describe the Duchess's retinue and how the people in it were dressed: *"The Duchess of Devonshire was dazzingly [sic] magnificent as 'Zenobia,' arrayed in the glistening fabrics and massive jewels in which artists have delighted to depict the Warrior Queen, the costume in this case being specially designed by the clever French artist, M. Comelli, who was also responsible for the splendid attire of the Queen's suite. This was composed of four children in white Assyrian robes, draped with pink shawls; four trumpeters in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel; and four fan-bearers attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold — a gloriously magnificent pageant."<ref name=":8" />{{rp|p. 5, Col. 9a}} *"Attending the hostess were four children, four fan-bearers, and four trumpeters, all magnificently arrayed in artistically embroidered Assyrian robes, helmets, and other accessories, correct in every detail."<ref name=":15" />{{rp|p. 5, Col. 6a}} *"The duchess was dressed as Zenobia, in gold cloth, gorgeously embroidered in gold, brilliants, and coloured stones, and opening over an under dress of white crêpe de Chine, worked finely in brilliants. The train of light green velvet was lined with blue, and sumptuously embroidered in jewels and gold, the colouring being particularly artistic. With this dress were worn splendid jewels, and a large horn crown, encrusted with diamonds, emeralds, and rubies. The duchess was attended by a suite of children, trumpeters, and fan-bearers, all picturesquely attired in Assyian [sic] costumes — the whole group being specially designed by M. Comelli."<ref name=":5" /> *"The host was dressed as Charles V. of Germany, in black velvet, satin, and fur; and the Duchess made the most gorgeous of Zenobias, in a gown of gold gauze, and a green velvet train — both a mass of exquisite oriental embroidery. The crown and hanging ropes of pearls, the jewelled girdle, and the train of children, fan-bearers, and trumpeters — all in Babylonish garb — as designed by M. Comelli, made a gloriously imposing and picturesque group."<ref name=":9" /> === Details of the Costumes in the Entourage === The Archives of the Duke of Devonshire (Devonshire Collections, Chatsworth) has "receipts" or invoices that functioned as receipts for several commercial concerns that were involved in making costumes or accessories for costumes for this ball. They are the following: * [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|M. (Attillo Giuseppe) Comelli]] * [[Social Victorians/People/Louisa Montagu Cavendish#B. Burnet & Co.|B. Burnet & Co.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Arthur Millward, Theatrical Jeweller|Arthur Millward, Theatrical Jeweller]] * [[Social Victorians/People/Louisa Montagu Cavendish#Liberty & Co., Ltd.|Liberty & Co., Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Lafayette, Ltd.|Lafayette, Ltd.]] * [[Social Victorians/People/Louisa Montagu Cavendish#Goldsmith, Pearl & Diamond Merchant, & Silversmith|Goldsmith, Pearl & Diamond Merchant, & Silversmith]] This list of commercial concerns almost certainly cannot be the complete list of all concerns that contributed to the costumes. These are the only receipts or invoices about expenses for the ball, however, that the Chatsworth Archive contains; similar documents were likely not even kept or were destroyed with other papers not retained at some point in time. The business concerns listed above were specialized and likely used for different elements of the costumes. As a theatrical designer, Comelli would have depended on the suppliers he knew and arranged with them for the construction of these costumes. The Chatsworth Archive calls these documents ''receipts'', which indeed they are because they were returned to Devonshire House as receipts for payment. From our perspective, though, they are invoices that contain specifics about what was used to make the costumes. The analysis of these invoices has led to an understanding of what the people who attended the Duchess in her entourage wore and a clearer sense, perhaps, of how many people walked in that entourage. This analysis is based on the items listed on the invoices and their pricing, most of which is included in the section for each invoice, below. While the Belfast ''News-Letter'' says that each group contained four members,<ref name=":8" />{{rp|p. 5, Col. 9a}} the invoices and receipts suggest that the newspaper, the single source for this information, was wrong about the number of people in each group. It is theoretically possible, of course, that suppliers other than the ones in the Chatsworth Archive made some of these costumes and that other invoices and receipts must have existed at that time. But the [[Social Victorians/People/Louisa Montagu Cavendish#M. (Attillo Giuseppe) Comelli|Comelli memo, below]], seems definitive: he designed and seems to have overseen the construction of the costumes, which numbered six rather than twelve. '''[Collier discussion?]''' Besides providing welcome detail about the costumes of the people in the Duchess's entourage, which is available nowhere else, these invoices also raise at least as many questions as they answer. ==== M. (Attillo Giuseppe) Comelli ==== Attillo Giuseppe Comelli was a designer for opera, ballet and theatre in Europe, the UK and North America.<ref name=":13" /> The receipt in the Chatsworth Archive was sent from Covent Garden. The invoice lists £4 for "Making six costumes," 3''s'' for "Extras" and 12s for "Cab fares for men paid by the request of M. Comelli."<ref>M. Comelli, Covent Garden, to Duchess of Devonshire. Date of invoice 13 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Three other names are on this invoice and receipt: * L. L[?] Collier [?], written under and perhaps as part of the direction to the Duchess of Devonshire * Mr. Strong ("Forwarded to Mr Strong. [sic] by the instructions of M. Comelli," written in the same hand as wrote the majority of the memo) * Floyd [?] Collier [??] ("Received with Thanks," presumably thanking for the payment, in a different hand) ==== B. Burnet & Co. ==== An invoice and receipt from B. Burnet & Co., held in the Archives of the Duke of Devonshire, has specific information about some of the fabrics, trims and accessories purchased for the costumes of the Duchess's retinue.<ref name=":11">B. Burnet & Co. to Louise, Duchess of Devonshire. Date of invoice 2 and 6 July 1897; postmarked 25 October 1897, for receipt of payment(?). The Devonshire Collections, Chatsworth, Reference number L/109/4(3).</ref> Besides itemizing some costume or accessory elements that seem clearly to be for the groups, the invoice also lists items not easy to associate with particular costumes, like the following: * 12 yards of White silk fringe 8in deep<ref name=":11" />{{rp|back left}} * 12 1/2 yards of "wht cloth"<ref name=":11" />{{rp|back left}} * 9 yards of "[[Social Victorians/Terminology#Selesia|Selesia]]"<ref name=":11" />{{rp|back left}} * 2 yards of Canvas<ref name=":11" />{{rp|back right}} * 4 Tan Wool Tights<ref name=":11" />{{rp|back right}} * 2 Tan Boys Tights<ref name=":11" />{{rp|back right}} At this time, we are not sure which costumes these elements were used for. Possibly the white silk fringe and the white cloth would have been used to construct the robes for the children and trumpeters in the entourage. The number of tights suggests that the six costumes on this invoice all included tights. With other elements of the trumpeters' costumes, the Burnet invoice also lists "6 prs Assyrian Buskins." Probably, to a late Victorian, buskins would have been "defensive leggings"<ref>Demmin, Auguste. An illustrated History of Arms and Armour: From the Earliest Period to the Present Time. George Bell, 1894. Google Books https://books.google.com/books?id=ArRCAAAAYAAJ: 106.</ref> laced together and covering the lower leg and often feet of a soldier. To a clothing and military historian, buskins (or greaves) were worn by people in a number of cultures over millennia and varied widely in style and construction. Buskins appear in Assyrian art held at the time by the British Museum. Listing six pairs of buskins suggests that every costume in the Duchess's entrourage included buskins, possibly worn over the tan tights. The Burnet invoice lists "4 Broad Belts," which may have held "4 Skins Fleshers."<ref name=":11" />{{rp|p. 1, front of invoice}} (A skin flesher is a kind of knife used to separate the skin from the meat in animals.) If each group included only two members, then perhaps the belts and fleshers were worn not only by the trumpeters but also by the fan-bearers. The Millward invoice (specifics in the section on the Millward invoice below) lists "8 Doz 'Plaques' for Belts'" with a drawing of an upright rectangle with a circle in the middle, which might have been a jewel. Double lines around the rectangle suggest that the plaques were not flat or the metal was not thin. The drawing does not give any ideas about how the plaques were attached to the belts, if they were. It is impossible to tell if the plaques were attached to the "4 Broad Belts" (likely for the trumpeters and fan-bearers), but unless they were quite tiny, "8 Doz 'Plaques'" would be far too many for the belts of only the two children. A different hand, probably "[L.??] L. Collier," wrote the following sentence at the end of the invoice and receipt, above the postmark:<blockquote>All the above named articles were used for the six [?] dresses made for the Devonshire Ball.<ref name=":11" />{{rp|back right}}</blockquote> This same hand, signing what is possibly "Floyd Collier," also signed the postmark of the Comelli invoice and receipt. On the Burnet document, this writer, possibly an assistant or employee of the Duchess of Devonshire, says that "six dresses" were made (if in fact, that word is ''six''). (No "Collier" is listed among the staff or servants of the Duke of Devonshire at the end of the 19th century.<ref name=":19">"Servants A-H." ''Historic Servants and Staff. Servants and Staff Database''. Retrieved 18 July 2022 https://www.chatsworth.org/media/11528/servants-a-h.pdf.</ref> The invoice appears to itemize materials used for six costumes: two children, two trumpeters and two fan-bearers. ==== Arthur Millward, Theatrical Jeweller ==== An invoice and request for payment from Arthur Millward, Theatrical Jeweller, held in the Archives of Chatsworth House, has more specifics about some of the fabrics, trims and accessories for the costumes of the Duchess's retinue.<ref name=":12">Memorandum. Arthur Millward, Theatrical Jeweller, to Louise, Duchess of Devonshire. Date of itemized invoice 1 July 1897; date of request for payment(?) 27 August 1897. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> This invoice lists the following, which could have been used in any of the costumes for the entourage: * 8 Doz 'Plaques' for Belts [discussed with the belts in the section on the Burnet invoice, above] * 4 Large Armlets * 4 Bracelets * 8 Armlets<ref name=":12" />{{rp|p. 2, back}} Because Millward was a Theatrical Jeweller, it seems likely that most (if not all) of the items listed on the invoice were made of metal and the jewels mentioned were artificial, made of glass or paste. Other items on the invoice seem to belong to the costumes of the trumpeters, which the Belfast ''News-Letter'' says included helmets: * 2 Helmets * 2 Centre pieces The Millward invoice shows tiny line drawings next to the words ''2 Helmets'' and ''2 Centre pieces''. These drawings suggest that the Centre pieces were attached to the helmets rather than being anything that would have been put on a table as decoration. Other items seem to belong to the costumes of the fan-bearers: * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers The "Pearl & Gold Headdresses" were likely the "jewelled diadems" mentioned in the Belfast ''News-Letter''. The "Fan properties with Feathers" are likely to have been the "long-handled fans of white feathers, mounted in blue and gold" mentioned in the newspaper report.<ref name=":8" />{{rp|p. 5, Col. 9a}} At the end of the Millward invoice, a "reduction as agreed with M [Mr?] Commelli [sic]" of £1 10''s'' is subtracted from a total of £22 3''s''. No reason for this reduction is given.<ref name=":12" />{{rp|p. 2, back}} ==== Liberty & Co., Ltd. ==== One invoice and receipt from the Chatsworth Archive, dated 12 July, to the Duchess of Devonshire, lists "13 yds S&W Satin[?]," 7 yards of blue and 6 of purple.<ref>Invoice and receipt. Liberty & Co. Ltd. To Her Grace, The Duchess of Devonshire. Date of itemized invoice 12 July [1897]. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Because the fabric is satin and from Liberty, it is possible that it was not used in the costumes of the people in the entourage but perhaps for the costume of the Duchess herself??? ==== Lafayette, Ltd. ==== The invoice and receipt from Lafayette, Ltd., the photographer that set up a temporary studio in the garden to take portraits of people at the ball in their costumes, may not be related to the ball at all.<ref>Invoice and receipt. Lafayette, Ltd. To His Grace The Duke of Devonshire. Date of itemized invoice 12 April 1897; addressed to the Duke, 18 February 1898; receipt and thanks for payment, 7 April 1898. The Devonshire Collections, Chatsworth, Reference number L/109/4(?).</ref> Three dates are written on the preprinted stationery: # 18/2/98 (18 February 1898), under the direction to "His Grace The Duke of Devonshire" # 4/12/97 (4 December 1897), next to the single item on the invoice for which a charge is listed: "6 [??] £1.10.0" # 7/4/98 (7 April 1898), in a different hand, with "Recd by cheque 7/4/98 Lafayette Ltd pp[?] [??] thanks" At the bottom of the page, in the hand that wrote all of the invoice except the receipt and thanks, is "With Lafayette Ltds Compliments." ==== Details for the Children in the Entourage ==== According to the ''Belfast News-Letter'', four children were "in white Assyrian robes, draped with pink shawls."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for "White Cloth Dresses":<ref name=":11" />{{rp|p. 2, back left of invoice}} * "2 Terra Gown draperies with Stars 200 in all" * "2 Cloth fronts embroidered with Square Medallions down centre" * "2 do do [ditto ditto, that is, cloth fronts] embroidered double border down front each side and collar" * "4 Sleeves embroidered Small Medallions" The Burnet & Co. invoice lists 6 yards of "Terra" Silk Fringe, which perhaps was used to trim the "terra draperies," or shawls, made from 3 1/4 yards of "Light Terra Satinette" worn by the children? ==== Details for the Trumpeters in the Entourage ==== According to the ''Belfast News-Letter'', four trumpeters were "in white cloth robes, embroidered in subdued tones of silks, with a purple shawl draped over, beautifully ornamented with embroidery, and wearing fringed steel helmets and leather cuirasses embossed in steel."<ref name=":8" />{{rp|p. 5, Col. 9a}} The trumpeters appear to have been dressed as soldiers or military men. According to the B. Burnet invoice, the following was purchased for the trumpeters' costumes:<ref name=":11" />{{rp|p. 1, front of invoice}} * 7 '''units (yards?)''' of purple silk [probably used for shawls?] * "2 skirt fronts with border alround [sic]" * "2 sets of Leather Cuarasses [sic] Embroidered front & back" * "4 Sleeves embroidered loop stitch" The Millward invoice lists * 2 Helmets * 2 Centre Pieces [probably for helmets rather than table decorations] ==== Details for the Fan-bearers in the Entourage ==== According to the ''Belfast News-Letter'', four fan-bearers were "attired in pale blue robes, with crimson shawls, enriched with gold and jewelled embroidery, adorned with jewelled diadems, and holding long-handled fans of white feathers, mounted in blue and gold."<ref name=":8" />{{rp|p. 5, Col. 9a}} According to the B. Burnet invoice, the following was purchased for the fan bearers's costumes:<ref name=":11" />{{rp|pp. 1–2, front and left-back of invoice}} * "Embroidering 2 Crimson draperies with Stars 334 in all" * "2 Top [?] fronts embroidered & round necks" * "4 Sleeves embroidered long stitch"The Millward invoice lists * 2 Pearl & Gold Headdresses * 2 Fan properties with Feathers<ref name=":12" />{{rp|p. 2, back}} The Burnet & Co. invoice lists 12 yards of "Red Silk Fringe," which perhaps was used to trim the "crimson shawls" or "Crimson draperies," which may have been made from the 5 yards of "Red Satinette." Again, this list suggests two rather than four costumes. == Demographics == *Nationality: born in Hanover, in what is now Germany<ref name=":0">{{Cite journal|date=2020-07-27|title=Louisa Cavendish, Duchess of Devonshire|url=https://en.wikipedia.org/w/index.php?title=Louisa_Cavendish,_Duchess_of_Devonshire&oldid=969824214|journal=Wikipedia|language=en}}</ref> === Residences === ==== As Duchess of Manchester ==== *Kimbolton Castle, Huntingdonshire *Manchester House, London ==== As Duchess of Devonshire ==== *Devonshire House, London (mid-April until mid-July, for the Season) *Compton Place, Eastbourne (mid-July until 12 August<ref name=":1" />{{rp|p. 32}}) *Bolton Abbey, Yorkshire (12 August until the middle of September<ref name=":1" />{{rp|p. 32}}) *Chatsworth, Derbyshire (middle of September until early Spring<ref name=":1" />{{rp|p. 32}}) *Lismore Castle, County Waterford (early Spring until the middle of April<ref name=":1" />{{rp|p. 32}}) == Family == *Louisa (or Luise) Friederike Auguste Gräfin von Alten Montagu Cavendish (15 January 1832 – 15 November 1911)<ref name=":2" /><ref name=":0" /> *William Drogo Montagu, 7th Duke of Manchester (15 October 1823 – 22 March 1890)<ref name=":3" /><ref>{{Cite journal|date=2020-09-07|title=William Montagu, 7th Duke of Manchester|url=https://en.wikipedia.org/w/index.php?title=William_Montagu,_7th_Duke_of_Manchester&oldid=977197445|journal=Wikipedia|language=en}}</ref> #George Victor Drogo Montagu, 8th Duke of Manchester (17 June 1853 – 18 August 1892) #Mary Louise [Louisa?] Elizabeth Montagu Douglas-Hamilton Forster (27 December 1854 – 10 February 1934) #Louisa Augusta Beatrice Montagu Acheson (c. 1856 – 3 March 1944) #Charles William Augustus Montagu (23 November 1860 – 10 November 1939) #Alice Maude Olivia Montagu Stanley (15 August 1862 – 23 July 1957) *[[Social Victorians/People/Spencer Compton Cavendish|Spencer Compton Cavendish]], 8th Duke of Devonshire (23 July 1833 – 24 March 1908) == Notes and Questions == #As Duchess of Manchester Luise was not invited to the wedding between Bertie and Alix, Victoria's punishment for Luise's having gotten the Duke of Derby to promise her the position of Mistress of the Robes (and then exacting that promise).<ref>Leslie, Anita. ''The Marlborough House Set''. New York: Doubleday, 1973.</ref>{{rp|pp. 47–48}} #"As a young woman she was extremely beautiful; Princess Catherine Radziwill saw her at a reception given by the Empress of Germany and recalls on being introduced to her 'how she struck me as the loveliest creature I had ever set eyes upon. Indeed I have only met three women in my whole existence who could be compared to her.'"<ref name=":1" />{{rp|p. 21}} == Footnotes == {{reflist}} 3zb9q39my46wslki4thahknqgcajd9a 0 267586 2416147 2415776 2022-08-18T12:59:12Z D.H 52339 /* {{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas */ wikitext text/x-wiki {{../Lorentz transformation (header)}} ==Most general Lorentz transformations== ===General quadratic form=== The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as<ref>Bôcher (1907), chapter X</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}\begin{align}q=\sum_{0}^{n}A_{ij}x_{i}x_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align} & =q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\ b=\sum_{0}^{n}A_{ij}x_{i}y_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y} & =b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}' \end{align} \quad\left(A_{ij}=A_{ji}\right)\\ \hline \left.\begin{align}x_{i}^{\prime} & =\sum_{j=0}^{n}g_{ij}x_{j}=\mathbf{g}\cdot\mathbf{x}\\ x_{i} & =\sum_{j=0}^{n}g_{ij}^{(-1)}x_{j}^{\prime}=\mathbf{g}^{-1}\cdot\mathbf{x}' \end{align} \right|\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}' \end{matrix}</math>|{{equationRef|Q1}}}} The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc. ===Most general Lorentz transformation=== The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:<ref name=ratcliffe>Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1</ref><ref>Naimark (1964), 2 in four dimensions</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} & =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\ -x_{0}y_{0}+\cdots+x_{n}y_{n} & =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime} \end{align} \\ \hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\ \downarrow\\ \begin{align}x_{0}^{\prime} & =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\ x_{1}^{\prime} & =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\ & \dots\\ x_{n}^{\prime} & =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn} \end{align} \\ \\ \mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\ \downarrow\\ \begin{align}x_{0} & =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\ x_{1} & =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\ & \dots\\ x_{n} & =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn} \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} & =\mathbf{g}^{-1}\\ \mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} & =\mathbf{A}\\ \mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} & =\mathbf{A}\\ \\ \end{align} \\ \begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \end{matrix} \end{matrix}</math>|{{equationRef|1a}}}} The invariance of the Lorentz interval with ''n''=1 between axes and conjugate diameters of hyperbolas was known for a long time since [[#Apo|Apollonius (ca. 200 BC)]]. Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.<ref>Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.</ref><ref>Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.</ref> They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881/82)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the "absolute" form <math>-x_{0}^{2}+\cdots+x_{n}^{2}=0</math> as discussed by [[#Klein|Klein (1871-73)]].<ref group=M>Killing (1885), p. 71</ref><ref>Müller (1910), p. 661, in particular footnote 247.</ref><ref>Sommerville (1911), p. 286, section K6.</ref> In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> by [[#Killing3|Killing (1888-1897)]]. ===Most general Lorentz transformation of velocity=== If <math>x_{i},\ x_{i}^{\prime}</math> in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates <math>u_{s},\ u_{s}^{\prime}</math> follow by :<math>\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)</math> defined by <math>u_{1}^{2}+u_{2}^{2}+\dots+u_{n}^{2}\le1</math> so that the Lorentz transformation becomes a [[w:homography]] inside the [[w:unit hypersphere]], which [[w:John Lighton Synge]] called "the most general formula for the composition of velocities" in terms of special relativity<ref>Synge (1955), p. 129 for ''n''=3</ref> (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})): {{NumBlk|:|<math>\begin{align}u_{s}^{\prime} & =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\ \\ u_{s} & =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}} \end{align} \left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \right.</math>|{{equationRef|1b}}}} Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere <math>-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0</math> were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]]. ==Historical notation== ==={{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas === [[w:Apollonius of Perga]] (c. 240–190 BC) defined the following proposition related to conjugate diameters of hyperbolas, which was translated and adapted to the modern reader by [[w:Thomas Heath (classicist)|w:Thomas Heath]] as follows:<ref group=M>Apollonius/Heath (1896), Proposition 129; (Apollonius, Book VII, Prop. 13).</ref> :[..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. <p style="background-color:Beige;border:1px solid black">Apollonius' result indicates the invariance of the Lorentz interval <math>-x_{0}^{\prime2}+x_{1}^{\prime2}=-x_{0}^{2}+x_{1}^{2}</math>, by which Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' can be interpreted as mapping from one pair of axes of a hyperbola to a pair of conjugate diameters.</p> This result was demonstrated by [[w:Guillaume de l'Hôpital]] (1707) in the framework of [[w:analytic geomety]]:<ref group=M>l'Hôpital (1707), Third book, Prop. XII, p. 76.</ref> :The difference of the squares of any two conjugate diameters ''Mm, Ss'' is equal to the difference of the squares of the two axes ''Aa, Bb.'' We have to prove that <math>\overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}</math>, or that <math>\overline{CM}^{2}-\overline{CS}^{2}=\overline{CA}^{2}-\overline{CB}^{2}</math>. ===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}=== After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows<ref group=M>Lagrange (1773/75), section 22</ref> :<math>\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\ \hline \begin{align}y & =Ms+Nx\\ z & =ms+nx \end{align} \left|\begin{matrix}\begin{align}P & =pM^{2}+2qMm+rm^{2}\\ Q & =pMN+q(Mn+Nm)+rmn\\ R & =pN^{2}+2qNn+rn^{2} \end{align} \\ \downarrow\\ PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2} \end{matrix}\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.</p> ==={{anchor|Gauss}} Gauss (1798–1818)=== ===={{anchor|Gauss1}} Binary quadratic forms==== The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:<ref group=M>Gauss (1798/1801), articles 157–158;</ref> :<math>\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\ F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\ \hline \begin{align}x & =\alpha x'+\beta y'\\ y & =\gamma x'+\delta y'\\ \\ x' & =\delta x-\beta y\\ y' & =-\gamma x+\alpha y \end{align} \left|\begin{matrix}\begin{align}a' & =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\ b' & =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\ c' & =a\beta^{2}+2b\beta\delta+c\delta^{2} \end{align} \\ \downarrow\\ b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2} \end{matrix}\right. \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called "proper equivalent" if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.<ref group=M>Gauss (1798/1801), section 159</ref> <p style="background-color:Beige;border:1px solid black">The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.</p> ===={{anchor|Gauss2}} Ternary quadratic forms==== Gauss (1798/1801)<ref group=M>Gauss (1798/1801), articles 266–285</ref> also discussed ternary quadratic forms with the general expression :<math>\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\\ g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)\\ \hline \begin{align}x & =\alpha y+\beta y'+\gamma y''\\ x' & =\alpha'y+\beta'y'+\gamma'y''\\ x'' & =\alpha''y+\beta''y'+\gamma''y'' \end{align} \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x²+y²+z²'', or indefinite in the case of different signs such as ''x²+y²-z²''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:<ref group=M>Gauss (1798/1801), article 277</ref> :<math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\\ \left(\begin{matrix}1, & -1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval <math>x^{2}+x^{\prime2}-x^{\prime\prime2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting: <br><math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)=\left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right)</math></p> ===={{anchor|Gauss4}} Homogeneous coordinates==== Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression :<math>(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E</math> into :<math>G+G'\cos T^{2}+G''\sin T^{2}</math> in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:<ref group=M>Gauss (1818), pp. 5–10</ref> :<math>\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\ k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\ \hline \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =k & \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =-k\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-k & \beta\beta-\beta'\beta'-\beta''\beta'' & =-k\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-k & \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =+k\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} }\right.\\ \hline k=1\\ \begin{align}t\cos E & =\alpha+\alpha'\cos T+\alpha''\sin T\\ t\sin E & =\beta+\beta'\cos T+\beta''\sin T\\ t & =\gamma+\gamma'\cos T+\gamma''\sin T \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}}) ''(n=2)''. Further setting <math>[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that :<math>aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz</math> can be transformed into :<math>Guu+G'u'u'+G''u''u''</math>, in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:<ref group=M>Gauss (1818), pp. 9–10</ref> :<math>\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =-\alpha x-\beta y+\gamma z\\ u' & =\alpha'x+\beta'y-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right.</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' satisfying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>, and can be related to Gauss' previous equations in terms of homogeneous coordinates <math>\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]</math>.</p> ==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation in 1827:<ref group=M>Jacobi (1827), p. 235, 239–240</ref> :<math>{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\ k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\ \hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P & =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\cos\vartheta & =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\sin\vartheta & =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \\ \cos\psi & =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\cos\varphi & =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\sin\varphi & =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta} \end{align} \\ \\ \hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z & =m\\ \alpha'\mu+\beta'x+\gamma'y+\delta'z & =m'\\ \alpha''\mu+\beta''x+\gamma''y+\delta''z & =m''\\ \alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z & =m'''\\ \\ Am+A'm'+A''m''+A'''m''' & =\mu\\ Bm+B'm'+B''m''+B'''m''' & =x\\ Cm+C'm'+C''m''+C'''m''' & =y\\ Dm+D'm'+D''m''+D'''m''' & =z\\ \\ \end{align} \\ \begin{align}\alpha & =-kA, & \beta & =-kB, & \gamma & =-kC, & \delta & =kD,\\ \alpha' & =kA', & \beta' & =kB', & \gamma' & =kC', & \delta' & =-kD',\\ \alpha'' & =kA'', & \beta'' & =kB'', & \gamma'' & =kC'', & \delta'' & =-kD'',\\ \alpha''' & =kA''', & \beta''' & =kB''', & \gamma''' & =kC''', & \delta''' & =-kD''', \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =-k\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' & =k\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' & =k\\ \alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' & =k\\ \alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' & =0\\ \alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' & =0\\ \alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' & =0\\ \alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' & =0\\ \alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' & =0\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' & =0\\ \\ -\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' & =k\\ -\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' & =k\\ -\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' & =k\\ -\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' & =-k\\ -\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' & =0\\ -\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' & =0\\ -\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' & =0\\ -\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' & =0\\ -\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' & =0\\ -\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' & =0 \end{align} \end{matrix}} \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] & =\left[u_{1},\ u_{2},\ u_{3}\right]\\{} [\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] & =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right] \end{align} }</math> and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing <math>m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}</math>.</p> Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):<ref group=M>The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.</ref> :<math>{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\ \mathbf{(1)}\begin{align}x & =\alpha s+\alpha's'+\alpha''s''\\ y & =\beta s+\beta's'+\beta''s''\\ z & =\gamma s+\gamma's'+\gamma''s''\\ \\ s & =\alpha x+\beta y+\gamma z\\ s' & =\alpha'x+\beta'y+\gamma'z\\ s'' & =\alpha''x+\beta''y+\gamma''z \end{align} \left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma & =1 & \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' & =1\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma' & =1 & \beta\beta+\beta'\beta'+\beta''\beta'' & =1\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' & =1 & \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' & =1\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma+\beta'\gamma'+\beta''\gamma'' & =0\\ \alpha''\alpha+\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' & =0\\ \alpha\alpha'+\beta\beta'+\gamma\gamma' & =0 & \alpha\beta+\alpha'\beta'+\alpha''\beta'' & =0 \end{align} \right.\\ \hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\ \left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\ \hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\ \left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\ \hline \begin{align}\cos\phi & =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \cos\eta & =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\ \sin\phi & =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \sin\eta & =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi} \end{align} \end{matrix}\\ \hline \\ \mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\ \hline \begin{align}y & =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, & s' & =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\ z & =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, & s'' & =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z}, \end{align} \end{matrix} \end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma & =1\\ \alpha'\alpha'-\beta'\beta'-\gamma'\gamma' & =-1\\ \alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' & =-1\\ \alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' & =0\\ \alpha''\alpha-\beta''\beta-\gamma''\gamma & =0\\ \alpha\alpha'-\beta\beta'-\gamma\gamma' & =0\\ \\ \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =1\\ \beta\beta-\beta'\beta'-\beta''\beta'' & =-1\\ \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =-1\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} \right. \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.</p> Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:<ref group =M>Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.</ref> :<math>{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\ \hline \mathbf{(1)\ }\begin{align}y_{\varkappa} & =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\ x_{\varkappa} & =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\ \\ \frac{y_{\varkappa}}{y_{n}} & =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\ \frac{x_{\varkappa}}{x_{n}} & =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}} \end{align} \left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} & =0\\ \alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} & =1\\ \\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} & =0\\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} & =1 \end{align} \right.\\ \hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\ \alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\ \hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} & =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\ \\ \xi_{\varkappa} & =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}} \end{align} \\ \hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1 \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.</p> He also stated the following transformation leaving invariant the Lorentz interval:<ref group=M>Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.</ref> :<math>\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\ \hline {\scriptstyle \begin{align}u & =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\ u_{1} & =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\ & \dots\\ u_{n-1} & =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\ \\ w & =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\ w_{1} & =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\ & \dots\\ w_{n-1} & =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1} \end{align} \left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =-1\\ \alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =0\\ \alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} & =0\\ \\ \alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} & =-1\\ \alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} & =0\\ \alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} & =0 \end{align} \text{ }\right.} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.</p> ==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids === [[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:<ref group=M>Chasles (1829), p. 139</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(a)\\ \\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(b)\\ \\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(c)\\ \\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(d) \end{matrix}</math> He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:<ref group=M>Chasles (1829), p. 141</ref> :<math>\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\ \gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\ \alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\ \beta''\quad\Rightarrow\quad\beta''\sqrt{-1} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:<ref group=M>Chasles (1829), pp. 143-144</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}x & =lx'+l'y'+l''z'\\ y & =mx'+m'y'+m''z'\\ z & =nx'+n'y'+n''z' \end{align} \\ \left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} & =0\\ \frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} & =0\\ \frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} & =0 \end{align} \right\} \\ \hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\ \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1 \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Chasles defined the conditional equations of ''l,m,n'' in the same way as those of <math>\alpha,\beta,\gamma</math> in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.</p> He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:<ref group=M>Chasles (1829), pp. 145-146</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =a^{2}\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =b^{2}\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(A)\\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(B)\\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =a^{2}\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =b^{2}\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(C)\\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(D) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates=== [[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation<ref group=M>Lebesgue (1837), pp. 338-341</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\ \hline {\scriptstyle \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\ \dots\\ y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} & =1 & (10)\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} & =0 & (11)\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} & =1 & (13)\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} & =0 & (14) \end{align} \right.} \end{matrix}</math> In order to achieve the invariance of the Lorentz interval<ref group=M>Lebesgue (1837), pp. 353–354</ref> :<math>x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}</math> he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign. <p style="background-color:Beige;border:1px solid black">These instructions give Lorentz transformation ({{equationNote|1a}}) in the form: <br><math>{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \hline \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\ \dots\\ -y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} & =1\\ a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} & =-1\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} & =0\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} & =1\\ a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} & =-1\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} & =0 \end{align} \right. \end{matrix}}</math></p> He went on to redefine the variables of the Lorentz interval and its transformation:<ref group=M>Lebesgue (1837), pp. 353–355</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \downarrow\\ \begin{align}x_{1} & =x_{n}\cos\theta_{1}, & x_{2} & =x_{n}\cos\theta_{2},\dots & x_{n-1} & =x_{n}\cos\theta_{n-1}\\ y_{1} & =y_{n}\cos\phi_{1}, & y_{2} & =y_{n}\cos\phi_{2},\dots & y_{n-1} & =y_{n}\cos\phi_{n-1} \end{align} \\ \downarrow\\ \cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\ \cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\ \hline \\ \cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\ (i=1,2,3\dots n) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Setting <math>[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]</math> it is equivalent to Lorentz transformation ({{equationNote|1b}}).</p> ==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids=== Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&alpha;), from which he derived equations (&beta;) and others:<ref group=M>Weddle (1847), p. 274</ref> :<math>\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1, & l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1, & l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} & =0\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1, & l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0 \end{align} \right\} & \dots(\alpha)\\ \\ \left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1, & l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1, & l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} & =0\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1, & m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0 \end{align} \right\} & \dots(\beta) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>, he defined three conjugate points <math>(x_{1}\dots,y_{1}\dots,z_{1}\dots)</math> on those two conjugate hyperboloids, related to each other in the same way as equations (&alpha;, &beta;) stated above:<ref group=M>Weddle (1847), pp. 275-276</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} & =0\\ \frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} & =0\\ \frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} & =0 \end{align} \quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} & =1\\ \frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} & =1\\ \frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} & =-1 \end{align} \\ \begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} & =a^{2}\\ y_{1}^{2}+y_{2}^{2}-y_{3}^{2} & =b^{2}\\ z_{1}^{2}+z_{2}^{2}-z_{3}^{2} & =-c^{2} \end{align} \quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} & =0\\ x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} & =0\\ y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} & =0 \end{align} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black"> These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:<ref group=M>Bour (1856), pp. 61; 64–65</ref> :<math>\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\ \hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \\ \hline \\ k=+1\\ t=\gamma+\gamma'\cos T+\gamma''\sin T,\\ 1=u,\ \cos T=u',\ \sin T=u',\\ t=z,\ t\cos E=x,\ t\sin E=y\\ \downarrow\\ \mathbf{(2)}\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =\gamma z-\alpha x-\beta y\\ u' & =\alpha'x+\beta'y'-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} & =k\\ -\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} & =-k\\ -\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} & =-k\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =k\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} } \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>. Furthermore, setting <math>[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math> in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> === {{anchor|Somov}} Somov (1863) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:<ref group=M>Somov (1863), pp. 12–14; p. 18 for differentials.</ref> :<math>\begin{matrix}\left.\begin{align}\cos\phi & =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\ \sin\phi & =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''} \end{align} \right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\ \cos^{2}\psi+\cos^{2}\psi=1 \end{matrix}\\ \hline \mathbf{(1)}\ \begin{align}\cos\phi & =x, & \cos\psi & =x'\\ \sin\phi & =y, & \sin\psi & =y' \end{align} \ \left|\begin{align}x & =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\ y & =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=1\\ x^{\prime2}+y^{\prime2}=1 \end{matrix}\\ \hline \begin{align}\cos\phi & =\frac{x}{z}, & \cos\psi & =\frac{x'}{z'}\\ \sin\phi & =\frac{y}{z}, & \sin\psi & =\frac{y'}{z'} \end{align} \ \left|\begin{align}\frac{x}{z} & =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\ \frac{y}{z} & =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\ x^{\prime2}+y^{\prime2}=z^{\prime2} \end{matrix}\\ \hline \mathbf{(2)}\ \left.\begin{align}x & =mx'+ny'+sz'\\ y & =m'x'+n'y'+s'z'\\ z & =m''x'+n''y'+s''z'\\ \\ x' & =mx+m'y-m''z\\ y' & =nx+n'y-n''z\\ z' & =-sx-s'y+s''z\\ \\ dx & =mdx'+ndy'+sdz'\\ dy & =m'dx'+n'dy'+s'dz'\\ dz & =m''dx'+n''dy'+s''dz' \end{align} \right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} & =1\\ n^{2}+n^{\prime2}-n^{\prime\prime2} & =1\\ -s^{2}-s^{\prime2}+s^{\prime\prime2} & =1\\ ns+n's'-n''s'' & =0\\ sm+s'm'-s''m'' & =0\\ mn+m'n'-m''n'' & =0\\ \\ m^{2}+n^{2}-s^{2} & =1\\ m^{\prime2}+n^{\prime2}-s^{\prime2} & =1\\ -m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} & =1\\ -m'm''-n'n''+s's'' & =0\\ -m''m-n''n+s''s & =0\\ mm'+nn'-ss' & =0 \end{align} }\\ dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Klein}} Klein (1871-73) – Cayley absolute and non-Euclidean geometry=== Elaborating on [[w:Arthur Cayley]]'s (1859) definition of an "absolute" ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a "fundamental [[w:conic section]]" in order to discuss motions such as rotation and translation in the non-Euclidean plane.<ref group=M>Klein (1871), pp. 601–602</ref> This was elaborated in (1873) when he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to a two-sheet [[w:hyperboloid]].<ref group=M>Klein (1873), pp. 127-128</ref> <p style="background-color:Beige;border:1px solid black">Klein's representation of hyperbolic space in terms of a two-sheet hyperboloid and its accompanied quadratic form suggests that Lorentz transformations can be geometrically interpreted as motions or isometries in hyperbolic space.</p> ==={{anchor|Killing}} Killing (1878–1893)=== ===={{anchor|Killing1}} Weierstrass coordinates==== [[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form :<math>k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}</math><ref group=M>Killing (1877/78), p. 74; Killing (1880), p. 279</ref> with <math>ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}</math><ref group=M>Killing (1880), eq. 25 on p. 283</ref> or<ref group=M>Killing (1880), p. 283</ref> :<math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> where ''k'' is the reciprocal measure of curvature, <math>k^{2}=\infty</math> denotes [[w:Euclidean geometry]], <math>k^{2}>0</math> [[w:elliptic geometry]], and <math>k^{2}<0</math> hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.<ref group=M>Killing (1877/78), eq. 25 on p. 283</ref> In (1879/80) he tried to formulate the corresponding transformations by plugging <math>k^{2}</math> into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:<ref group=M>Killing (1879/80), p. 274</ref> <math>\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\ \hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\ -k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\ k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, & -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}} \end{matrix}\\ \left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right) \end{matrix}</math> In (1885) he wrote the Weierstrass coordinates and their transformation as follows:<ref group=M>Killing (1885), pp. 18, 28–30, 53</ref> :<math>\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\ k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\ ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\ \hline \begin{align}k^{2}p' & =k^{2}wp+w'x+w''y\\ x' & =ap+a'x+a''y\\ y' & =bp+b'x+b''y\\ \\ k^{2}p & =k^{2}wp'+ax'+by'\\ x & =w'p'+a'x+b'y'\\ y & =w''p'+a''x'+b''y' \end{align} \left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} & =k^{2}\\ \frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} & =1\\ \frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} & =1\\ aw+a'w'+a''w'' & =0\\ bw+b'w'+b''w'' & =0\\ \frac{ab}{k^{2}}+a'b'+a''b'' & =0\\ \\ k^{2}w^{2}+a^{2}+b^{2} & =k^{2}\\ \frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} & =1\\ \frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} & =1\\ ww'+aa'+bb' & =0\\ ww''+aa''+bb'' & =0\\ \frac{w'w''}{k^{2}}+a'a''+b'b'' & =0 \end{align} }\right. \end{matrix}</math> In (1885) he also gave the transformation for ''n'' dimensions:<ref group=M>Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222</ref><ref>Ratcliffe (1994), § 3.6</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\ \hline \left.\begin{align}k^{2}\xi_{0} & =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\ \xi_{\varkappa} & =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\ \\ k^{2}x_{0} & =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\ x_{\varkappa} & =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n} \end{align} \right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ \frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} & =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa) \end{align} } \end{matrix}</math> In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.<ref group=M>Killing (1884/85), pp. 4–5</ref> Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting <math>k^{2}=-1</math> and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid <math>p^{2}-x^{2}-y^{2}=1</math> (known as [[w:hyperboloid model]]),<ref group=M>Killing (1885), Note 9 on p. 260</ref><ref name=rey /> by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space. <p style="background-color:Beige;border:1px solid black">All of Killing's transformations between 1879 and 1885 don't work when <math>k^{2}</math> is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> Finally, in (1893) he wrote:<ref group=M>Killing (1893), see pp. 144, 327–328</ref> :<math>\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\ \hline \begin{align}t' & =at+bu+cv\\ u' & =a't+b'u+c'v\\ v' & =a''t+b''u+c''v \end{align} \left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} & =k^{2}\\ k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} & =1\\ k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} & =1\\ k^{2}ab+a'b'+a''b'' & =0\\ k^{2}ac+a'c'+a''c'' & =0\\ k^{2}bc+b'c'+b''c'' & =0 \end{align} \right. \end{matrix}</math> and in ''n'' dimensions<ref group=M>Killing (1893), pp. 314–316, 216–217</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\ ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\ \hline \begin{align}y_{0} & =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\ y_{1} & =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\ & \,\,\,\vdots\\ y_{n} & =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n} \end{align} \left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} & =1\\ k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} & =0\\ (\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> ===={{anchor|Killing3}} Infinitesimal transformations and Lie group==== After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions, Killing (1887/88)<ref group=M>Killing (1887/88a), pp. 274–275</ref> defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere: :<math>\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\ \hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\ \text{where}\\ \left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\ \left[\iota\ne\varkappa\ne\lambda\ne\mu\right] \end{matrix}</math> and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:<ref group=M>Killing (1892), p. 177</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ \hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\ \text{where}\\ \left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f; \end{matrix}</math> In (1897/98) he showed the relation between Weierstrass coordinates <math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> and corrdinates <math>k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0</math> used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:<ref group=M>Killing (1897/98), pp. 255–256</ref> :<math>\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} & (a)\\ k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} & (b) \end{matrix}\\ \hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\ \text{where}\\ \left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\ \left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\ \left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\ \hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\ \downarrow\\ k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\ \hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota} \end{matrix} \end{matrix}</math> He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b). <p style="background-color:Beige;border:1px solid black">Setting <math>k^{2}=-1</math> denotes the group of hyperbolic motions and thus the Lorentz group.</p> === {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates === [[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).<ref>Gray (1997)</ref> He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:<ref group=M name=p1>Poincaré (1881a), pp. 133–134</ref><ref>Dickson (1923), pp. 220–221</ref> :<math>\begin{matrix}\begin{align}F & =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\ & =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\ F & =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\ & =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1 \end{align} \\ \hline \begin{align}\xi' & =\alpha\xi+\beta\eta+\gamma\zeta\\ \eta' & =\alpha'\xi+\beta'\eta+\gamma'\zeta\\ \zeta' & =\alpha''\xi+\beta''\eta+\gamma''\zeta \end{align} \left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1\\ \alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right. \end{matrix}</math> He went on to describe the properties of "hyperbolic coordinates".<ref group=M name=poinc>Poincaré (1881b), p. 333</ref><ref name=rey>Reynolds (1993)</ref> Poincaré mentioned the hyperboloid model also in (1887).<ref group=M>Poincaré (1887), p. 206</ref> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> === {{anchor|Cox}} Cox (1881–1883) – Weierstrass coordinates === [[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)<ref name=guder group=M>Gudermann (1830), §1–3, §18–19</ref> and [[w:George Salmon]] (1862)<ref group=M>Salmon (1862), section 212, p. 165</ref> on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)<ref group=M>Frischauf (1876), pp. 86–87</ref> in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:<ref group=M>Cox (1881), p. 186 for Weierstrass coordinates; (1881/82), pp. 193–194 for Lorentz transformation. On p. 193, the misprinted expression <math>x^{2}-y^{2}-z^{2}</math> should read <math>z^{2}-y^{2}-x^{2}</math></ref> :<math>\begin{matrix}z^{2}-x^{2}-y^{2}=1\\ z^{2}-y^{2}-x^{2}=Z^{2}-Y^{2}-X^{2}\\ \hline \begin{align}x & =l_{1}X+l_{2}Y+l_{3}Z\\ y & =m_{1}X+m_{2}Y+m_{3}Z\\ z & =n_{1}X+n_{2}Y+n_{3}Z\\ \\ X & =l_{1}x+m_{1}y-n_{1}z\\ Y & =l_{2}x+m_{2}y-n_{2}z\\ Z & =l_{3}x+m_{3}y-n_{3}z \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} & =0\\ \\ l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1\\ l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0\\ n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1 \end{align} }</math> with <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1 \end{align} }</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' up to a sign change in the inverse transformation.</p> Cox also gave the Weierstrass coordinates and their transformation in hyperbolic space:<ref group=M>Cox (1881), pp. 199, 206–207</ref> :<math>\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\ w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\ \hline \begin{align}x & =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\ y & =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\ z & =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\ w & =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\ \\ x' & =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\ y' & =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\ z' & =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\ w' & =l_{4}x+m_{4}y+n_{4}z-r_{4}w \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} & =1\\ l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} & =1\\ l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} & =0\\ l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} & =0\\ l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} & =0\\ l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}</math> with <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' up to a sign change in both the first as well as inverse transformation.</p> === {{anchor|Hill}} Hill (1882) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations<ref group=M>Hill (1882), pp. 323–325</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\ k\left(\sin^{2}E+\cos^{2}E-1\right)\\ \hline \begin{align} & & \cos E' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ & \mathbf{(1)} & \sin E' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \hline \\ & & x & =\alpha u+\alpha'u'+\alpha''u''\\ & & y & =\beta u+\beta'u'+\beta''u''\\ & & z & =\gamma u+\gamma'u'+\gamma''u''\\ & \mathbf{(2)}\\ & & u & =-\alpha x-\beta y+\gamma z\\ & & u' & =\alpha'x+\beta'y'-\gamma'z\\ & & u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =-1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =1\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ (k=-1)\\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =-k\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .</p> === {{anchor|Picard}} Picard (1882-1884) – Quadratic forms === [[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to "hyperfuchsian" functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:<ref group=M>Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system</ref><ref>Dickson (1923), pp. 280-281</ref> :<math>\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\ \\ \mathbf{(1)}\ \begin{align}x & =M_{1}X+P_{1}Y+R_{1}Z\\ y & =M_{2}X+P_{2}Y+R_{2}Z\\ z & =M_{3}X+P_{3}Y+R_{3}Z \end{align} \\ \\ \left[\begin{align}[][x,y,z]=\text{complex}\\ \left[x_{0},y_{0},z_{0}\right]=\text{conjugate} \end{align} \right]\\ \\ \hline \\ x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\ x=x'+ix'',\quad y=y'+iy''\\ \\ \mathbf{(2)}\ \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} & =1\\ P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0\\ P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ \\ M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} & =1\\ M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} & =1\\ M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} & =-1\\ \mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} & =0\\ \mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} & =0\\ \mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} & =0\\ \\ \left[\begin{align}[][M,P,R\dots]=\text{complex}\\ \left[\mu,\pi,\rho\dots\right]=\text{conjugate} \end{align} \right] \end{align} }\right.\end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x²+y²-z²=X²+Y²-Z²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:<ref group=M>Picard (1884a), p. 13</ref> :<math>\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\ \hline \begin{align}U & =\mathcal{A}u+\mathcal{B}v\\ V & =\mathcal{C}u+\mathcal{D}v \end{align} \left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} & =1\\ \mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} & =0\\ \mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} & =-1\\ \mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} & =1 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U²-V²=u²-v²''.</p> Or in (1884b):<ref group=M>Picard (1884b), p. 416</ref> :<math>\begin{matrix}xx_{0}+yy_{0}-1=0\\ \hline \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884c):<ref group=M>Picard (1884c), pp. 123–124; 163</ref> :<math>\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\ \hline \mathbf{(1)}\ \begin{align}U & =Mu+Pv+Rw\\ V & =M'u+P'v+R'w\\ W & =M''u+P''v+R''w\\ \\ u & =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\ v & =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\ w & =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W \end{align} \left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} & =1\\ PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} & =1\\ RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} & =-1\\ MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} & =0\\ MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} & =0\\ PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} & =0\\ \\ MM_{0}+PP_{0}-RR_{0} & =1\\ M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} & =1\\ M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} & =-1\\ M_{0}M'+P_{0}P'-R_{0}R' & =0\\ M_{0}M''+P_{0}P''-R_{0}R'' & =0\\ M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' & =0 \end{align} }\right.\\ \hline \text{Invariance of unit hypersphere:}\\ \mathbf{(2)}\ \begin{align}\xi' & =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\ \eta' & =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''} \end{align} \left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} & =1\\ BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} & =1\\ CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} & =-1\\ AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} & =0\\ AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} & =0\\ BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U²+V²-W²=u²+v²-w²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing <math>\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1</math>.</p> === {{anchor|Callandreau}} Callandreau (1885) – Homography === Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations<ref group=M>Callandreau (1885), pp. A.7; A.12</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\ {\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\ \hline \begin{align}\cos\varepsilon' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \sin\varepsilon' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T} \end{align} \left|{\scriptstyle \begin{align} & \left(k=1\right)\\ \alpha^{2}+\beta^{2}-\gamma^{2} & =-k & \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =+k & \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =+k & \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-1 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-1 & \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =+1 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>.</p> ==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations=== In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions.<ref group=M>Lie (1885/86), p. 411</ref> In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):<ref group=M>Werner (1889), pp. 4, 28</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\ \hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\ \text{where}\\ \left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\ \left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\ \left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right] \end{matrix}</math> More generally, Lie (1890)<ref group=M>Lie (1890a), p. 295;</ref> defined non-Euclidean motions in terms of two forms <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0</math> in which the imaginary form with <math>+1</math> denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:<ref group=M>Lie (1890a), p. 311</ref> :<math>\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\ \hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n) \end{matrix}</math> Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form: :<math>x^{2}+y^{2}-1=0</math><ref group=M>Lie (1893), p. 474</ref> or <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0</math><ref group=M>Lie (1893), p. 479</ref> or <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math>.<ref group=M>Lie (1893), p. 481</ref> <p style="background-color:Beige;border:1px solid black">The group of hyperbolic motions is isomorphic to the Lorentz group. The interval <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math> becomes the Lorentz interval <math>x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0</math> by setting <br><math>(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)</math></p> ==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates=== [[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':<ref group=M>Gérard (1892), pp. 40–41</ref> :<math>\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\ X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\ \hline \begin{align}X & =aX'+a'Y'+a''Z'\\ Y & =bX'+b'Y'+b''Z'\\ Z & =cX'+c'Y'+c''Z'\\ \\ X' & =aX+bY-cZ\\ Y' & =a'X+b'Y-c'Z\\ Z' & =-a''X-b''Y+c''Z \end{align} \left|\begin{align}a^{2}+b^{2}-c^{2} & =1\\ a^{\prime2}+b^{\prime2}-c^{\prime2} & =1\\ a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} & =-1\\ aa'+bb'-cc' & =0\\ a'a''+b'b''-c'c'' & =0\\ a''a+b''b-c''c & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> He gave the case of translation as follows:<ref group=M name=gerard>Gérard (1892), pp. 40–41</ref> :<math>\begin{align}X & =Z_{0}X'+X_{0}Z'\\ Y & =Y'\\ Z & =X_{0}X'+Z_{0}Z' \end{align} \ \text{with}\ \begin{align}X_{0} & =\operatorname{sh}OO'\\ Z_{0} & =\operatorname{ch}OO' \end{align} </math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz boost ({{equationNote|3b}}).</p> ==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates=== [[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:<ref group=M>Hausdorff (1899), p. 165, pp. 181-182</ref> :<math>\begin{matrix}p^{2}-x^{2}-y^{2}=1\\ \hline \begin{align}x & =a_{1}x'+a_{2}y'+x_{0}p'\\ y & =b_{1}x'+b{}_{2}y'+y_{0}p'\\ p & =e_{1}x'+e_{2}y'+p_{0}p'\\ \\ x' & =a_{1}x+b_{1}y-e_{1}p\\ y' & =a_{2}x+b_{2}y-e_{2}p\\ -p' & =x_{0}x+y_{0}y-p_{0}p \end{align} \left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} & =1\\ a_{2}^{2}+b_{2}^{2}-e_{2}^{2} & =1\\ -x_{0}^{2}-y_{0}^{2}+p_{0}^{2} & =1\\ a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} & =0\\ a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} & =0\\ a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} & =0\\ \\ a_{1}^{2}+a_{2}^{2}-x_{0}^{2} & =1\\ b_{1}^{2}+b_{2}^{2}-y_{0}^{2} & =1\\ -e_{1}^{2}-e_{2}^{2}+p_{0}^{2} & =1\\ b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} & =0\\ a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} & =0\\ a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates === In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting <math>k=\sqrt{-1}R</math> with ''R'' as real quantity):<ref group=M>Woods (1901/02), p. 98, 104</ref> :<math>\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\ \hline \begin{align}u' & =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ v' & =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ w' & =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}} \end{align} \left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} & =k^{2}\\ (i=1,2,3)\\ k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} & =1\\ \alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} & =0\\ (i,h=1,2,3,4;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''²=-1.</p> Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with <math>k=\sqrt{-1}k</math> for hyperbolic space):<ref group=M>Woods (1903/05), pp. 45–46; p. 48)</ref> :<math>\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\ ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\ \hline \begin{align}x_{1}^{\prime} & =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\ x_{2}^{\prime} & =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\ x_{3}^{\prime} & =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\ x_{0}^{\prime} & =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0} \end{align} \left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) & =1\\ \delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) & =k^{2}\\ (i=1,2,3)\\ \delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) & =0\\ (i,h=0,1,2,3;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''²=-1.</p> ==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates=== [[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''<ref group=M>Liebmann (1904/05), p. 168; pp. 175–176</ref> :<math>\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\ \hline \begin{align}x_{1} & =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\ y_{1} & =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\ x_{1} & =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\ \\ x & =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\ y & =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\ p & =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1} \end{align} \left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} & =1\\ -\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} & =1\\ -\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} & =1\\ \alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} & =0\\ \alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} & =0\\ \alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==References== ===Historical mathematical sources=== {{reflist|3|group=M}} *{{#section:History of Topics in Special Relativity/mathsource|apo}} *{{#section:History of Topics in Special Relativity/mathsource|bour56att}} *{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} *{{#section:History of Topics in Special Relativity/mathsource|chas29}} *{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} *{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} *{{#section:History of Topics in Special Relativity/mathsource|fris76}} *{{#section:History of Topics in Special Relativity/mathsource|gau98}} *{{#section:History of Topics in Special Relativity/mathsource|gau18}} *{{#section:History of Topics in Special Relativity/mathsource|ger92}} *{{#section:History of Topics in Special Relativity/mathsource|gud30}} *{{#section:History of Topics in Special Relativity/mathsource|haus99}} *{{#section:History of Topics in Special Relativity/mathsource|hill82}} *{{#section:History of Topics in Special Relativity/mathsource|jac27}} *{{#section:History of Topics in Special Relativity/mathsource|jac32a}} *{{#section:History of Topics in Special Relativity/mathsource|jac32b}} *{{#section:History of Topics in Special Relativity/mathsource|jac33}} *{{#section:History of Topics in Special Relativity/mathsource|kil77}} *{{#section:History of Topics in Special Relativity/mathsource|kil79}} *{{#section:History of Topics in Special Relativity/mathsource|kil84}} *{{#section:History of Topics in Special Relativity/mathsource|kil85}} *{{#section:History of Topics in Special Relativity/mathsource|kil93}} *{{#section:History of Topics in Special Relativity/mathsource|kil97}} *{{#section:History of Topics in Special Relativity/mathsource|klei71}} *{{#section:History of Topics in Special Relativity/mathsource|klei73}} *{{#section:History of Topics in Special Relativity/mathsource|lag73}} *{{#section:History of Topics in Special Relativity/mathsource|leb37}} *{{#section:History of Topics in Special Relativity/mathsource|lie85}} *{{#section:History of Topics in Special Relativity/mathsource|lie90}} *{{#section:History of Topics in Special Relativity/mathsource|lie93}} *{{#section:History of Topics in Special Relativity/mathsource|lieb04}} *{{#section:History of Topics in Special Relativity/mathsource|pic82}} *{{#section:History of Topics in Special Relativity/mathsource|pic84a}} *{{#section:History of Topics in Special Relativity/mathsource|pic84b}} *{{#section:History of Topics in Special Relativity/mathsource|pic84c}} *{{#section:History of Topics in Special Relativity/mathsource|poin81a}} *{{#section:History of Topics in Special Relativity/mathsource|poin81b}} *{{#section:History of Topics in Special Relativity/mathsource|poin87}} *{{#section:History of Topics in Special Relativity/mathsource|sal62}} *{{#section:History of Topics in Special Relativity/mathsource|som63}} *{{#section:History of Topics in Special Relativity/mathsource|wedd47}} *{{#section:History of Topics in Special Relativity/mathsource|wern89}} *{{#section:History of Topics in Special Relativity/mathsource|woo01}} *{{#section:History of Topics in Special Relativity/mathsource|woo03}} ===Secondary sources=== {{reflist|3}} {{#section:History of Topics in Special Relativity/secsource|L1}} [[Category:Special Relativity]] [[Category:History of Physics]] 4kx7xz9qa3m8hp02p27ma5zxqifyh9u 2416148 2416147 2022-08-18T13:05:12Z D.H 52339 /* {{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas */ wikitext text/x-wiki {{../Lorentz transformation (header)}} ==Most general Lorentz transformations== ===General quadratic form=== The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as<ref>Bôcher (1907), chapter X</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}\begin{align}q=\sum_{0}^{n}A_{ij}x_{i}x_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align} & =q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\ b=\sum_{0}^{n}A_{ij}x_{i}y_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y} & =b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}' \end{align} \quad\left(A_{ij}=A_{ji}\right)\\ \hline \left.\begin{align}x_{i}^{\prime} & =\sum_{j=0}^{n}g_{ij}x_{j}=\mathbf{g}\cdot\mathbf{x}\\ x_{i} & =\sum_{j=0}^{n}g_{ij}^{(-1)}x_{j}^{\prime}=\mathbf{g}^{-1}\cdot\mathbf{x}' \end{align} \right|\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}' \end{matrix}</math>|{{equationRef|Q1}}}} The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc. ===Most general Lorentz transformation=== The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:<ref name=ratcliffe>Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1</ref><ref>Naimark (1964), 2 in four dimensions</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} & =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\ -x_{0}y_{0}+\cdots+x_{n}y_{n} & =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime} \end{align} \\ \hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\ \downarrow\\ \begin{align}x_{0}^{\prime} & =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\ x_{1}^{\prime} & =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\ & \dots\\ x_{n}^{\prime} & =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn} \end{align} \\ \\ \mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\ \downarrow\\ \begin{align}x_{0} & =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\ x_{1} & =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\ & \dots\\ x_{n} & =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn} \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} & =\mathbf{g}^{-1}\\ \mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} & =\mathbf{A}\\ \mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} & =\mathbf{A}\\ \\ \end{align} \\ \begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \end{matrix} \end{matrix}</math>|{{equationRef|1a}}}} The invariance of the Lorentz interval with ''n''=1 between axes and conjugate diameters of hyperbolas was known for a long time since [[#Apo|Apollonius (ca. 200 BC)]]. Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.<ref>Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.</ref><ref>Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.</ref> They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881/82)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the "absolute" form <math>-x_{0}^{2}+\cdots+x_{n}^{2}=0</math> as discussed by [[#Klein|Klein (1871-73)]].<ref group=M>Killing (1885), p. 71</ref><ref>Müller (1910), p. 661, in particular footnote 247.</ref><ref>Sommerville (1911), p. 286, section K6.</ref> In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> by [[#Killing3|Killing (1888-1897)]]. ===Most general Lorentz transformation of velocity=== If <math>x_{i},\ x_{i}^{\prime}</math> in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates <math>u_{s},\ u_{s}^{\prime}</math> follow by :<math>\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)</math> defined by <math>u_{1}^{2}+u_{2}^{2}+\dots+u_{n}^{2}\le1</math> so that the Lorentz transformation becomes a [[w:homography]] inside the [[w:unit hypersphere]], which [[w:John Lighton Synge]] called "the most general formula for the composition of velocities" in terms of special relativity<ref>Synge (1955), p. 129 for ''n''=3</ref> (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})): {{NumBlk|:|<math>\begin{align}u_{s}^{\prime} & =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\ \\ u_{s} & =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}} \end{align} \left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \right.</math>|{{equationRef|1b}}}} Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere <math>-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0</math> were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]]. ==Historical notation== ==={{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas === [[w:Apollonius of Perga]] (c. 240–190 BC) defined the following proposition related to conjugate diameters of hyperbolas, which was translated and adapted to the modern reader by [[w:Thomas Heath (classicist)|w:Thomas Heath]] as follows:<ref group=M>Apollonius/Heath (1896), Proposition 129; (Apollonius, Book VII, Prop. 13).</ref> :[..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. <p style="background-color:Beige;border:1px solid black">Apollonius' result indicates the invariance of the Lorentz interval <math>-x_{0}^{\prime2}+x_{1}^{\prime2}=-x_{0}^{2}+x_{1}^{2}</math>, by which Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' can be interpreted as mapping from one pair of axes of a hyperbola to a pair of conjugate diameters.</p> This result was well known to modern authors, such as [[w:Guillaume de l'Hôpital]] (1707) who demonstrated it in the framework of [[w:analytic geomety]]:<ref group=M>l'Hôpital (1707), Third book, Prop. XII, p. 76.</ref> :The difference of the squares of any two conjugate diameters ''Mm, Ss'' is equal to the difference of the squares of the two axes ''Aa, Bb.'' We have to prove that <math>\overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}</math>, or that <math>\overline{CM}^{2}-\overline{CS}^{2}=\overline{CA}^{2}-\overline{CB}^{2}</math>. ===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}=== After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows<ref group=M>Lagrange (1773/75), section 22</ref> :<math>\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\ \hline \begin{align}y & =Ms+Nx\\ z & =ms+nx \end{align} \left|\begin{matrix}\begin{align}P & =pM^{2}+2qMm+rm^{2}\\ Q & =pMN+q(Mn+Nm)+rmn\\ R & =pN^{2}+2qNn+rn^{2} \end{align} \\ \downarrow\\ PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2} \end{matrix}\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.</p> ==={{anchor|Gauss}} Gauss (1798–1818)=== ===={{anchor|Gauss1}} Binary quadratic forms==== The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:<ref group=M>Gauss (1798/1801), articles 157–158;</ref> :<math>\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\ F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\ \hline \begin{align}x & =\alpha x'+\beta y'\\ y & =\gamma x'+\delta y'\\ \\ x' & =\delta x-\beta y\\ y' & =-\gamma x+\alpha y \end{align} \left|\begin{matrix}\begin{align}a' & =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\ b' & =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\ c' & =a\beta^{2}+2b\beta\delta+c\delta^{2} \end{align} \\ \downarrow\\ b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2} \end{matrix}\right. \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called "proper equivalent" if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.<ref group=M>Gauss (1798/1801), section 159</ref> <p style="background-color:Beige;border:1px solid black">The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.</p> ===={{anchor|Gauss2}} Ternary quadratic forms==== Gauss (1798/1801)<ref group=M>Gauss (1798/1801), articles 266–285</ref> also discussed ternary quadratic forms with the general expression :<math>\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\\ g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)\\ \hline \begin{align}x & =\alpha y+\beta y'+\gamma y''\\ x' & =\alpha'y+\beta'y'+\gamma'y''\\ x'' & =\alpha''y+\beta''y'+\gamma''y'' \end{align} \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x²+y²+z²'', or indefinite in the case of different signs such as ''x²+y²-z²''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:<ref group=M>Gauss (1798/1801), article 277</ref> :<math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\\ \left(\begin{matrix}1, & -1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval <math>x^{2}+x^{\prime2}-x^{\prime\prime2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting: <br><math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)=\left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right)</math></p> ===={{anchor|Gauss4}} Homogeneous coordinates==== Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression :<math>(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E</math> into :<math>G+G'\cos T^{2}+G''\sin T^{2}</math> in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:<ref group=M>Gauss (1818), pp. 5–10</ref> :<math>\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\ k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\ \hline \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =k & \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =-k\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-k & \beta\beta-\beta'\beta'-\beta''\beta'' & =-k\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-k & \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =+k\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} }\right.\\ \hline k=1\\ \begin{align}t\cos E & =\alpha+\alpha'\cos T+\alpha''\sin T\\ t\sin E & =\beta+\beta'\cos T+\beta''\sin T\\ t & =\gamma+\gamma'\cos T+\gamma''\sin T \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}}) ''(n=2)''. Further setting <math>[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that :<math>aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz</math> can be transformed into :<math>Guu+G'u'u'+G''u''u''</math>, in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:<ref group=M>Gauss (1818), pp. 9–10</ref> :<math>\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =-\alpha x-\beta y+\gamma z\\ u' & =\alpha'x+\beta'y-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right.</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' satisfying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>, and can be related to Gauss' previous equations in terms of homogeneous coordinates <math>\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]</math>.</p> ==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation in 1827:<ref group=M>Jacobi (1827), p. 235, 239–240</ref> :<math>{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\ k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\ \hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P & =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\cos\vartheta & =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\sin\vartheta & =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \\ \cos\psi & =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\cos\varphi & =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\sin\varphi & =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta} \end{align} \\ \\ \hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z & =m\\ \alpha'\mu+\beta'x+\gamma'y+\delta'z & =m'\\ \alpha''\mu+\beta''x+\gamma''y+\delta''z & =m''\\ \alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z & =m'''\\ \\ Am+A'm'+A''m''+A'''m''' & =\mu\\ Bm+B'm'+B''m''+B'''m''' & =x\\ Cm+C'm'+C''m''+C'''m''' & =y\\ Dm+D'm'+D''m''+D'''m''' & =z\\ \\ \end{align} \\ \begin{align}\alpha & =-kA, & \beta & =-kB, & \gamma & =-kC, & \delta & =kD,\\ \alpha' & =kA', & \beta' & =kB', & \gamma' & =kC', & \delta' & =-kD',\\ \alpha'' & =kA'', & \beta'' & =kB'', & \gamma'' & =kC'', & \delta'' & =-kD'',\\ \alpha''' & =kA''', & \beta''' & =kB''', & \gamma''' & =kC''', & \delta''' & =-kD''', \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =-k\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' & =k\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' & =k\\ \alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' & =k\\ \alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' & =0\\ \alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' & =0\\ \alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' & =0\\ \alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' & =0\\ \alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' & =0\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' & =0\\ \\ -\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' & =k\\ -\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' & =k\\ -\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' & =k\\ -\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' & =-k\\ -\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' & =0\\ -\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' & =0\\ -\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' & =0\\ -\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' & =0\\ -\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' & =0\\ -\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' & =0 \end{align} \end{matrix}} \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] & =\left[u_{1},\ u_{2},\ u_{3}\right]\\{} [\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] & =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right] \end{align} }</math> and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing <math>m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}</math>.</p> Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):<ref group=M>The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.</ref> :<math>{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\ \mathbf{(1)}\begin{align}x & =\alpha s+\alpha's'+\alpha''s''\\ y & =\beta s+\beta's'+\beta''s''\\ z & =\gamma s+\gamma's'+\gamma''s''\\ \\ s & =\alpha x+\beta y+\gamma z\\ s' & =\alpha'x+\beta'y+\gamma'z\\ s'' & =\alpha''x+\beta''y+\gamma''z \end{align} \left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma & =1 & \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' & =1\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma' & =1 & \beta\beta+\beta'\beta'+\beta''\beta'' & =1\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' & =1 & \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' & =1\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma+\beta'\gamma'+\beta''\gamma'' & =0\\ \alpha''\alpha+\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' & =0\\ \alpha\alpha'+\beta\beta'+\gamma\gamma' & =0 & \alpha\beta+\alpha'\beta'+\alpha''\beta'' & =0 \end{align} \right.\\ \hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\ \left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\ \hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\ \left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\ \hline \begin{align}\cos\phi & =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \cos\eta & =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\ \sin\phi & =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \sin\eta & =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi} \end{align} \end{matrix}\\ \hline \\ \mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\ \hline \begin{align}y & =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, & s' & =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\ z & =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, & s'' & =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z}, \end{align} \end{matrix} \end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma & =1\\ \alpha'\alpha'-\beta'\beta'-\gamma'\gamma' & =-1\\ \alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' & =-1\\ \alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' & =0\\ \alpha''\alpha-\beta''\beta-\gamma''\gamma & =0\\ \alpha\alpha'-\beta\beta'-\gamma\gamma' & =0\\ \\ \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =1\\ \beta\beta-\beta'\beta'-\beta''\beta'' & =-1\\ \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =-1\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} \right. \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.</p> Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:<ref group =M>Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.</ref> :<math>{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\ \hline \mathbf{(1)\ }\begin{align}y_{\varkappa} & =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\ x_{\varkappa} & =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\ \\ \frac{y_{\varkappa}}{y_{n}} & =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\ \frac{x_{\varkappa}}{x_{n}} & =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}} \end{align} \left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} & =0\\ \alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} & =1\\ \\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} & =0\\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} & =1 \end{align} \right.\\ \hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\ \alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\ \hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} & =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\ \\ \xi_{\varkappa} & =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}} \end{align} \\ \hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1 \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.</p> He also stated the following transformation leaving invariant the Lorentz interval:<ref group=M>Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.</ref> :<math>\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\ \hline {\scriptstyle \begin{align}u & =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\ u_{1} & =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\ & \dots\\ u_{n-1} & =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\ \\ w & =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\ w_{1} & =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\ & \dots\\ w_{n-1} & =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1} \end{align} \left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =-1\\ \alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =0\\ \alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} & =0\\ \\ \alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} & =-1\\ \alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} & =0\\ \alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} & =0 \end{align} \text{ }\right.} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.</p> ==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids === [[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:<ref group=M>Chasles (1829), p. 139</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(a)\\ \\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(b)\\ \\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(c)\\ \\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(d) \end{matrix}</math> He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:<ref group=M>Chasles (1829), p. 141</ref> :<math>\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\ \gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\ \alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\ \beta''\quad\Rightarrow\quad\beta''\sqrt{-1} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:<ref group=M>Chasles (1829), pp. 143-144</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}x & =lx'+l'y'+l''z'\\ y & =mx'+m'y'+m''z'\\ z & =nx'+n'y'+n''z' \end{align} \\ \left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} & =0\\ \frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} & =0\\ \frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} & =0 \end{align} \right\} \\ \hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\ \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1 \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Chasles defined the conditional equations of ''l,m,n'' in the same way as those of <math>\alpha,\beta,\gamma</math> in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.</p> He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:<ref group=M>Chasles (1829), pp. 145-146</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =a^{2}\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =b^{2}\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(A)\\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(B)\\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =a^{2}\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =b^{2}\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(C)\\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(D) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates=== [[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation<ref group=M>Lebesgue (1837), pp. 338-341</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\ \hline {\scriptstyle \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\ \dots\\ y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} & =1 & (10)\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} & =0 & (11)\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} & =1 & (13)\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} & =0 & (14) \end{align} \right.} \end{matrix}</math> In order to achieve the invariance of the Lorentz interval<ref group=M>Lebesgue (1837), pp. 353–354</ref> :<math>x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}</math> he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign. <p style="background-color:Beige;border:1px solid black">These instructions give Lorentz transformation ({{equationNote|1a}}) in the form: <br><math>{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \hline \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\ \dots\\ -y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} & =1\\ a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} & =-1\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} & =0\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} & =1\\ a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} & =-1\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} & =0 \end{align} \right. \end{matrix}}</math></p> He went on to redefine the variables of the Lorentz interval and its transformation:<ref group=M>Lebesgue (1837), pp. 353–355</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \downarrow\\ \begin{align}x_{1} & =x_{n}\cos\theta_{1}, & x_{2} & =x_{n}\cos\theta_{2},\dots & x_{n-1} & =x_{n}\cos\theta_{n-1}\\ y_{1} & =y_{n}\cos\phi_{1}, & y_{2} & =y_{n}\cos\phi_{2},\dots & y_{n-1} & =y_{n}\cos\phi_{n-1} \end{align} \\ \downarrow\\ \cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\ \cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\ \hline \\ \cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\ (i=1,2,3\dots n) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Setting <math>[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]</math> it is equivalent to Lorentz transformation ({{equationNote|1b}}).</p> ==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids=== Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&alpha;), from which he derived equations (&beta;) and others:<ref group=M>Weddle (1847), p. 274</ref> :<math>\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1, & l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1, & l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} & =0\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1, & l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0 \end{align} \right\} & \dots(\alpha)\\ \\ \left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1, & l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1, & l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} & =0\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1, & m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0 \end{align} \right\} & \dots(\beta) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>, he defined three conjugate points <math>(x_{1}\dots,y_{1}\dots,z_{1}\dots)</math> on those two conjugate hyperboloids, related to each other in the same way as equations (&alpha;, &beta;) stated above:<ref group=M>Weddle (1847), pp. 275-276</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} & =0\\ \frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} & =0\\ \frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} & =0 \end{align} \quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} & =1\\ \frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} & =1\\ \frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} & =-1 \end{align} \\ \begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} & =a^{2}\\ y_{1}^{2}+y_{2}^{2}-y_{3}^{2} & =b^{2}\\ z_{1}^{2}+z_{2}^{2}-z_{3}^{2} & =-c^{2} \end{align} \quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} & =0\\ x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} & =0\\ y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} & =0 \end{align} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black"> These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:<ref group=M>Bour (1856), pp. 61; 64–65</ref> :<math>\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\ \hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \\ \hline \\ k=+1\\ t=\gamma+\gamma'\cos T+\gamma''\sin T,\\ 1=u,\ \cos T=u',\ \sin T=u',\\ t=z,\ t\cos E=x,\ t\sin E=y\\ \downarrow\\ \mathbf{(2)}\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =\gamma z-\alpha x-\beta y\\ u' & =\alpha'x+\beta'y'-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} & =k\\ -\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} & =-k\\ -\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} & =-k\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =k\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} } \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>. Furthermore, setting <math>[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math> in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> === {{anchor|Somov}} Somov (1863) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:<ref group=M>Somov (1863), pp. 12–14; p. 18 for differentials.</ref> :<math>\begin{matrix}\left.\begin{align}\cos\phi & =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\ \sin\phi & =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''} \end{align} \right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\ \cos^{2}\psi+\cos^{2}\psi=1 \end{matrix}\\ \hline \mathbf{(1)}\ \begin{align}\cos\phi & =x, & \cos\psi & =x'\\ \sin\phi & =y, & \sin\psi & =y' \end{align} \ \left|\begin{align}x & =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\ y & =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=1\\ x^{\prime2}+y^{\prime2}=1 \end{matrix}\\ \hline \begin{align}\cos\phi & =\frac{x}{z}, & \cos\psi & =\frac{x'}{z'}\\ \sin\phi & =\frac{y}{z}, & \sin\psi & =\frac{y'}{z'} \end{align} \ \left|\begin{align}\frac{x}{z} & =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\ \frac{y}{z} & =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\ x^{\prime2}+y^{\prime2}=z^{\prime2} \end{matrix}\\ \hline \mathbf{(2)}\ \left.\begin{align}x & =mx'+ny'+sz'\\ y & =m'x'+n'y'+s'z'\\ z & =m''x'+n''y'+s''z'\\ \\ x' & =mx+m'y-m''z\\ y' & =nx+n'y-n''z\\ z' & =-sx-s'y+s''z\\ \\ dx & =mdx'+ndy'+sdz'\\ dy & =m'dx'+n'dy'+s'dz'\\ dz & =m''dx'+n''dy'+s''dz' \end{align} \right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} & =1\\ n^{2}+n^{\prime2}-n^{\prime\prime2} & =1\\ -s^{2}-s^{\prime2}+s^{\prime\prime2} & =1\\ ns+n's'-n''s'' & =0\\ sm+s'm'-s''m'' & =0\\ mn+m'n'-m''n'' & =0\\ \\ m^{2}+n^{2}-s^{2} & =1\\ m^{\prime2}+n^{\prime2}-s^{\prime2} & =1\\ -m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} & =1\\ -m'm''-n'n''+s's'' & =0\\ -m''m-n''n+s''s & =0\\ mm'+nn'-ss' & =0 \end{align} }\\ dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Klein}} Klein (1871-73) – Cayley absolute and non-Euclidean geometry=== Elaborating on [[w:Arthur Cayley]]'s (1859) definition of an "absolute" ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a "fundamental [[w:conic section]]" in order to discuss motions such as rotation and translation in the non-Euclidean plane.<ref group=M>Klein (1871), pp. 601–602</ref> This was elaborated in (1873) when he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to a two-sheet [[w:hyperboloid]].<ref group=M>Klein (1873), pp. 127-128</ref> <p style="background-color:Beige;border:1px solid black">Klein's representation of hyperbolic space in terms of a two-sheet hyperboloid and its accompanied quadratic form suggests that Lorentz transformations can be geometrically interpreted as motions or isometries in hyperbolic space.</p> ==={{anchor|Killing}} Killing (1878–1893)=== ===={{anchor|Killing1}} Weierstrass coordinates==== [[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form :<math>k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}</math><ref group=M>Killing (1877/78), p. 74; Killing (1880), p. 279</ref> with <math>ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}</math><ref group=M>Killing (1880), eq. 25 on p. 283</ref> or<ref group=M>Killing (1880), p. 283</ref> :<math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> where ''k'' is the reciprocal measure of curvature, <math>k^{2}=\infty</math> denotes [[w:Euclidean geometry]], <math>k^{2}>0</math> [[w:elliptic geometry]], and <math>k^{2}<0</math> hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.<ref group=M>Killing (1877/78), eq. 25 on p. 283</ref> In (1879/80) he tried to formulate the corresponding transformations by plugging <math>k^{2}</math> into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:<ref group=M>Killing (1879/80), p. 274</ref> <math>\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\ \hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\ -k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\ k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, & -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}} \end{matrix}\\ \left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right) \end{matrix}</math> In (1885) he wrote the Weierstrass coordinates and their transformation as follows:<ref group=M>Killing (1885), pp. 18, 28–30, 53</ref> :<math>\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\ k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\ ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\ \hline \begin{align}k^{2}p' & =k^{2}wp+w'x+w''y\\ x' & =ap+a'x+a''y\\ y' & =bp+b'x+b''y\\ \\ k^{2}p & =k^{2}wp'+ax'+by'\\ x & =w'p'+a'x+b'y'\\ y & =w''p'+a''x'+b''y' \end{align} \left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} & =k^{2}\\ \frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} & =1\\ \frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} & =1\\ aw+a'w'+a''w'' & =0\\ bw+b'w'+b''w'' & =0\\ \frac{ab}{k^{2}}+a'b'+a''b'' & =0\\ \\ k^{2}w^{2}+a^{2}+b^{2} & =k^{2}\\ \frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} & =1\\ \frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} & =1\\ ww'+aa'+bb' & =0\\ ww''+aa''+bb'' & =0\\ \frac{w'w''}{k^{2}}+a'a''+b'b'' & =0 \end{align} }\right. \end{matrix}</math> In (1885) he also gave the transformation for ''n'' dimensions:<ref group=M>Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222</ref><ref>Ratcliffe (1994), § 3.6</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\ \hline \left.\begin{align}k^{2}\xi_{0} & =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\ \xi_{\varkappa} & =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\ \\ k^{2}x_{0} & =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\ x_{\varkappa} & =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n} \end{align} \right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ \frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} & =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa) \end{align} } \end{matrix}</math> In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.<ref group=M>Killing (1884/85), pp. 4–5</ref> Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting <math>k^{2}=-1</math> and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid <math>p^{2}-x^{2}-y^{2}=1</math> (known as [[w:hyperboloid model]]),<ref group=M>Killing (1885), Note 9 on p. 260</ref><ref name=rey /> by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space. <p style="background-color:Beige;border:1px solid black">All of Killing's transformations between 1879 and 1885 don't work when <math>k^{2}</math> is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> Finally, in (1893) he wrote:<ref group=M>Killing (1893), see pp. 144, 327–328</ref> :<math>\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\ \hline \begin{align}t' & =at+bu+cv\\ u' & =a't+b'u+c'v\\ v' & =a''t+b''u+c''v \end{align} \left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} & =k^{2}\\ k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} & =1\\ k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} & =1\\ k^{2}ab+a'b'+a''b'' & =0\\ k^{2}ac+a'c'+a''c'' & =0\\ k^{2}bc+b'c'+b''c'' & =0 \end{align} \right. \end{matrix}</math> and in ''n'' dimensions<ref group=M>Killing (1893), pp. 314–316, 216–217</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\ ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\ \hline \begin{align}y_{0} & =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\ y_{1} & =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\ & \,\,\,\vdots\\ y_{n} & =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n} \end{align} \left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} & =1\\ k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} & =0\\ (\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> ===={{anchor|Killing3}} Infinitesimal transformations and Lie group==== After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions, Killing (1887/88)<ref group=M>Killing (1887/88a), pp. 274–275</ref> defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere: :<math>\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\ \hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\ \text{where}\\ \left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\ \left[\iota\ne\varkappa\ne\lambda\ne\mu\right] \end{matrix}</math> and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:<ref group=M>Killing (1892), p. 177</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ \hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\ \text{where}\\ \left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f; \end{matrix}</math> In (1897/98) he showed the relation between Weierstrass coordinates <math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> and corrdinates <math>k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0</math> used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:<ref group=M>Killing (1897/98), pp. 255–256</ref> :<math>\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} & (a)\\ k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} & (b) \end{matrix}\\ \hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\ \text{where}\\ \left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\ \left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\ \left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\ \hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\ \downarrow\\ k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\ \hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota} \end{matrix} \end{matrix}</math> He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b). <p style="background-color:Beige;border:1px solid black">Setting <math>k^{2}=-1</math> denotes the group of hyperbolic motions and thus the Lorentz group.</p> === {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates === [[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).<ref>Gray (1997)</ref> He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:<ref group=M name=p1>Poincaré (1881a), pp. 133–134</ref><ref>Dickson (1923), pp. 220–221</ref> :<math>\begin{matrix}\begin{align}F & =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\ & =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\ F & =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\ & =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1 \end{align} \\ \hline \begin{align}\xi' & =\alpha\xi+\beta\eta+\gamma\zeta\\ \eta' & =\alpha'\xi+\beta'\eta+\gamma'\zeta\\ \zeta' & =\alpha''\xi+\beta''\eta+\gamma''\zeta \end{align} \left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1\\ \alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right. \end{matrix}</math> He went on to describe the properties of "hyperbolic coordinates".<ref group=M name=poinc>Poincaré (1881b), p. 333</ref><ref name=rey>Reynolds (1993)</ref> Poincaré mentioned the hyperboloid model also in (1887).<ref group=M>Poincaré (1887), p. 206</ref> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> === {{anchor|Cox}} Cox (1881–1883) – Weierstrass coordinates === [[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)<ref name=guder group=M>Gudermann (1830), §1–3, §18–19</ref> and [[w:George Salmon]] (1862)<ref group=M>Salmon (1862), section 212, p. 165</ref> on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)<ref group=M>Frischauf (1876), pp. 86–87</ref> in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:<ref group=M>Cox (1881), p. 186 for Weierstrass coordinates; (1881/82), pp. 193–194 for Lorentz transformation. On p. 193, the misprinted expression <math>x^{2}-y^{2}-z^{2}</math> should read <math>z^{2}-y^{2}-x^{2}</math></ref> :<math>\begin{matrix}z^{2}-x^{2}-y^{2}=1\\ z^{2}-y^{2}-x^{2}=Z^{2}-Y^{2}-X^{2}\\ \hline \begin{align}x & =l_{1}X+l_{2}Y+l_{3}Z\\ y & =m_{1}X+m_{2}Y+m_{3}Z\\ z & =n_{1}X+n_{2}Y+n_{3}Z\\ \\ X & =l_{1}x+m_{1}y-n_{1}z\\ Y & =l_{2}x+m_{2}y-n_{2}z\\ Z & =l_{3}x+m_{3}y-n_{3}z \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} & =0\\ \\ l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1\\ l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0\\ n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1 \end{align} }</math> with <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1 \end{align} }</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' up to a sign change in the inverse transformation.</p> Cox also gave the Weierstrass coordinates and their transformation in hyperbolic space:<ref group=M>Cox (1881), pp. 199, 206–207</ref> :<math>\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\ w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\ \hline \begin{align}x & =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\ y & =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\ z & =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\ w & =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\ \\ x' & =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\ y' & =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\ z' & =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\ w' & =l_{4}x+m_{4}y+n_{4}z-r_{4}w \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} & =1\\ l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} & =1\\ l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} & =0\\ l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} & =0\\ l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} & =0\\ l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}</math> with <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' up to a sign change in both the first as well as inverse transformation.</p> === {{anchor|Hill}} Hill (1882) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations<ref group=M>Hill (1882), pp. 323–325</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\ k\left(\sin^{2}E+\cos^{2}E-1\right)\\ \hline \begin{align} & & \cos E' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ & \mathbf{(1)} & \sin E' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \hline \\ & & x & =\alpha u+\alpha'u'+\alpha''u''\\ & & y & =\beta u+\beta'u'+\beta''u''\\ & & z & =\gamma u+\gamma'u'+\gamma''u''\\ & \mathbf{(2)}\\ & & u & =-\alpha x-\beta y+\gamma z\\ & & u' & =\alpha'x+\beta'y'-\gamma'z\\ & & u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =-1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =1\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ (k=-1)\\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =-k\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .</p> === {{anchor|Picard}} Picard (1882-1884) – Quadratic forms === [[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to "hyperfuchsian" functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:<ref group=M>Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system</ref><ref>Dickson (1923), pp. 280-281</ref> :<math>\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\ \\ \mathbf{(1)}\ \begin{align}x & =M_{1}X+P_{1}Y+R_{1}Z\\ y & =M_{2}X+P_{2}Y+R_{2}Z\\ z & =M_{3}X+P_{3}Y+R_{3}Z \end{align} \\ \\ \left[\begin{align}[][x,y,z]=\text{complex}\\ \left[x_{0},y_{0},z_{0}\right]=\text{conjugate} \end{align} \right]\\ \\ \hline \\ x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\ x=x'+ix'',\quad y=y'+iy''\\ \\ \mathbf{(2)}\ \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} & =1\\ P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0\\ P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ \\ M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} & =1\\ M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} & =1\\ M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} & =-1\\ \mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} & =0\\ \mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} & =0\\ \mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} & =0\\ \\ \left[\begin{align}[][M,P,R\dots]=\text{complex}\\ \left[\mu,\pi,\rho\dots\right]=\text{conjugate} \end{align} \right] \end{align} }\right.\end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x²+y²-z²=X²+Y²-Z²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:<ref group=M>Picard (1884a), p. 13</ref> :<math>\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\ \hline \begin{align}U & =\mathcal{A}u+\mathcal{B}v\\ V & =\mathcal{C}u+\mathcal{D}v \end{align} \left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} & =1\\ \mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} & =0\\ \mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} & =-1\\ \mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} & =1 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U²-V²=u²-v²''.</p> Or in (1884b):<ref group=M>Picard (1884b), p. 416</ref> :<math>\begin{matrix}xx_{0}+yy_{0}-1=0\\ \hline \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884c):<ref group=M>Picard (1884c), pp. 123–124; 163</ref> :<math>\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\ \hline \mathbf{(1)}\ \begin{align}U & =Mu+Pv+Rw\\ V & =M'u+P'v+R'w\\ W & =M''u+P''v+R''w\\ \\ u & =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\ v & =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\ w & =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W \end{align} \left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} & =1\\ PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} & =1\\ RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} & =-1\\ MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} & =0\\ MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} & =0\\ PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} & =0\\ \\ MM_{0}+PP_{0}-RR_{0} & =1\\ M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} & =1\\ M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} & =-1\\ M_{0}M'+P_{0}P'-R_{0}R' & =0\\ M_{0}M''+P_{0}P''-R_{0}R'' & =0\\ M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' & =0 \end{align} }\right.\\ \hline \text{Invariance of unit hypersphere:}\\ \mathbf{(2)}\ \begin{align}\xi' & =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\ \eta' & =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''} \end{align} \left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} & =1\\ BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} & =1\\ CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} & =-1\\ AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} & =0\\ AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} & =0\\ BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U²+V²-W²=u²+v²-w²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing <math>\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1</math>.</p> === {{anchor|Callandreau}} Callandreau (1885) – Homography === Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations<ref group=M>Callandreau (1885), pp. A.7; A.12</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\ {\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\ \hline \begin{align}\cos\varepsilon' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \sin\varepsilon' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T} \end{align} \left|{\scriptstyle \begin{align} & \left(k=1\right)\\ \alpha^{2}+\beta^{2}-\gamma^{2} & =-k & \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =+k & \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =+k & \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-1 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-1 & \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =+1 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>.</p> ==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations=== In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions.<ref group=M>Lie (1885/86), p. 411</ref> In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):<ref group=M>Werner (1889), pp. 4, 28</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\ \hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\ \text{where}\\ \left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\ \left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\ \left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right] \end{matrix}</math> More generally, Lie (1890)<ref group=M>Lie (1890a), p. 295;</ref> defined non-Euclidean motions in terms of two forms <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0</math> in which the imaginary form with <math>+1</math> denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:<ref group=M>Lie (1890a), p. 311</ref> :<math>\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\ \hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n) \end{matrix}</math> Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form: :<math>x^{2}+y^{2}-1=0</math><ref group=M>Lie (1893), p. 474</ref> or <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0</math><ref group=M>Lie (1893), p. 479</ref> or <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math>.<ref group=M>Lie (1893), p. 481</ref> <p style="background-color:Beige;border:1px solid black">The group of hyperbolic motions is isomorphic to the Lorentz group. The interval <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math> becomes the Lorentz interval <math>x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0</math> by setting <br><math>(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)</math></p> ==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates=== [[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':<ref group=M>Gérard (1892), pp. 40–41</ref> :<math>\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\ X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\ \hline \begin{align}X & =aX'+a'Y'+a''Z'\\ Y & =bX'+b'Y'+b''Z'\\ Z & =cX'+c'Y'+c''Z'\\ \\ X' & =aX+bY-cZ\\ Y' & =a'X+b'Y-c'Z\\ Z' & =-a''X-b''Y+c''Z \end{align} \left|\begin{align}a^{2}+b^{2}-c^{2} & =1\\ a^{\prime2}+b^{\prime2}-c^{\prime2} & =1\\ a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} & =-1\\ aa'+bb'-cc' & =0\\ a'a''+b'b''-c'c'' & =0\\ a''a+b''b-c''c & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> He gave the case of translation as follows:<ref group=M name=gerard>Gérard (1892), pp. 40–41</ref> :<math>\begin{align}X & =Z_{0}X'+X_{0}Z'\\ Y & =Y'\\ Z & =X_{0}X'+Z_{0}Z' \end{align} \ \text{with}\ \begin{align}X_{0} & =\operatorname{sh}OO'\\ Z_{0} & =\operatorname{ch}OO' \end{align} </math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz boost ({{equationNote|3b}}).</p> ==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates=== [[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:<ref group=M>Hausdorff (1899), p. 165, pp. 181-182</ref> :<math>\begin{matrix}p^{2}-x^{2}-y^{2}=1\\ \hline \begin{align}x & =a_{1}x'+a_{2}y'+x_{0}p'\\ y & =b_{1}x'+b{}_{2}y'+y_{0}p'\\ p & =e_{1}x'+e_{2}y'+p_{0}p'\\ \\ x' & =a_{1}x+b_{1}y-e_{1}p\\ y' & =a_{2}x+b_{2}y-e_{2}p\\ -p' & =x_{0}x+y_{0}y-p_{0}p \end{align} \left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} & =1\\ a_{2}^{2}+b_{2}^{2}-e_{2}^{2} & =1\\ -x_{0}^{2}-y_{0}^{2}+p_{0}^{2} & =1\\ a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} & =0\\ a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} & =0\\ a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} & =0\\ \\ a_{1}^{2}+a_{2}^{2}-x_{0}^{2} & =1\\ b_{1}^{2}+b_{2}^{2}-y_{0}^{2} & =1\\ -e_{1}^{2}-e_{2}^{2}+p_{0}^{2} & =1\\ b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} & =0\\ a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} & =0\\ a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates === In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting <math>k=\sqrt{-1}R</math> with ''R'' as real quantity):<ref group=M>Woods (1901/02), p. 98, 104</ref> :<math>\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\ \hline \begin{align}u' & =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ v' & =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ w' & =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}} \end{align} \left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} & =k^{2}\\ (i=1,2,3)\\ k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} & =1\\ \alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} & =0\\ (i,h=1,2,3,4;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''²=-1.</p> Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with <math>k=\sqrt{-1}k</math> for hyperbolic space):<ref group=M>Woods (1903/05), pp. 45–46; p. 48)</ref> :<math>\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\ ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\ \hline \begin{align}x_{1}^{\prime} & =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\ x_{2}^{\prime} & =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\ x_{3}^{\prime} & =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\ x_{0}^{\prime} & =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0} \end{align} \left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) & =1\\ \delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) & =k^{2}\\ (i=1,2,3)\\ \delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) & =0\\ (i,h=0,1,2,3;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''²=-1.</p> ==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates=== [[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''<ref group=M>Liebmann (1904/05), p. 168; pp. 175–176</ref> :<math>\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\ \hline \begin{align}x_{1} & =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\ y_{1} & =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\ x_{1} & =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\ \\ x & =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\ y & =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\ p & =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1} \end{align} \left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} & =1\\ -\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} & =1\\ -\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} & =1\\ \alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} & =0\\ \alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} & =0\\ \alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==References== ===Historical mathematical sources=== {{reflist|3|group=M}} *{{#section:History of Topics in Special Relativity/mathsource|apo}} *{{#section:History of Topics in Special Relativity/mathsource|bour56att}} *{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} *{{#section:History of Topics in Special Relativity/mathsource|chas29}} *{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} *{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} *{{#section:History of Topics in Special Relativity/mathsource|fris76}} *{{#section:History of Topics in Special Relativity/mathsource|gau98}} *{{#section:History of Topics in Special Relativity/mathsource|gau18}} *{{#section:History of Topics in Special Relativity/mathsource|ger92}} *{{#section:History of Topics in Special Relativity/mathsource|gud30}} *{{#section:History of Topics in Special Relativity/mathsource|haus99}} *{{#section:History of Topics in Special Relativity/mathsource|hill82}} *{{#section:History of Topics in Special Relativity/mathsource|jac27}} *{{#section:History of Topics in Special Relativity/mathsource|jac32a}} *{{#section:History of Topics in Special Relativity/mathsource|jac32b}} *{{#section:History of Topics in Special Relativity/mathsource|jac33}} *{{#section:History of Topics in Special Relativity/mathsource|kil77}} *{{#section:History of Topics in Special Relativity/mathsource|kil79}} *{{#section:History of Topics in Special Relativity/mathsource|kil84}} *{{#section:History of Topics in Special Relativity/mathsource|kil85}} *{{#section:History of Topics in Special Relativity/mathsource|kil93}} *{{#section:History of Topics in Special Relativity/mathsource|kil97}} *{{#section:History of Topics in Special Relativity/mathsource|klei71}} *{{#section:History of Topics in Special Relativity/mathsource|klei73}} *{{#section:History of Topics in Special Relativity/mathsource|lag73}} *{{#section:History of Topics in Special Relativity/mathsource|leb37}} *{{#section:History of Topics in Special Relativity/mathsource|lie85}} *{{#section:History of Topics in Special Relativity/mathsource|lie90}} *{{#section:History of Topics in Special Relativity/mathsource|lie93}} *{{#section:History of Topics in Special Relativity/mathsource|lieb04}} *{{#section:History of Topics in Special Relativity/mathsource|pic82}} *{{#section:History of Topics in Special Relativity/mathsource|pic84a}} *{{#section:History of Topics in Special Relativity/mathsource|pic84b}} *{{#section:History of Topics in Special Relativity/mathsource|pic84c}} *{{#section:History of Topics in Special Relativity/mathsource|poin81a}} *{{#section:History of Topics in Special Relativity/mathsource|poin81b}} *{{#section:History of Topics in Special Relativity/mathsource|poin87}} *{{#section:History of Topics in Special Relativity/mathsource|sal62}} *{{#section:History of Topics in Special Relativity/mathsource|som63}} *{{#section:History of Topics in Special Relativity/mathsource|wedd47}} *{{#section:History of Topics in Special Relativity/mathsource|wern89}} *{{#section:History of Topics in Special Relativity/mathsource|woo01}} *{{#section:History of Topics in Special Relativity/mathsource|woo03}} ===Secondary sources=== {{reflist|3}} {{#section:History of Topics in Special Relativity/secsource|L1}} [[Category:Special Relativity]] [[Category:History of Physics]] qg4of98tv61g2e3ekvf6zy5cwdduae5 2416149 2416148 2022-08-18T13:08:01Z D.H 52339 /* {{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas */ wikitext text/x-wiki {{../Lorentz transformation (header)}} ==Most general Lorentz transformations== ===General quadratic form=== The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as<ref>Bôcher (1907), chapter X</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}\begin{align}q=\sum_{0}^{n}A_{ij}x_{i}x_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align} & =q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\ b=\sum_{0}^{n}A_{ij}x_{i}y_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y} & =b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}' \end{align} \quad\left(A_{ij}=A_{ji}\right)\\ \hline \left.\begin{align}x_{i}^{\prime} & =\sum_{j=0}^{n}g_{ij}x_{j}=\mathbf{g}\cdot\mathbf{x}\\ x_{i} & =\sum_{j=0}^{n}g_{ij}^{(-1)}x_{j}^{\prime}=\mathbf{g}^{-1}\cdot\mathbf{x}' \end{align} \right|\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}' \end{matrix}</math>|{{equationRef|Q1}}}} The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc. ===Most general Lorentz transformation=== The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:<ref name=ratcliffe>Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1</ref><ref>Naimark (1964), 2 in four dimensions</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} & =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\ -x_{0}y_{0}+\cdots+x_{n}y_{n} & =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime} \end{align} \\ \hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\ \downarrow\\ \begin{align}x_{0}^{\prime} & =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\ x_{1}^{\prime} & =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\ & \dots\\ x_{n}^{\prime} & =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn} \end{align} \\ \\ \mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\ \downarrow\\ \begin{align}x_{0} & =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\ x_{1} & =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\ & \dots\\ x_{n} & =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn} \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} & =\mathbf{g}^{-1}\\ \mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} & =\mathbf{A}\\ \mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} & =\mathbf{A}\\ \\ \end{align} \\ \begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \end{matrix} \end{matrix}</math>|{{equationRef|1a}}}} The invariance of the Lorentz interval with ''n''=1 between axes and conjugate diameters of hyperbolas was known for a long time since [[#Apo|Apollonius (ca. 200 BC)]]. Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.<ref>Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.</ref><ref>Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.</ref> They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881/82)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the "absolute" form <math>-x_{0}^{2}+\cdots+x_{n}^{2}=0</math> as discussed by [[#Klein|Klein (1871-73)]].<ref group=M>Killing (1885), p. 71</ref><ref>Müller (1910), p. 661, in particular footnote 247.</ref><ref>Sommerville (1911), p. 286, section K6.</ref> In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> by [[#Killing3|Killing (1888-1897)]]. ===Most general Lorentz transformation of velocity=== If <math>x_{i},\ x_{i}^{\prime}</math> in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates <math>u_{s},\ u_{s}^{\prime}</math> follow by :<math>\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)</math> defined by <math>u_{1}^{2}+u_{2}^{2}+\dots+u_{n}^{2}\le1</math> so that the Lorentz transformation becomes a [[w:homography]] inside the [[w:unit hypersphere]], which [[w:John Lighton Synge]] called "the most general formula for the composition of velocities" in terms of special relativity<ref>Synge (1955), p. 129 for ''n''=3</ref> (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})): {{NumBlk|:|<math>\begin{align}u_{s}^{\prime} & =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\ \\ u_{s} & =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}} \end{align} \left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \right.</math>|{{equationRef|1b}}}} Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere <math>-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0</math> were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]]. ==Historical notation== ==={{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas === [[w:Apollonius of Perga]] (c. 240–190 BC) defined the following proposition related to conjugate diameters of hyperbolas, which was translated and adapted to the modern reader by [[w:Thomas Heath (classicist)|w:Thomas Heath]] as follows:<ref group=M>Apollonius/Heath (1896), Proposition 129; (Apollonius, Book VII, Prop. 13).</ref> :[..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. <p style="background-color:Beige;border:1px solid black">Apollonius' result indicates the invariance of the Lorentz interval <math>-x_{0}^{\prime2}+x_{1}^{\prime2}=-x_{0}^{2}+x_{1}^{2}</math>, by which Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' can be interpreted as mapping from one pair of axes of a hyperbola to a pair of conjugate diameters.</p> This result was well known to modern authors, such as [[w:Guillaume de l'Hôpital]] (1707) who used [[w:analytic geomety]] in order to demonstrate the following theorem:<ref group=M>l'Hôpital (1707), Third book, Prop. XII, p. 76.</ref> :The difference of the squares of any two conjugate diameters ''Mm, Ss'' is equal to the difference of the squares of the two axes ''Aa, Bb.'' We have to prove that <math>\overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}</math>, or that <math>\overline{CM}^{2}-\overline{CS}^{2}=\overline{CA}^{2}-\overline{CB}^{2}</math>. ===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}=== After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows<ref group=M>Lagrange (1773/75), section 22</ref> :<math>\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\ \hline \begin{align}y & =Ms+Nx\\ z & =ms+nx \end{align} \left|\begin{matrix}\begin{align}P & =pM^{2}+2qMm+rm^{2}\\ Q & =pMN+q(Mn+Nm)+rmn\\ R & =pN^{2}+2qNn+rn^{2} \end{align} \\ \downarrow\\ PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2} \end{matrix}\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.</p> ==={{anchor|Gauss}} Gauss (1798–1818)=== ===={{anchor|Gauss1}} Binary quadratic forms==== The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:<ref group=M>Gauss (1798/1801), articles 157–158;</ref> :<math>\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\ F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\ \hline \begin{align}x & =\alpha x'+\beta y'\\ y & =\gamma x'+\delta y'\\ \\ x' & =\delta x-\beta y\\ y' & =-\gamma x+\alpha y \end{align} \left|\begin{matrix}\begin{align}a' & =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\ b' & =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\ c' & =a\beta^{2}+2b\beta\delta+c\delta^{2} \end{align} \\ \downarrow\\ b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2} \end{matrix}\right. \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called "proper equivalent" if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.<ref group=M>Gauss (1798/1801), section 159</ref> <p style="background-color:Beige;border:1px solid black">The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.</p> ===={{anchor|Gauss2}} Ternary quadratic forms==== Gauss (1798/1801)<ref group=M>Gauss (1798/1801), articles 266–285</ref> also discussed ternary quadratic forms with the general expression :<math>\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\\ g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)\\ \hline \begin{align}x & =\alpha y+\beta y'+\gamma y''\\ x' & =\alpha'y+\beta'y'+\gamma'y''\\ x'' & =\alpha''y+\beta''y'+\gamma''y'' \end{align} \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x²+y²+z²'', or indefinite in the case of different signs such as ''x²+y²-z²''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:<ref group=M>Gauss (1798/1801), article 277</ref> :<math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\\ \left(\begin{matrix}1, & -1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval <math>x^{2}+x^{\prime2}-x^{\prime\prime2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting: <br><math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)=\left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right)</math></p> ===={{anchor|Gauss4}} Homogeneous coordinates==== Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression :<math>(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E</math> into :<math>G+G'\cos T^{2}+G''\sin T^{2}</math> in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:<ref group=M>Gauss (1818), pp. 5–10</ref> :<math>\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\ k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\ \hline \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =k & \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =-k\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-k & \beta\beta-\beta'\beta'-\beta''\beta'' & =-k\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-k & \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =+k\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} }\right.\\ \hline k=1\\ \begin{align}t\cos E & =\alpha+\alpha'\cos T+\alpha''\sin T\\ t\sin E & =\beta+\beta'\cos T+\beta''\sin T\\ t & =\gamma+\gamma'\cos T+\gamma''\sin T \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}}) ''(n=2)''. Further setting <math>[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that :<math>aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz</math> can be transformed into :<math>Guu+G'u'u'+G''u''u''</math>, in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:<ref group=M>Gauss (1818), pp. 9–10</ref> :<math>\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =-\alpha x-\beta y+\gamma z\\ u' & =\alpha'x+\beta'y-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right.</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' satisfying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>, and can be related to Gauss' previous equations in terms of homogeneous coordinates <math>\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]</math>.</p> ==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation in 1827:<ref group=M>Jacobi (1827), p. 235, 239–240</ref> :<math>{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\ k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\ \hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P & =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\cos\vartheta & =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\sin\vartheta & =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \\ \cos\psi & =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\cos\varphi & =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\sin\varphi & =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta} \end{align} \\ \\ \hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z & =m\\ \alpha'\mu+\beta'x+\gamma'y+\delta'z & =m'\\ \alpha''\mu+\beta''x+\gamma''y+\delta''z & =m''\\ \alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z & =m'''\\ \\ Am+A'm'+A''m''+A'''m''' & =\mu\\ Bm+B'm'+B''m''+B'''m''' & =x\\ Cm+C'm'+C''m''+C'''m''' & =y\\ Dm+D'm'+D''m''+D'''m''' & =z\\ \\ \end{align} \\ \begin{align}\alpha & =-kA, & \beta & =-kB, & \gamma & =-kC, & \delta & =kD,\\ \alpha' & =kA', & \beta' & =kB', & \gamma' & =kC', & \delta' & =-kD',\\ \alpha'' & =kA'', & \beta'' & =kB'', & \gamma'' & =kC'', & \delta'' & =-kD'',\\ \alpha''' & =kA''', & \beta''' & =kB''', & \gamma''' & =kC''', & \delta''' & =-kD''', \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =-k\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' & =k\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' & =k\\ \alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' & =k\\ \alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' & =0\\ \alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' & =0\\ \alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' & =0\\ \alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' & =0\\ \alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' & =0\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' & =0\\ \\ -\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' & =k\\ -\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' & =k\\ -\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' & =k\\ -\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' & =-k\\ -\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' & =0\\ -\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' & =0\\ -\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' & =0\\ -\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' & =0\\ -\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' & =0\\ -\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' & =0 \end{align} \end{matrix}} \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] & =\left[u_{1},\ u_{2},\ u_{3}\right]\\{} [\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] & =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right] \end{align} }</math> and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing <math>m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}</math>.</p> Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):<ref group=M>The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.</ref> :<math>{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\ \mathbf{(1)}\begin{align}x & =\alpha s+\alpha's'+\alpha''s''\\ y & =\beta s+\beta's'+\beta''s''\\ z & =\gamma s+\gamma's'+\gamma''s''\\ \\ s & =\alpha x+\beta y+\gamma z\\ s' & =\alpha'x+\beta'y+\gamma'z\\ s'' & =\alpha''x+\beta''y+\gamma''z \end{align} \left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma & =1 & \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' & =1\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma' & =1 & \beta\beta+\beta'\beta'+\beta''\beta'' & =1\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' & =1 & \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' & =1\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma+\beta'\gamma'+\beta''\gamma'' & =0\\ \alpha''\alpha+\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' & =0\\ \alpha\alpha'+\beta\beta'+\gamma\gamma' & =0 & \alpha\beta+\alpha'\beta'+\alpha''\beta'' & =0 \end{align} \right.\\ \hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\ \left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\ \hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\ \left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\ \hline \begin{align}\cos\phi & =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \cos\eta & =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\ \sin\phi & =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \sin\eta & =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi} \end{align} \end{matrix}\\ \hline \\ \mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\ \hline \begin{align}y & =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, & s' & =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\ z & =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, & s'' & =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z}, \end{align} \end{matrix} \end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma & =1\\ \alpha'\alpha'-\beta'\beta'-\gamma'\gamma' & =-1\\ \alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' & =-1\\ \alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' & =0\\ \alpha''\alpha-\beta''\beta-\gamma''\gamma & =0\\ \alpha\alpha'-\beta\beta'-\gamma\gamma' & =0\\ \\ \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =1\\ \beta\beta-\beta'\beta'-\beta''\beta'' & =-1\\ \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =-1\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} \right. \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.</p> Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:<ref group =M>Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.</ref> :<math>{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\ \hline \mathbf{(1)\ }\begin{align}y_{\varkappa} & =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\ x_{\varkappa} & =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\ \\ \frac{y_{\varkappa}}{y_{n}} & =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\ \frac{x_{\varkappa}}{x_{n}} & =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}} \end{align} \left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} & =0\\ \alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} & =1\\ \\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} & =0\\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} & =1 \end{align} \right.\\ \hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\ \alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\ \hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} & =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\ \\ \xi_{\varkappa} & =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}} \end{align} \\ \hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1 \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.</p> He also stated the following transformation leaving invariant the Lorentz interval:<ref group=M>Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.</ref> :<math>\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\ \hline {\scriptstyle \begin{align}u & =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\ u_{1} & =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\ & \dots\\ u_{n-1} & =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\ \\ w & =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\ w_{1} & =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\ & \dots\\ w_{n-1} & =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1} \end{align} \left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =-1\\ \alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =0\\ \alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} & =0\\ \\ \alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} & =-1\\ \alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} & =0\\ \alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} & =0 \end{align} \text{ }\right.} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.</p> ==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids === [[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:<ref group=M>Chasles (1829), p. 139</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(a)\\ \\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(b)\\ \\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(c)\\ \\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(d) \end{matrix}</math> He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:<ref group=M>Chasles (1829), p. 141</ref> :<math>\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\ \gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\ \alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\ \beta''\quad\Rightarrow\quad\beta''\sqrt{-1} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:<ref group=M>Chasles (1829), pp. 143-144</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}x & =lx'+l'y'+l''z'\\ y & =mx'+m'y'+m''z'\\ z & =nx'+n'y'+n''z' \end{align} \\ \left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} & =0\\ \frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} & =0\\ \frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} & =0 \end{align} \right\} \\ \hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\ \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1 \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Chasles defined the conditional equations of ''l,m,n'' in the same way as those of <math>\alpha,\beta,\gamma</math> in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.</p> He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:<ref group=M>Chasles (1829), pp. 145-146</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =a^{2}\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =b^{2}\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(A)\\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(B)\\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =a^{2}\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =b^{2}\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(C)\\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(D) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates=== [[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation<ref group=M>Lebesgue (1837), pp. 338-341</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\ \hline {\scriptstyle \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\ \dots\\ y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} & =1 & (10)\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} & =0 & (11)\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} & =1 & (13)\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} & =0 & (14) \end{align} \right.} \end{matrix}</math> In order to achieve the invariance of the Lorentz interval<ref group=M>Lebesgue (1837), pp. 353–354</ref> :<math>x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}</math> he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign. <p style="background-color:Beige;border:1px solid black">These instructions give Lorentz transformation ({{equationNote|1a}}) in the form: <br><math>{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \hline \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\ \dots\\ -y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} & =1\\ a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} & =-1\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} & =0\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} & =1\\ a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} & =-1\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} & =0 \end{align} \right. \end{matrix}}</math></p> He went on to redefine the variables of the Lorentz interval and its transformation:<ref group=M>Lebesgue (1837), pp. 353–355</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \downarrow\\ \begin{align}x_{1} & =x_{n}\cos\theta_{1}, & x_{2} & =x_{n}\cos\theta_{2},\dots & x_{n-1} & =x_{n}\cos\theta_{n-1}\\ y_{1} & =y_{n}\cos\phi_{1}, & y_{2} & =y_{n}\cos\phi_{2},\dots & y_{n-1} & =y_{n}\cos\phi_{n-1} \end{align} \\ \downarrow\\ \cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\ \cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\ \hline \\ \cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\ (i=1,2,3\dots n) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Setting <math>[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]</math> it is equivalent to Lorentz transformation ({{equationNote|1b}}).</p> ==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids=== Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&alpha;), from which he derived equations (&beta;) and others:<ref group=M>Weddle (1847), p. 274</ref> :<math>\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1, & l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1, & l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} & =0\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1, & l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0 \end{align} \right\} & \dots(\alpha)\\ \\ \left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1, & l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1, & l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} & =0\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1, & m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0 \end{align} \right\} & \dots(\beta) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>, he defined three conjugate points <math>(x_{1}\dots,y_{1}\dots,z_{1}\dots)</math> on those two conjugate hyperboloids, related to each other in the same way as equations (&alpha;, &beta;) stated above:<ref group=M>Weddle (1847), pp. 275-276</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} & =0\\ \frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} & =0\\ \frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} & =0 \end{align} \quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} & =1\\ \frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} & =1\\ \frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} & =-1 \end{align} \\ \begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} & =a^{2}\\ y_{1}^{2}+y_{2}^{2}-y_{3}^{2} & =b^{2}\\ z_{1}^{2}+z_{2}^{2}-z_{3}^{2} & =-c^{2} \end{align} \quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} & =0\\ x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} & =0\\ y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} & =0 \end{align} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black"> These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:<ref group=M>Bour (1856), pp. 61; 64–65</ref> :<math>\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\ \hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \\ \hline \\ k=+1\\ t=\gamma+\gamma'\cos T+\gamma''\sin T,\\ 1=u,\ \cos T=u',\ \sin T=u',\\ t=z,\ t\cos E=x,\ t\sin E=y\\ \downarrow\\ \mathbf{(2)}\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =\gamma z-\alpha x-\beta y\\ u' & =\alpha'x+\beta'y'-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} & =k\\ -\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} & =-k\\ -\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} & =-k\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =k\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} } \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>. Furthermore, setting <math>[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math> in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> === {{anchor|Somov}} Somov (1863) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:<ref group=M>Somov (1863), pp. 12–14; p. 18 for differentials.</ref> :<math>\begin{matrix}\left.\begin{align}\cos\phi & =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\ \sin\phi & =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''} \end{align} \right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\ \cos^{2}\psi+\cos^{2}\psi=1 \end{matrix}\\ \hline \mathbf{(1)}\ \begin{align}\cos\phi & =x, & \cos\psi & =x'\\ \sin\phi & =y, & \sin\psi & =y' \end{align} \ \left|\begin{align}x & =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\ y & =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=1\\ x^{\prime2}+y^{\prime2}=1 \end{matrix}\\ \hline \begin{align}\cos\phi & =\frac{x}{z}, & \cos\psi & =\frac{x'}{z'}\\ \sin\phi & =\frac{y}{z}, & \sin\psi & =\frac{y'}{z'} \end{align} \ \left|\begin{align}\frac{x}{z} & =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\ \frac{y}{z} & =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\ x^{\prime2}+y^{\prime2}=z^{\prime2} \end{matrix}\\ \hline \mathbf{(2)}\ \left.\begin{align}x & =mx'+ny'+sz'\\ y & =m'x'+n'y'+s'z'\\ z & =m''x'+n''y'+s''z'\\ \\ x' & =mx+m'y-m''z\\ y' & =nx+n'y-n''z\\ z' & =-sx-s'y+s''z\\ \\ dx & =mdx'+ndy'+sdz'\\ dy & =m'dx'+n'dy'+s'dz'\\ dz & =m''dx'+n''dy'+s''dz' \end{align} \right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} & =1\\ n^{2}+n^{\prime2}-n^{\prime\prime2} & =1\\ -s^{2}-s^{\prime2}+s^{\prime\prime2} & =1\\ ns+n's'-n''s'' & =0\\ sm+s'm'-s''m'' & =0\\ mn+m'n'-m''n'' & =0\\ \\ m^{2}+n^{2}-s^{2} & =1\\ m^{\prime2}+n^{\prime2}-s^{\prime2} & =1\\ -m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} & =1\\ -m'm''-n'n''+s's'' & =0\\ -m''m-n''n+s''s & =0\\ mm'+nn'-ss' & =0 \end{align} }\\ dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Klein}} Klein (1871-73) – Cayley absolute and non-Euclidean geometry=== Elaborating on [[w:Arthur Cayley]]'s (1859) definition of an "absolute" ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a "fundamental [[w:conic section]]" in order to discuss motions such as rotation and translation in the non-Euclidean plane.<ref group=M>Klein (1871), pp. 601–602</ref> This was elaborated in (1873) when he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to a two-sheet [[w:hyperboloid]].<ref group=M>Klein (1873), pp. 127-128</ref> <p style="background-color:Beige;border:1px solid black">Klein's representation of hyperbolic space in terms of a two-sheet hyperboloid and its accompanied quadratic form suggests that Lorentz transformations can be geometrically interpreted as motions or isometries in hyperbolic space.</p> ==={{anchor|Killing}} Killing (1878–1893)=== ===={{anchor|Killing1}} Weierstrass coordinates==== [[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form :<math>k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}</math><ref group=M>Killing (1877/78), p. 74; Killing (1880), p. 279</ref> with <math>ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}</math><ref group=M>Killing (1880), eq. 25 on p. 283</ref> or<ref group=M>Killing (1880), p. 283</ref> :<math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> where ''k'' is the reciprocal measure of curvature, <math>k^{2}=\infty</math> denotes [[w:Euclidean geometry]], <math>k^{2}>0</math> [[w:elliptic geometry]], and <math>k^{2}<0</math> hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.<ref group=M>Killing (1877/78), eq. 25 on p. 283</ref> In (1879/80) he tried to formulate the corresponding transformations by plugging <math>k^{2}</math> into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:<ref group=M>Killing (1879/80), p. 274</ref> <math>\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\ \hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\ -k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\ k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, & -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}} \end{matrix}\\ \left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right) \end{matrix}</math> In (1885) he wrote the Weierstrass coordinates and their transformation as follows:<ref group=M>Killing (1885), pp. 18, 28–30, 53</ref> :<math>\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\ k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\ ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\ \hline \begin{align}k^{2}p' & =k^{2}wp+w'x+w''y\\ x' & =ap+a'x+a''y\\ y' & =bp+b'x+b''y\\ \\ k^{2}p & =k^{2}wp'+ax'+by'\\ x & =w'p'+a'x+b'y'\\ y & =w''p'+a''x'+b''y' \end{align} \left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} & =k^{2}\\ \frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} & =1\\ \frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} & =1\\ aw+a'w'+a''w'' & =0\\ bw+b'w'+b''w'' & =0\\ \frac{ab}{k^{2}}+a'b'+a''b'' & =0\\ \\ k^{2}w^{2}+a^{2}+b^{2} & =k^{2}\\ \frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} & =1\\ \frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} & =1\\ ww'+aa'+bb' & =0\\ ww''+aa''+bb'' & =0\\ \frac{w'w''}{k^{2}}+a'a''+b'b'' & =0 \end{align} }\right. \end{matrix}</math> In (1885) he also gave the transformation for ''n'' dimensions:<ref group=M>Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222</ref><ref>Ratcliffe (1994), § 3.6</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\ \hline \left.\begin{align}k^{2}\xi_{0} & =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\ \xi_{\varkappa} & =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\ \\ k^{2}x_{0} & =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\ x_{\varkappa} & =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n} \end{align} \right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ \frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} & =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa) \end{align} } \end{matrix}</math> In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.<ref group=M>Killing (1884/85), pp. 4–5</ref> Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting <math>k^{2}=-1</math> and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid <math>p^{2}-x^{2}-y^{2}=1</math> (known as [[w:hyperboloid model]]),<ref group=M>Killing (1885), Note 9 on p. 260</ref><ref name=rey /> by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space. <p style="background-color:Beige;border:1px solid black">All of Killing's transformations between 1879 and 1885 don't work when <math>k^{2}</math> is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> Finally, in (1893) he wrote:<ref group=M>Killing (1893), see pp. 144, 327–328</ref> :<math>\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\ \hline \begin{align}t' & =at+bu+cv\\ u' & =a't+b'u+c'v\\ v' & =a''t+b''u+c''v \end{align} \left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} & =k^{2}\\ k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} & =1\\ k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} & =1\\ k^{2}ab+a'b'+a''b'' & =0\\ k^{2}ac+a'c'+a''c'' & =0\\ k^{2}bc+b'c'+b''c'' & =0 \end{align} \right. \end{matrix}</math> and in ''n'' dimensions<ref group=M>Killing (1893), pp. 314–316, 216–217</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\ ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\ \hline \begin{align}y_{0} & =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\ y_{1} & =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\ & \,\,\,\vdots\\ y_{n} & =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n} \end{align} \left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} & =1\\ k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} & =0\\ (\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> ===={{anchor|Killing3}} Infinitesimal transformations and Lie group==== After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions, Killing (1887/88)<ref group=M>Killing (1887/88a), pp. 274–275</ref> defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere: :<math>\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\ \hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\ \text{where}\\ \left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\ \left[\iota\ne\varkappa\ne\lambda\ne\mu\right] \end{matrix}</math> and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:<ref group=M>Killing (1892), p. 177</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ \hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\ \text{where}\\ \left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f; \end{matrix}</math> In (1897/98) he showed the relation between Weierstrass coordinates <math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> and corrdinates <math>k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0</math> used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:<ref group=M>Killing (1897/98), pp. 255–256</ref> :<math>\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} & (a)\\ k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} & (b) \end{matrix}\\ \hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\ \text{where}\\ \left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\ \left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\ \left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\ \hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\ \downarrow\\ k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\ \hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota} \end{matrix} \end{matrix}</math> He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b). <p style="background-color:Beige;border:1px solid black">Setting <math>k^{2}=-1</math> denotes the group of hyperbolic motions and thus the Lorentz group.</p> === {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates === [[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).<ref>Gray (1997)</ref> He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:<ref group=M name=p1>Poincaré (1881a), pp. 133–134</ref><ref>Dickson (1923), pp. 220–221</ref> :<math>\begin{matrix}\begin{align}F & =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\ & =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\ F & =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\ & =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1 \end{align} \\ \hline \begin{align}\xi' & =\alpha\xi+\beta\eta+\gamma\zeta\\ \eta' & =\alpha'\xi+\beta'\eta+\gamma'\zeta\\ \zeta' & =\alpha''\xi+\beta''\eta+\gamma''\zeta \end{align} \left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1\\ \alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right. \end{matrix}</math> He went on to describe the properties of "hyperbolic coordinates".<ref group=M name=poinc>Poincaré (1881b), p. 333</ref><ref name=rey>Reynolds (1993)</ref> Poincaré mentioned the hyperboloid model also in (1887).<ref group=M>Poincaré (1887), p. 206</ref> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> === {{anchor|Cox}} Cox (1881–1883) – Weierstrass coordinates === [[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)<ref name=guder group=M>Gudermann (1830), §1–3, §18–19</ref> and [[w:George Salmon]] (1862)<ref group=M>Salmon (1862), section 212, p. 165</ref> on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)<ref group=M>Frischauf (1876), pp. 86–87</ref> in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:<ref group=M>Cox (1881), p. 186 for Weierstrass coordinates; (1881/82), pp. 193–194 for Lorentz transformation. On p. 193, the misprinted expression <math>x^{2}-y^{2}-z^{2}</math> should read <math>z^{2}-y^{2}-x^{2}</math></ref> :<math>\begin{matrix}z^{2}-x^{2}-y^{2}=1\\ z^{2}-y^{2}-x^{2}=Z^{2}-Y^{2}-X^{2}\\ \hline \begin{align}x & =l_{1}X+l_{2}Y+l_{3}Z\\ y & =m_{1}X+m_{2}Y+m_{3}Z\\ z & =n_{1}X+n_{2}Y+n_{3}Z\\ \\ X & =l_{1}x+m_{1}y-n_{1}z\\ Y & =l_{2}x+m_{2}y-n_{2}z\\ Z & =l_{3}x+m_{3}y-n_{3}z \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} & =0\\ \\ l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1\\ l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0\\ n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1 \end{align} }</math> with <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1 \end{align} }</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' up to a sign change in the inverse transformation.</p> Cox also gave the Weierstrass coordinates and their transformation in hyperbolic space:<ref group=M>Cox (1881), pp. 199, 206–207</ref> :<math>\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\ w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\ \hline \begin{align}x & =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\ y & =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\ z & =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\ w & =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\ \\ x' & =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\ y' & =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\ z' & =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\ w' & =l_{4}x+m_{4}y+n_{4}z-r_{4}w \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} & =1\\ l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} & =1\\ l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} & =0\\ l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} & =0\\ l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} & =0\\ l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}</math> with <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' up to a sign change in both the first as well as inverse transformation.</p> === {{anchor|Hill}} Hill (1882) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations<ref group=M>Hill (1882), pp. 323–325</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\ k\left(\sin^{2}E+\cos^{2}E-1\right)\\ \hline \begin{align} & & \cos E' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ & \mathbf{(1)} & \sin E' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \hline \\ & & x & =\alpha u+\alpha'u'+\alpha''u''\\ & & y & =\beta u+\beta'u'+\beta''u''\\ & & z & =\gamma u+\gamma'u'+\gamma''u''\\ & \mathbf{(2)}\\ & & u & =-\alpha x-\beta y+\gamma z\\ & & u' & =\alpha'x+\beta'y'-\gamma'z\\ & & u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =-1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =1\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ (k=-1)\\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =-k\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .</p> === {{anchor|Picard}} Picard (1882-1884) – Quadratic forms === [[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to "hyperfuchsian" functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:<ref group=M>Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system</ref><ref>Dickson (1923), pp. 280-281</ref> :<math>\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\ \\ \mathbf{(1)}\ \begin{align}x & =M_{1}X+P_{1}Y+R_{1}Z\\ y & =M_{2}X+P_{2}Y+R_{2}Z\\ z & =M_{3}X+P_{3}Y+R_{3}Z \end{align} \\ \\ \left[\begin{align}[][x,y,z]=\text{complex}\\ \left[x_{0},y_{0},z_{0}\right]=\text{conjugate} \end{align} \right]\\ \\ \hline \\ x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\ x=x'+ix'',\quad y=y'+iy''\\ \\ \mathbf{(2)}\ \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} & =1\\ P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0\\ P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ \\ M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} & =1\\ M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} & =1\\ M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} & =-1\\ \mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} & =0\\ \mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} & =0\\ \mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} & =0\\ \\ \left[\begin{align}[][M,P,R\dots]=\text{complex}\\ \left[\mu,\pi,\rho\dots\right]=\text{conjugate} \end{align} \right] \end{align} }\right.\end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x²+y²-z²=X²+Y²-Z²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:<ref group=M>Picard (1884a), p. 13</ref> :<math>\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\ \hline \begin{align}U & =\mathcal{A}u+\mathcal{B}v\\ V & =\mathcal{C}u+\mathcal{D}v \end{align} \left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} & =1\\ \mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} & =0\\ \mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} & =-1\\ \mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} & =1 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U²-V²=u²-v²''.</p> Or in (1884b):<ref group=M>Picard (1884b), p. 416</ref> :<math>\begin{matrix}xx_{0}+yy_{0}-1=0\\ \hline \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884c):<ref group=M>Picard (1884c), pp. 123–124; 163</ref> :<math>\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\ \hline \mathbf{(1)}\ \begin{align}U & =Mu+Pv+Rw\\ V & =M'u+P'v+R'w\\ W & =M''u+P''v+R''w\\ \\ u & =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\ v & =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\ w & =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W \end{align} \left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} & =1\\ PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} & =1\\ RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} & =-1\\ MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} & =0\\ MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} & =0\\ PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} & =0\\ \\ MM_{0}+PP_{0}-RR_{0} & =1\\ M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} & =1\\ M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} & =-1\\ M_{0}M'+P_{0}P'-R_{0}R' & =0\\ M_{0}M''+P_{0}P''-R_{0}R'' & =0\\ M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' & =0 \end{align} }\right.\\ \hline \text{Invariance of unit hypersphere:}\\ \mathbf{(2)}\ \begin{align}\xi' & =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\ \eta' & =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''} \end{align} \left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} & =1\\ BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} & =1\\ CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} & =-1\\ AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} & =0\\ AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} & =0\\ BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U²+V²-W²=u²+v²-w²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing <math>\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1</math>.</p> === {{anchor|Callandreau}} Callandreau (1885) – Homography === Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations<ref group=M>Callandreau (1885), pp. A.7; A.12</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\ {\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\ \hline \begin{align}\cos\varepsilon' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \sin\varepsilon' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T} \end{align} \left|{\scriptstyle \begin{align} & \left(k=1\right)\\ \alpha^{2}+\beta^{2}-\gamma^{2} & =-k & \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =+k & \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =+k & \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-1 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-1 & \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =+1 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>.</p> ==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations=== In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions.<ref group=M>Lie (1885/86), p. 411</ref> In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):<ref group=M>Werner (1889), pp. 4, 28</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\ \hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\ \text{where}\\ \left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\ \left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\ \left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right] \end{matrix}</math> More generally, Lie (1890)<ref group=M>Lie (1890a), p. 295;</ref> defined non-Euclidean motions in terms of two forms <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0</math> in which the imaginary form with <math>+1</math> denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:<ref group=M>Lie (1890a), p. 311</ref> :<math>\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\ \hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n) \end{matrix}</math> Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form: :<math>x^{2}+y^{2}-1=0</math><ref group=M>Lie (1893), p. 474</ref> or <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0</math><ref group=M>Lie (1893), p. 479</ref> or <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math>.<ref group=M>Lie (1893), p. 481</ref> <p style="background-color:Beige;border:1px solid black">The group of hyperbolic motions is isomorphic to the Lorentz group. The interval <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math> becomes the Lorentz interval <math>x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0</math> by setting <br><math>(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)</math></p> ==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates=== [[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':<ref group=M>Gérard (1892), pp. 40–41</ref> :<math>\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\ X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\ \hline \begin{align}X & =aX'+a'Y'+a''Z'\\ Y & =bX'+b'Y'+b''Z'\\ Z & =cX'+c'Y'+c''Z'\\ \\ X' & =aX+bY-cZ\\ Y' & =a'X+b'Y-c'Z\\ Z' & =-a''X-b''Y+c''Z \end{align} \left|\begin{align}a^{2}+b^{2}-c^{2} & =1\\ a^{\prime2}+b^{\prime2}-c^{\prime2} & =1\\ a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} & =-1\\ aa'+bb'-cc' & =0\\ a'a''+b'b''-c'c'' & =0\\ a''a+b''b-c''c & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> He gave the case of translation as follows:<ref group=M name=gerard>Gérard (1892), pp. 40–41</ref> :<math>\begin{align}X & =Z_{0}X'+X_{0}Z'\\ Y & =Y'\\ Z & =X_{0}X'+Z_{0}Z' \end{align} \ \text{with}\ \begin{align}X_{0} & =\operatorname{sh}OO'\\ Z_{0} & =\operatorname{ch}OO' \end{align} </math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz boost ({{equationNote|3b}}).</p> ==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates=== [[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:<ref group=M>Hausdorff (1899), p. 165, pp. 181-182</ref> :<math>\begin{matrix}p^{2}-x^{2}-y^{2}=1\\ \hline \begin{align}x & =a_{1}x'+a_{2}y'+x_{0}p'\\ y & =b_{1}x'+b{}_{2}y'+y_{0}p'\\ p & =e_{1}x'+e_{2}y'+p_{0}p'\\ \\ x' & =a_{1}x+b_{1}y-e_{1}p\\ y' & =a_{2}x+b_{2}y-e_{2}p\\ -p' & =x_{0}x+y_{0}y-p_{0}p \end{align} \left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} & =1\\ a_{2}^{2}+b_{2}^{2}-e_{2}^{2} & =1\\ -x_{0}^{2}-y_{0}^{2}+p_{0}^{2} & =1\\ a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} & =0\\ a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} & =0\\ a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} & =0\\ \\ a_{1}^{2}+a_{2}^{2}-x_{0}^{2} & =1\\ b_{1}^{2}+b_{2}^{2}-y_{0}^{2} & =1\\ -e_{1}^{2}-e_{2}^{2}+p_{0}^{2} & =1\\ b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} & =0\\ a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} & =0\\ a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates === In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting <math>k=\sqrt{-1}R</math> with ''R'' as real quantity):<ref group=M>Woods (1901/02), p. 98, 104</ref> :<math>\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\ \hline \begin{align}u' & =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ v' & =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ w' & =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}} \end{align} \left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} & =k^{2}\\ (i=1,2,3)\\ k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} & =1\\ \alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} & =0\\ (i,h=1,2,3,4;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''²=-1.</p> Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with <math>k=\sqrt{-1}k</math> for hyperbolic space):<ref group=M>Woods (1903/05), pp. 45–46; p. 48)</ref> :<math>\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\ ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\ \hline \begin{align}x_{1}^{\prime} & =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\ x_{2}^{\prime} & =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\ x_{3}^{\prime} & =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\ x_{0}^{\prime} & =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0} \end{align} \left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) & =1\\ \delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) & =k^{2}\\ (i=1,2,3)\\ \delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) & =0\\ (i,h=0,1,2,3;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''²=-1.</p> ==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates=== [[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''<ref group=M>Liebmann (1904/05), p. 168; pp. 175–176</ref> :<math>\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\ \hline \begin{align}x_{1} & =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\ y_{1} & =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\ x_{1} & =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\ \\ x & =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\ y & =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\ p & =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1} \end{align} \left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} & =1\\ -\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} & =1\\ -\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} & =1\\ \alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} & =0\\ \alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} & =0\\ \alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==References== ===Historical mathematical sources=== {{reflist|3|group=M}} *{{#section:History of Topics in Special Relativity/mathsource|apo}} *{{#section:History of Topics in Special Relativity/mathsource|bour56att}} *{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} *{{#section:History of Topics in Special Relativity/mathsource|chas29}} *{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} *{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} *{{#section:History of Topics in Special Relativity/mathsource|fris76}} *{{#section:History of Topics in Special Relativity/mathsource|gau98}} *{{#section:History of Topics in Special Relativity/mathsource|gau18}} *{{#section:History of Topics in Special Relativity/mathsource|ger92}} *{{#section:History of Topics in Special Relativity/mathsource|gud30}} *{{#section:History of Topics in Special Relativity/mathsource|haus99}} *{{#section:History of Topics in Special Relativity/mathsource|hill82}} *{{#section:History of Topics in Special Relativity/mathsource|jac27}} *{{#section:History of Topics in Special Relativity/mathsource|jac32a}} *{{#section:History of Topics in Special Relativity/mathsource|jac32b}} *{{#section:History of Topics in Special Relativity/mathsource|jac33}} *{{#section:History of Topics in Special Relativity/mathsource|kil77}} *{{#section:History of Topics in Special Relativity/mathsource|kil79}} *{{#section:History of Topics in Special Relativity/mathsource|kil84}} *{{#section:History of Topics in Special Relativity/mathsource|kil85}} *{{#section:History of Topics in Special Relativity/mathsource|kil93}} *{{#section:History of Topics in Special Relativity/mathsource|kil97}} *{{#section:History of Topics in Special Relativity/mathsource|klei71}} *{{#section:History of Topics in Special Relativity/mathsource|klei73}} *{{#section:History of Topics in Special Relativity/mathsource|lag73}} *{{#section:History of Topics in Special Relativity/mathsource|leb37}} *{{#section:History of Topics in Special Relativity/mathsource|lie85}} *{{#section:History of Topics in Special Relativity/mathsource|lie90}} *{{#section:History of Topics in Special Relativity/mathsource|lie93}} *{{#section:History of Topics in Special Relativity/mathsource|lieb04}} *{{#section:History of Topics in Special Relativity/mathsource|pic82}} *{{#section:History of Topics in Special Relativity/mathsource|pic84a}} *{{#section:History of Topics in Special Relativity/mathsource|pic84b}} *{{#section:History of Topics in Special Relativity/mathsource|pic84c}} *{{#section:History of Topics in Special Relativity/mathsource|poin81a}} *{{#section:History of Topics in Special Relativity/mathsource|poin81b}} *{{#section:History of Topics in Special Relativity/mathsource|poin87}} *{{#section:History of Topics in Special Relativity/mathsource|sal62}} *{{#section:History of Topics in Special Relativity/mathsource|som63}} *{{#section:History of Topics in Special Relativity/mathsource|wedd47}} *{{#section:History of Topics in Special Relativity/mathsource|wern89}} *{{#section:History of Topics in Special Relativity/mathsource|woo01}} *{{#section:History of Topics in Special Relativity/mathsource|woo03}} ===Secondary sources=== {{reflist|3}} {{#section:History of Topics in Special Relativity/secsource|L1}} [[Category:Special Relativity]] [[Category:History of Physics]] q6o09wre9gmtdx10jzvpen6ge8fpsul 2416151 2416149 2022-08-18T13:17:17Z D.H 52339 /* Historical mathematical sources */ wikitext text/x-wiki {{../Lorentz transformation (header)}} ==Most general Lorentz transformations== ===General quadratic form=== The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as<ref>Bôcher (1907), chapter X</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}\begin{align}q=\sum_{0}^{n}A_{ij}x_{i}x_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align} & =q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\ b=\sum_{0}^{n}A_{ij}x_{i}y_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y} & =b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}' \end{align} \quad\left(A_{ij}=A_{ji}\right)\\ \hline \left.\begin{align}x_{i}^{\prime} & =\sum_{j=0}^{n}g_{ij}x_{j}=\mathbf{g}\cdot\mathbf{x}\\ x_{i} & =\sum_{j=0}^{n}g_{ij}^{(-1)}x_{j}^{\prime}=\mathbf{g}^{-1}\cdot\mathbf{x}' \end{align} \right|\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}' \end{matrix}</math>|{{equationRef|Q1}}}} The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc. ===Most general Lorentz transformation=== The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:<ref name=ratcliffe>Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1</ref><ref>Naimark (1964), 2 in four dimensions</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} & =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\ -x_{0}y_{0}+\cdots+x_{n}y_{n} & =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime} \end{align} \\ \hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\ \downarrow\\ \begin{align}x_{0}^{\prime} & =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\ x_{1}^{\prime} & =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\ & \dots\\ x_{n}^{\prime} & =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn} \end{align} \\ \\ \mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\ \downarrow\\ \begin{align}x_{0} & =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\ x_{1} & =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\ & \dots\\ x_{n} & =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn} \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} & =\mathbf{g}^{-1}\\ \mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} & =\mathbf{A}\\ \mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} & =\mathbf{A}\\ \\ \end{align} \\ \begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \end{matrix} \end{matrix}</math>|{{equationRef|1a}}}} The invariance of the Lorentz interval with ''n''=1 between axes and conjugate diameters of hyperbolas was known for a long time since [[#Apo|Apollonius (ca. 200 BC)]]. Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.<ref>Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.</ref><ref>Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.</ref> They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881/82)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the "absolute" form <math>-x_{0}^{2}+\cdots+x_{n}^{2}=0</math> as discussed by [[#Klein|Klein (1871-73)]].<ref group=M>Killing (1885), p. 71</ref><ref>Müller (1910), p. 661, in particular footnote 247.</ref><ref>Sommerville (1911), p. 286, section K6.</ref> In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> by [[#Killing3|Killing (1888-1897)]]. ===Most general Lorentz transformation of velocity=== If <math>x_{i},\ x_{i}^{\prime}</math> in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates <math>u_{s},\ u_{s}^{\prime}</math> follow by :<math>\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)</math> defined by <math>u_{1}^{2}+u_{2}^{2}+\dots+u_{n}^{2}\le1</math> so that the Lorentz transformation becomes a [[w:homography]] inside the [[w:unit hypersphere]], which [[w:John Lighton Synge]] called "the most general formula for the composition of velocities" in terms of special relativity<ref>Synge (1955), p. 129 for ''n''=3</ref> (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})): {{NumBlk|:|<math>\begin{align}u_{s}^{\prime} & =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\ \\ u_{s} & =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}} \end{align} \left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \right.</math>|{{equationRef|1b}}}} Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere <math>-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0</math> were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]]. ==Historical notation== ==={{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas === [[w:Apollonius of Perga]] (c. 240–190 BC) defined the following proposition related to conjugate diameters of hyperbolas, which was translated and adapted to the modern reader by [[w:Thomas Heath (classicist)|w:Thomas Heath]] as follows:<ref group=M>Apollonius/Heath (1896), Proposition 129; (Apollonius, Book VII, Prop. 13).</ref> :[..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. <p style="background-color:Beige;border:1px solid black">Apollonius' result indicates the invariance of the Lorentz interval <math>-x_{0}^{\prime2}+x_{1}^{\prime2}=-x_{0}^{2}+x_{1}^{2}</math>, by which Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' can be interpreted as mapping from one pair of axes of a hyperbola to a pair of conjugate diameters.</p> This result was well known to modern authors, such as [[w:Guillaume de l'Hôpital]] (1707) who used [[w:analytic geomety]] in order to demonstrate the following theorem:<ref group=M>l'Hôpital (1707), Third book, Prop. XII, p. 76.</ref> :The difference of the squares of any two conjugate diameters ''Mm, Ss'' is equal to the difference of the squares of the two axes ''Aa, Bb.'' We have to prove that <math>\overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}</math>, or that <math>\overline{CM}^{2}-\overline{CS}^{2}=\overline{CA}^{2}-\overline{CB}^{2}</math>. ===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}=== After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows<ref group=M>Lagrange (1773/75), section 22</ref> :<math>\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\ \hline \begin{align}y & =Ms+Nx\\ z & =ms+nx \end{align} \left|\begin{matrix}\begin{align}P & =pM^{2}+2qMm+rm^{2}\\ Q & =pMN+q(Mn+Nm)+rmn\\ R & =pN^{2}+2qNn+rn^{2} \end{align} \\ \downarrow\\ PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2} \end{matrix}\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.</p> ==={{anchor|Gauss}} Gauss (1798–1818)=== ===={{anchor|Gauss1}} Binary quadratic forms==== The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:<ref group=M>Gauss (1798/1801), articles 157–158;</ref> :<math>\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\ F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\ \hline \begin{align}x & =\alpha x'+\beta y'\\ y & =\gamma x'+\delta y'\\ \\ x' & =\delta x-\beta y\\ y' & =-\gamma x+\alpha y \end{align} \left|\begin{matrix}\begin{align}a' & =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\ b' & =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\ c' & =a\beta^{2}+2b\beta\delta+c\delta^{2} \end{align} \\ \downarrow\\ b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2} \end{matrix}\right. \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called "proper equivalent" if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.<ref group=M>Gauss (1798/1801), section 159</ref> <p style="background-color:Beige;border:1px solid black">The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.</p> ===={{anchor|Gauss2}} Ternary quadratic forms==== Gauss (1798/1801)<ref group=M>Gauss (1798/1801), articles 266–285</ref> also discussed ternary quadratic forms with the general expression :<math>\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\\ g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)\\ \hline \begin{align}x & =\alpha y+\beta y'+\gamma y''\\ x' & =\alpha'y+\beta'y'+\gamma'y''\\ x'' & =\alpha''y+\beta''y'+\gamma''y'' \end{align} \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x²+y²+z²'', or indefinite in the case of different signs such as ''x²+y²-z²''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:<ref group=M>Gauss (1798/1801), article 277</ref> :<math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\\ \left(\begin{matrix}1, & -1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval <math>x^{2}+x^{\prime2}-x^{\prime\prime2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting: <br><math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)=\left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right)</math></p> ===={{anchor|Gauss4}} Homogeneous coordinates==== Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression :<math>(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E</math> into :<math>G+G'\cos T^{2}+G''\sin T^{2}</math> in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:<ref group=M>Gauss (1818), pp. 5–10</ref> :<math>\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\ k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\ \hline \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =k & \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =-k\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-k & \beta\beta-\beta'\beta'-\beta''\beta'' & =-k\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-k & \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =+k\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} }\right.\\ \hline k=1\\ \begin{align}t\cos E & =\alpha+\alpha'\cos T+\alpha''\sin T\\ t\sin E & =\beta+\beta'\cos T+\beta''\sin T\\ t & =\gamma+\gamma'\cos T+\gamma''\sin T \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}}) ''(n=2)''. Further setting <math>[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that :<math>aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz</math> can be transformed into :<math>Guu+G'u'u'+G''u''u''</math>, in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:<ref group=M>Gauss (1818), pp. 9–10</ref> :<math>\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =-\alpha x-\beta y+\gamma z\\ u' & =\alpha'x+\beta'y-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right.</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' satisfying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>, and can be related to Gauss' previous equations in terms of homogeneous coordinates <math>\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]</math>.</p> ==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation in 1827:<ref group=M>Jacobi (1827), p. 235, 239–240</ref> :<math>{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\ k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\ \hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P & =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\cos\vartheta & =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\sin\vartheta & =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \\ \cos\psi & =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\cos\varphi & =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\sin\varphi & =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta} \end{align} \\ \\ \hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z & =m\\ \alpha'\mu+\beta'x+\gamma'y+\delta'z & =m'\\ \alpha''\mu+\beta''x+\gamma''y+\delta''z & =m''\\ \alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z & =m'''\\ \\ Am+A'm'+A''m''+A'''m''' & =\mu\\ Bm+B'm'+B''m''+B'''m''' & =x\\ Cm+C'm'+C''m''+C'''m''' & =y\\ Dm+D'm'+D''m''+D'''m''' & =z\\ \\ \end{align} \\ \begin{align}\alpha & =-kA, & \beta & =-kB, & \gamma & =-kC, & \delta & =kD,\\ \alpha' & =kA', & \beta' & =kB', & \gamma' & =kC', & \delta' & =-kD',\\ \alpha'' & =kA'', & \beta'' & =kB'', & \gamma'' & =kC'', & \delta'' & =-kD'',\\ \alpha''' & =kA''', & \beta''' & =kB''', & \gamma''' & =kC''', & \delta''' & =-kD''', \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =-k\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' & =k\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' & =k\\ \alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' & =k\\ \alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' & =0\\ \alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' & =0\\ \alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' & =0\\ \alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' & =0\\ \alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' & =0\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' & =0\\ \\ -\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' & =k\\ -\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' & =k\\ -\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' & =k\\ -\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' & =-k\\ -\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' & =0\\ -\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' & =0\\ -\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' & =0\\ -\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' & =0\\ -\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' & =0\\ -\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' & =0 \end{align} \end{matrix}} \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] & =\left[u_{1},\ u_{2},\ u_{3}\right]\\{} [\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] & =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right] \end{align} }</math> and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing <math>m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}</math>.</p> Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):<ref group=M>The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.</ref> :<math>{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\ \mathbf{(1)}\begin{align}x & =\alpha s+\alpha's'+\alpha''s''\\ y & =\beta s+\beta's'+\beta''s''\\ z & =\gamma s+\gamma's'+\gamma''s''\\ \\ s & =\alpha x+\beta y+\gamma z\\ s' & =\alpha'x+\beta'y+\gamma'z\\ s'' & =\alpha''x+\beta''y+\gamma''z \end{align} \left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma & =1 & \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' & =1\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma' & =1 & \beta\beta+\beta'\beta'+\beta''\beta'' & =1\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' & =1 & \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' & =1\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma+\beta'\gamma'+\beta''\gamma'' & =0\\ \alpha''\alpha+\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' & =0\\ \alpha\alpha'+\beta\beta'+\gamma\gamma' & =0 & \alpha\beta+\alpha'\beta'+\alpha''\beta'' & =0 \end{align} \right.\\ \hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\ \left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\ \hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\ \left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\ \hline \begin{align}\cos\phi & =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \cos\eta & =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\ \sin\phi & =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \sin\eta & =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi} \end{align} \end{matrix}\\ \hline \\ \mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\ \hline \begin{align}y & =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, & s' & =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\ z & =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, & s'' & =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z}, \end{align} \end{matrix} \end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma & =1\\ \alpha'\alpha'-\beta'\beta'-\gamma'\gamma' & =-1\\ \alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' & =-1\\ \alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' & =0\\ \alpha''\alpha-\beta''\beta-\gamma''\gamma & =0\\ \alpha\alpha'-\beta\beta'-\gamma\gamma' & =0\\ \\ \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =1\\ \beta\beta-\beta'\beta'-\beta''\beta'' & =-1\\ \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =-1\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} \right. \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.</p> Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:<ref group =M>Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.</ref> :<math>{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\ \hline \mathbf{(1)\ }\begin{align}y_{\varkappa} & =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\ x_{\varkappa} & =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\ \\ \frac{y_{\varkappa}}{y_{n}} & =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\ \frac{x_{\varkappa}}{x_{n}} & =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}} \end{align} \left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} & =0\\ \alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} & =1\\ \\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} & =0\\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} & =1 \end{align} \right.\\ \hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\ \alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\ \hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} & =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\ \\ \xi_{\varkappa} & =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}} \end{align} \\ \hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1 \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.</p> He also stated the following transformation leaving invariant the Lorentz interval:<ref group=M>Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.</ref> :<math>\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\ \hline {\scriptstyle \begin{align}u & =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\ u_{1} & =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\ & \dots\\ u_{n-1} & =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\ \\ w & =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\ w_{1} & =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\ & \dots\\ w_{n-1} & =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1} \end{align} \left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =-1\\ \alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =0\\ \alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} & =0\\ \\ \alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} & =-1\\ \alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} & =0\\ \alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} & =0 \end{align} \text{ }\right.} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.</p> ==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids === [[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:<ref group=M>Chasles (1829), p. 139</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(a)\\ \\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(b)\\ \\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(c)\\ \\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(d) \end{matrix}</math> He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:<ref group=M>Chasles (1829), p. 141</ref> :<math>\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\ \gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\ \alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\ \beta''\quad\Rightarrow\quad\beta''\sqrt{-1} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:<ref group=M>Chasles (1829), pp. 143-144</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}x & =lx'+l'y'+l''z'\\ y & =mx'+m'y'+m''z'\\ z & =nx'+n'y'+n''z' \end{align} \\ \left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} & =0\\ \frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} & =0\\ \frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} & =0 \end{align} \right\} \\ \hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\ \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1 \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Chasles defined the conditional equations of ''l,m,n'' in the same way as those of <math>\alpha,\beta,\gamma</math> in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.</p> He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:<ref group=M>Chasles (1829), pp. 145-146</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =a^{2}\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =b^{2}\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(A)\\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(B)\\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =a^{2}\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =b^{2}\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(C)\\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(D) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates=== [[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation<ref group=M>Lebesgue (1837), pp. 338-341</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\ \hline {\scriptstyle \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\ \dots\\ y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} & =1 & (10)\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} & =0 & (11)\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} & =1 & (13)\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} & =0 & (14) \end{align} \right.} \end{matrix}</math> In order to achieve the invariance of the Lorentz interval<ref group=M>Lebesgue (1837), pp. 353–354</ref> :<math>x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}</math> he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign. <p style="background-color:Beige;border:1px solid black">These instructions give Lorentz transformation ({{equationNote|1a}}) in the form: <br><math>{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \hline \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\ \dots\\ -y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} & =1\\ a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} & =-1\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} & =0\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} & =1\\ a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} & =-1\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} & =0 \end{align} \right. \end{matrix}}</math></p> He went on to redefine the variables of the Lorentz interval and its transformation:<ref group=M>Lebesgue (1837), pp. 353–355</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \downarrow\\ \begin{align}x_{1} & =x_{n}\cos\theta_{1}, & x_{2} & =x_{n}\cos\theta_{2},\dots & x_{n-1} & =x_{n}\cos\theta_{n-1}\\ y_{1} & =y_{n}\cos\phi_{1}, & y_{2} & =y_{n}\cos\phi_{2},\dots & y_{n-1} & =y_{n}\cos\phi_{n-1} \end{align} \\ \downarrow\\ \cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\ \cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\ \hline \\ \cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\ (i=1,2,3\dots n) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Setting <math>[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]</math> it is equivalent to Lorentz transformation ({{equationNote|1b}}).</p> ==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids=== Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&alpha;), from which he derived equations (&beta;) and others:<ref group=M>Weddle (1847), p. 274</ref> :<math>\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1, & l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1, & l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} & =0\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1, & l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0 \end{align} \right\} & \dots(\alpha)\\ \\ \left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1, & l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1, & l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} & =0\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1, & m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0 \end{align} \right\} & \dots(\beta) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>, he defined three conjugate points <math>(x_{1}\dots,y_{1}\dots,z_{1}\dots)</math> on those two conjugate hyperboloids, related to each other in the same way as equations (&alpha;, &beta;) stated above:<ref group=M>Weddle (1847), pp. 275-276</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} & =0\\ \frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} & =0\\ \frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} & =0 \end{align} \quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} & =1\\ \frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} & =1\\ \frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} & =-1 \end{align} \\ \begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} & =a^{2}\\ y_{1}^{2}+y_{2}^{2}-y_{3}^{2} & =b^{2}\\ z_{1}^{2}+z_{2}^{2}-z_{3}^{2} & =-c^{2} \end{align} \quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} & =0\\ x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} & =0\\ y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} & =0 \end{align} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black"> These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:<ref group=M>Bour (1856), pp. 61; 64–65</ref> :<math>\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\ \hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \\ \hline \\ k=+1\\ t=\gamma+\gamma'\cos T+\gamma''\sin T,\\ 1=u,\ \cos T=u',\ \sin T=u',\\ t=z,\ t\cos E=x,\ t\sin E=y\\ \downarrow\\ \mathbf{(2)}\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =\gamma z-\alpha x-\beta y\\ u' & =\alpha'x+\beta'y'-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} & =k\\ -\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} & =-k\\ -\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} & =-k\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =k\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} } \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>. Furthermore, setting <math>[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math> in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> === {{anchor|Somov}} Somov (1863) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:<ref group=M>Somov (1863), pp. 12–14; p. 18 for differentials.</ref> :<math>\begin{matrix}\left.\begin{align}\cos\phi & =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\ \sin\phi & =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''} \end{align} \right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\ \cos^{2}\psi+\cos^{2}\psi=1 \end{matrix}\\ \hline \mathbf{(1)}\ \begin{align}\cos\phi & =x, & \cos\psi & =x'\\ \sin\phi & =y, & \sin\psi & =y' \end{align} \ \left|\begin{align}x & =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\ y & =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=1\\ x^{\prime2}+y^{\prime2}=1 \end{matrix}\\ \hline \begin{align}\cos\phi & =\frac{x}{z}, & \cos\psi & =\frac{x'}{z'}\\ \sin\phi & =\frac{y}{z}, & \sin\psi & =\frac{y'}{z'} \end{align} \ \left|\begin{align}\frac{x}{z} & =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\ \frac{y}{z} & =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\ x^{\prime2}+y^{\prime2}=z^{\prime2} \end{matrix}\\ \hline \mathbf{(2)}\ \left.\begin{align}x & =mx'+ny'+sz'\\ y & =m'x'+n'y'+s'z'\\ z & =m''x'+n''y'+s''z'\\ \\ x' & =mx+m'y-m''z\\ y' & =nx+n'y-n''z\\ z' & =-sx-s'y+s''z\\ \\ dx & =mdx'+ndy'+sdz'\\ dy & =m'dx'+n'dy'+s'dz'\\ dz & =m''dx'+n''dy'+s''dz' \end{align} \right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} & =1\\ n^{2}+n^{\prime2}-n^{\prime\prime2} & =1\\ -s^{2}-s^{\prime2}+s^{\prime\prime2} & =1\\ ns+n's'-n''s'' & =0\\ sm+s'm'-s''m'' & =0\\ mn+m'n'-m''n'' & =0\\ \\ m^{2}+n^{2}-s^{2} & =1\\ m^{\prime2}+n^{\prime2}-s^{\prime2} & =1\\ -m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} & =1\\ -m'm''-n'n''+s's'' & =0\\ -m''m-n''n+s''s & =0\\ mm'+nn'-ss' & =0 \end{align} }\\ dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Klein}} Klein (1871-73) – Cayley absolute and non-Euclidean geometry=== Elaborating on [[w:Arthur Cayley]]'s (1859) definition of an "absolute" ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a "fundamental [[w:conic section]]" in order to discuss motions such as rotation and translation in the non-Euclidean plane.<ref group=M>Klein (1871), pp. 601–602</ref> This was elaborated in (1873) when he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to a two-sheet [[w:hyperboloid]].<ref group=M>Klein (1873), pp. 127-128</ref> <p style="background-color:Beige;border:1px solid black">Klein's representation of hyperbolic space in terms of a two-sheet hyperboloid and its accompanied quadratic form suggests that Lorentz transformations can be geometrically interpreted as motions or isometries in hyperbolic space.</p> ==={{anchor|Killing}} Killing (1878–1893)=== ===={{anchor|Killing1}} Weierstrass coordinates==== [[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form :<math>k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}</math><ref group=M>Killing (1877/78), p. 74; Killing (1880), p. 279</ref> with <math>ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}</math><ref group=M>Killing (1880), eq. 25 on p. 283</ref> or<ref group=M>Killing (1880), p. 283</ref> :<math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> where ''k'' is the reciprocal measure of curvature, <math>k^{2}=\infty</math> denotes [[w:Euclidean geometry]], <math>k^{2}>0</math> [[w:elliptic geometry]], and <math>k^{2}<0</math> hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.<ref group=M>Killing (1877/78), eq. 25 on p. 283</ref> In (1879/80) he tried to formulate the corresponding transformations by plugging <math>k^{2}</math> into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:<ref group=M>Killing (1879/80), p. 274</ref> <math>\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\ \hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\ -k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\ k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, & -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}} \end{matrix}\\ \left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right) \end{matrix}</math> In (1885) he wrote the Weierstrass coordinates and their transformation as follows:<ref group=M>Killing (1885), pp. 18, 28–30, 53</ref> :<math>\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\ k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\ ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\ \hline \begin{align}k^{2}p' & =k^{2}wp+w'x+w''y\\ x' & =ap+a'x+a''y\\ y' & =bp+b'x+b''y\\ \\ k^{2}p & =k^{2}wp'+ax'+by'\\ x & =w'p'+a'x+b'y'\\ y & =w''p'+a''x'+b''y' \end{align} \left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} & =k^{2}\\ \frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} & =1\\ \frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} & =1\\ aw+a'w'+a''w'' & =0\\ bw+b'w'+b''w'' & =0\\ \frac{ab}{k^{2}}+a'b'+a''b'' & =0\\ \\ k^{2}w^{2}+a^{2}+b^{2} & =k^{2}\\ \frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} & =1\\ \frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} & =1\\ ww'+aa'+bb' & =0\\ ww''+aa''+bb'' & =0\\ \frac{w'w''}{k^{2}}+a'a''+b'b'' & =0 \end{align} }\right. \end{matrix}</math> In (1885) he also gave the transformation for ''n'' dimensions:<ref group=M>Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222</ref><ref>Ratcliffe (1994), § 3.6</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\ \hline \left.\begin{align}k^{2}\xi_{0} & =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\ \xi_{\varkappa} & =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\ \\ k^{2}x_{0} & =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\ x_{\varkappa} & =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n} \end{align} \right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ \frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} & =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa) \end{align} } \end{matrix}</math> In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.<ref group=M>Killing (1884/85), pp. 4–5</ref> Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting <math>k^{2}=-1</math> and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid <math>p^{2}-x^{2}-y^{2}=1</math> (known as [[w:hyperboloid model]]),<ref group=M>Killing (1885), Note 9 on p. 260</ref><ref name=rey /> by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space. <p style="background-color:Beige;border:1px solid black">All of Killing's transformations between 1879 and 1885 don't work when <math>k^{2}</math> is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> Finally, in (1893) he wrote:<ref group=M>Killing (1893), see pp. 144, 327–328</ref> :<math>\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\ \hline \begin{align}t' & =at+bu+cv\\ u' & =a't+b'u+c'v\\ v' & =a''t+b''u+c''v \end{align} \left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} & =k^{2}\\ k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} & =1\\ k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} & =1\\ k^{2}ab+a'b'+a''b'' & =0\\ k^{2}ac+a'c'+a''c'' & =0\\ k^{2}bc+b'c'+b''c'' & =0 \end{align} \right. \end{matrix}</math> and in ''n'' dimensions<ref group=M>Killing (1893), pp. 314–316, 216–217</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\ ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\ \hline \begin{align}y_{0} & =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\ y_{1} & =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\ & \,\,\,\vdots\\ y_{n} & =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n} \end{align} \left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} & =1\\ k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} & =0\\ (\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> ===={{anchor|Killing3}} Infinitesimal transformations and Lie group==== After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions, Killing (1887/88)<ref group=M>Killing (1887/88a), pp. 274–275</ref> defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere: :<math>\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\ \hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\ \text{where}\\ \left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\ \left[\iota\ne\varkappa\ne\lambda\ne\mu\right] \end{matrix}</math> and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:<ref group=M>Killing (1892), p. 177</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ \hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\ \text{where}\\ \left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f; \end{matrix}</math> In (1897/98) he showed the relation between Weierstrass coordinates <math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> and corrdinates <math>k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0</math> used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:<ref group=M>Killing (1897/98), pp. 255–256</ref> :<math>\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} & (a)\\ k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} & (b) \end{matrix}\\ \hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\ \text{where}\\ \left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\ \left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\ \left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\ \hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\ \downarrow\\ k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\ \hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota} \end{matrix} \end{matrix}</math> He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b). <p style="background-color:Beige;border:1px solid black">Setting <math>k^{2}=-1</math> denotes the group of hyperbolic motions and thus the Lorentz group.</p> === {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates === [[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).<ref>Gray (1997)</ref> He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:<ref group=M name=p1>Poincaré (1881a), pp. 133–134</ref><ref>Dickson (1923), pp. 220–221</ref> :<math>\begin{matrix}\begin{align}F & =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\ & =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\ F & =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\ & =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1 \end{align} \\ \hline \begin{align}\xi' & =\alpha\xi+\beta\eta+\gamma\zeta\\ \eta' & =\alpha'\xi+\beta'\eta+\gamma'\zeta\\ \zeta' & =\alpha''\xi+\beta''\eta+\gamma''\zeta \end{align} \left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1\\ \alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right. \end{matrix}</math> He went on to describe the properties of "hyperbolic coordinates".<ref group=M name=poinc>Poincaré (1881b), p. 333</ref><ref name=rey>Reynolds (1993)</ref> Poincaré mentioned the hyperboloid model also in (1887).<ref group=M>Poincaré (1887), p. 206</ref> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> === {{anchor|Cox}} Cox (1881–1883) – Weierstrass coordinates === [[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)<ref name=guder group=M>Gudermann (1830), §1–3, §18–19</ref> and [[w:George Salmon]] (1862)<ref group=M>Salmon (1862), section 212, p. 165</ref> on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)<ref group=M>Frischauf (1876), pp. 86–87</ref> in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:<ref group=M>Cox (1881), p. 186 for Weierstrass coordinates; (1881/82), pp. 193–194 for Lorentz transformation. On p. 193, the misprinted expression <math>x^{2}-y^{2}-z^{2}</math> should read <math>z^{2}-y^{2}-x^{2}</math></ref> :<math>\begin{matrix}z^{2}-x^{2}-y^{2}=1\\ z^{2}-y^{2}-x^{2}=Z^{2}-Y^{2}-X^{2}\\ \hline \begin{align}x & =l_{1}X+l_{2}Y+l_{3}Z\\ y & =m_{1}X+m_{2}Y+m_{3}Z\\ z & =n_{1}X+n_{2}Y+n_{3}Z\\ \\ X & =l_{1}x+m_{1}y-n_{1}z\\ Y & =l_{2}x+m_{2}y-n_{2}z\\ Z & =l_{3}x+m_{3}y-n_{3}z \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} & =0\\ \\ l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1\\ l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0\\ n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1 \end{align} }</math> with <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1 \end{align} }</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' up to a sign change in the inverse transformation.</p> Cox also gave the Weierstrass coordinates and their transformation in hyperbolic space:<ref group=M>Cox (1881), pp. 199, 206–207</ref> :<math>\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\ w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\ \hline \begin{align}x & =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\ y & =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\ z & =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\ w & =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\ \\ x' & =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\ y' & =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\ z' & =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\ w' & =l_{4}x+m_{4}y+n_{4}z-r_{4}w \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} & =1\\ l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} & =1\\ l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} & =0\\ l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} & =0\\ l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} & =0\\ l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}</math> with <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' up to a sign change in both the first as well as inverse transformation.</p> === {{anchor|Hill}} Hill (1882) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations<ref group=M>Hill (1882), pp. 323–325</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\ k\left(\sin^{2}E+\cos^{2}E-1\right)\\ \hline \begin{align} & & \cos E' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ & \mathbf{(1)} & \sin E' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \hline \\ & & x & =\alpha u+\alpha'u'+\alpha''u''\\ & & y & =\beta u+\beta'u'+\beta''u''\\ & & z & =\gamma u+\gamma'u'+\gamma''u''\\ & \mathbf{(2)}\\ & & u & =-\alpha x-\beta y+\gamma z\\ & & u' & =\alpha'x+\beta'y'-\gamma'z\\ & & u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =-1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =1\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ (k=-1)\\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =-k\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .</p> === {{anchor|Picard}} Picard (1882-1884) – Quadratic forms === [[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to "hyperfuchsian" functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:<ref group=M>Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system</ref><ref>Dickson (1923), pp. 280-281</ref> :<math>\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\ \\ \mathbf{(1)}\ \begin{align}x & =M_{1}X+P_{1}Y+R_{1}Z\\ y & =M_{2}X+P_{2}Y+R_{2}Z\\ z & =M_{3}X+P_{3}Y+R_{3}Z \end{align} \\ \\ \left[\begin{align}[][x,y,z]=\text{complex}\\ \left[x_{0},y_{0},z_{0}\right]=\text{conjugate} \end{align} \right]\\ \\ \hline \\ x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\ x=x'+ix'',\quad y=y'+iy''\\ \\ \mathbf{(2)}\ \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} & =1\\ P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0\\ P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ \\ M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} & =1\\ M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} & =1\\ M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} & =-1\\ \mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} & =0\\ \mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} & =0\\ \mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} & =0\\ \\ \left[\begin{align}[][M,P,R\dots]=\text{complex}\\ \left[\mu,\pi,\rho\dots\right]=\text{conjugate} \end{align} \right] \end{align} }\right.\end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x²+y²-z²=X²+Y²-Z²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:<ref group=M>Picard (1884a), p. 13</ref> :<math>\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\ \hline \begin{align}U & =\mathcal{A}u+\mathcal{B}v\\ V & =\mathcal{C}u+\mathcal{D}v \end{align} \left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} & =1\\ \mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} & =0\\ \mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} & =-1\\ \mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} & =1 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U²-V²=u²-v²''.</p> Or in (1884b):<ref group=M>Picard (1884b), p. 416</ref> :<math>\begin{matrix}xx_{0}+yy_{0}-1=0\\ \hline \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884c):<ref group=M>Picard (1884c), pp. 123–124; 163</ref> :<math>\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\ \hline \mathbf{(1)}\ \begin{align}U & =Mu+Pv+Rw\\ V & =M'u+P'v+R'w\\ W & =M''u+P''v+R''w\\ \\ u & =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\ v & =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\ w & =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W \end{align} \left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} & =1\\ PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} & =1\\ RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} & =-1\\ MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} & =0\\ MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} & =0\\ PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} & =0\\ \\ MM_{0}+PP_{0}-RR_{0} & =1\\ M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} & =1\\ M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} & =-1\\ M_{0}M'+P_{0}P'-R_{0}R' & =0\\ M_{0}M''+P_{0}P''-R_{0}R'' & =0\\ M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' & =0 \end{align} }\right.\\ \hline \text{Invariance of unit hypersphere:}\\ \mathbf{(2)}\ \begin{align}\xi' & =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\ \eta' & =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''} \end{align} \left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} & =1\\ BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} & =1\\ CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} & =-1\\ AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} & =0\\ AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} & =0\\ BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U²+V²-W²=u²+v²-w²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing <math>\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1</math>.</p> === {{anchor|Callandreau}} Callandreau (1885) – Homography === Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations<ref group=M>Callandreau (1885), pp. A.7; A.12</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\ {\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\ \hline \begin{align}\cos\varepsilon' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \sin\varepsilon' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T} \end{align} \left|{\scriptstyle \begin{align} & \left(k=1\right)\\ \alpha^{2}+\beta^{2}-\gamma^{2} & =-k & \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =+k & \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =+k & \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-1 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-1 & \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =+1 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>.</p> ==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations=== In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions.<ref group=M>Lie (1885/86), p. 411</ref> In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):<ref group=M>Werner (1889), pp. 4, 28</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\ \hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\ \text{where}\\ \left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\ \left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\ \left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right] \end{matrix}</math> More generally, Lie (1890)<ref group=M>Lie (1890a), p. 295;</ref> defined non-Euclidean motions in terms of two forms <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0</math> in which the imaginary form with <math>+1</math> denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:<ref group=M>Lie (1890a), p. 311</ref> :<math>\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\ \hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n) \end{matrix}</math> Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form: :<math>x^{2}+y^{2}-1=0</math><ref group=M>Lie (1893), p. 474</ref> or <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0</math><ref group=M>Lie (1893), p. 479</ref> or <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math>.<ref group=M>Lie (1893), p. 481</ref> <p style="background-color:Beige;border:1px solid black">The group of hyperbolic motions is isomorphic to the Lorentz group. The interval <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math> becomes the Lorentz interval <math>x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0</math> by setting <br><math>(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)</math></p> ==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates=== [[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':<ref group=M>Gérard (1892), pp. 40–41</ref> :<math>\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\ X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\ \hline \begin{align}X & =aX'+a'Y'+a''Z'\\ Y & =bX'+b'Y'+b''Z'\\ Z & =cX'+c'Y'+c''Z'\\ \\ X' & =aX+bY-cZ\\ Y' & =a'X+b'Y-c'Z\\ Z' & =-a''X-b''Y+c''Z \end{align} \left|\begin{align}a^{2}+b^{2}-c^{2} & =1\\ a^{\prime2}+b^{\prime2}-c^{\prime2} & =1\\ a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} & =-1\\ aa'+bb'-cc' & =0\\ a'a''+b'b''-c'c'' & =0\\ a''a+b''b-c''c & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> He gave the case of translation as follows:<ref group=M name=gerard>Gérard (1892), pp. 40–41</ref> :<math>\begin{align}X & =Z_{0}X'+X_{0}Z'\\ Y & =Y'\\ Z & =X_{0}X'+Z_{0}Z' \end{align} \ \text{with}\ \begin{align}X_{0} & =\operatorname{sh}OO'\\ Z_{0} & =\operatorname{ch}OO' \end{align} </math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz boost ({{equationNote|3b}}).</p> ==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates=== [[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:<ref group=M>Hausdorff (1899), p. 165, pp. 181-182</ref> :<math>\begin{matrix}p^{2}-x^{2}-y^{2}=1\\ \hline \begin{align}x & =a_{1}x'+a_{2}y'+x_{0}p'\\ y & =b_{1}x'+b{}_{2}y'+y_{0}p'\\ p & =e_{1}x'+e_{2}y'+p_{0}p'\\ \\ x' & =a_{1}x+b_{1}y-e_{1}p\\ y' & =a_{2}x+b_{2}y-e_{2}p\\ -p' & =x_{0}x+y_{0}y-p_{0}p \end{align} \left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} & =1\\ a_{2}^{2}+b_{2}^{2}-e_{2}^{2} & =1\\ -x_{0}^{2}-y_{0}^{2}+p_{0}^{2} & =1\\ a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} & =0\\ a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} & =0\\ a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} & =0\\ \\ a_{1}^{2}+a_{2}^{2}-x_{0}^{2} & =1\\ b_{1}^{2}+b_{2}^{2}-y_{0}^{2} & =1\\ -e_{1}^{2}-e_{2}^{2}+p_{0}^{2} & =1\\ b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} & =0\\ a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} & =0\\ a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates === In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting <math>k=\sqrt{-1}R</math> with ''R'' as real quantity):<ref group=M>Woods (1901/02), p. 98, 104</ref> :<math>\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\ \hline \begin{align}u' & =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ v' & =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ w' & =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}} \end{align} \left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} & =k^{2}\\ (i=1,2,3)\\ k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} & =1\\ \alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} & =0\\ (i,h=1,2,3,4;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''²=-1.</p> Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with <math>k=\sqrt{-1}k</math> for hyperbolic space):<ref group=M>Woods (1903/05), pp. 45–46; p. 48)</ref> :<math>\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\ ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\ \hline \begin{align}x_{1}^{\prime} & =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\ x_{2}^{\prime} & =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\ x_{3}^{\prime} & =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\ x_{0}^{\prime} & =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0} \end{align} \left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) & =1\\ \delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) & =k^{2}\\ (i=1,2,3)\\ \delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) & =0\\ (i,h=0,1,2,3;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''²=-1.</p> ==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates=== [[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''<ref group=M>Liebmann (1904/05), p. 168; pp. 175–176</ref> :<math>\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\ \hline \begin{align}x_{1} & =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\ y_{1} & =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\ x_{1} & =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\ \\ x & =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\ y & =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\ p & =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1} \end{align} \left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} & =1\\ -\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} & =1\\ -\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} & =1\\ \alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} & =0\\ \alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} & =0\\ \alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==References== ===Historical mathematical sources=== {{reflist|3|group=M}} *{{#section:History of Topics in Special Relativity/mathsource|apo}} *{{#section:History of Topics in Special Relativity/mathsource|bour56att}} *{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} *{{#section:History of Topics in Special Relativity/mathsource|chas29}} *{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} *{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} *{{#section:History of Topics in Special Relativity/mathsource|fris76}} *{{#section:History of Topics in Special Relativity/mathsource|gau98}} *{{#section:History of Topics in Special Relativity/mathsource|gau18}} *{{#section:History of Topics in Special Relativity/mathsource|ger92}} *{{#section:History of Topics in Special Relativity/mathsource|gud30}} *{{#section:History of Topics in Special Relativity/mathsource|haus99}} *{{#section:History of Topics in Special Relativity/mathsource|hill82}} *{{#section:History of Topics in Special Relativity/mathsource|jac27}} *{{#section:History of Topics in Special Relativity/mathsource|jac32a}} *{{#section:History of Topics in Special Relativity/mathsource|jac32b}} *{{#section:History of Topics in Special Relativity/mathsource|jac33}} *{{#section:History of Topics in Special Relativity/mathsource|kil77}} *{{#section:History of Topics in Special Relativity/mathsource|kil79}} *{{#section:History of Topics in Special Relativity/mathsource|kil84}} *{{#section:History of Topics in Special Relativity/mathsource|kil85}} *{{#section:History of Topics in Special Relativity/mathsource|kil93}} *{{#section:History of Topics in Special Relativity/mathsource|kil97}} *{{#section:History of Topics in Special Relativity/mathsource|klei71}} *{{#section:History of Topics in Special Relativity/mathsource|klei73}} *{{#section:History of Topics in Special Relativity/mathsource|lag73}} *{{#section:History of Topics in Special Relativity/mathsource|leb37}} *{{#section:History of Topics in Special Relativity/mathsource|lie85}} *{{#section:History of Topics in Special Relativity/mathsource|lie90}} *{{#section:History of Topics in Special Relativity/mathsource|lie93}} *{{#section:History of Topics in Special Relativity/mathsource|lieb04}} *{{#section:History of Topics in Special Relativity/mathsource|lop}} *{{#section:History of Topics in Special Relativity/mathsource|pic82}} *{{#section:History of Topics in Special Relativity/mathsource|pic84a}} *{{#section:History of Topics in Special Relativity/mathsource|pic84b}} *{{#section:History of Topics in Special Relativity/mathsource|pic84c}} *{{#section:History of Topics in Special Relativity/mathsource|poin81a}} *{{#section:History of Topics in Special Relativity/mathsource|poin81b}} *{{#section:History of Topics in Special Relativity/mathsource|poin87}} *{{#section:History of Topics in Special Relativity/mathsource|sal62}} *{{#section:History of Topics in Special Relativity/mathsource|som63}} *{{#section:History of Topics in Special Relativity/mathsource|wedd47}} *{{#section:History of Topics in Special Relativity/mathsource|wern89}} *{{#section:History of Topics in Special Relativity/mathsource|woo01}} *{{#section:History of Topics in Special Relativity/mathsource|woo03}} ===Secondary sources=== {{reflist|3}} {{#section:History of Topics in Special Relativity/secsource|L1}} [[Category:Special Relativity]] [[Category:History of Physics]] plc5eal0xt83ufm1qaw25k5tjalxb25 2416152 2416151 2022-08-18T13:18:30Z D.H 52339 /* {{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas */ wikitext text/x-wiki {{../Lorentz transformation (header)}} ==Most general Lorentz transformations== ===General quadratic form=== The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as<ref>Bôcher (1907), chapter X</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}\begin{align}q=\sum_{0}^{n}A_{ij}x_{i}x_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align} & =q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\ b=\sum_{0}^{n}A_{ij}x_{i}y_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y} & =b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}' \end{align} \quad\left(A_{ij}=A_{ji}\right)\\ \hline \left.\begin{align}x_{i}^{\prime} & =\sum_{j=0}^{n}g_{ij}x_{j}=\mathbf{g}\cdot\mathbf{x}\\ x_{i} & =\sum_{j=0}^{n}g_{ij}^{(-1)}x_{j}^{\prime}=\mathbf{g}^{-1}\cdot\mathbf{x}' \end{align} \right|\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}' \end{matrix}</math>|{{equationRef|Q1}}}} The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc. ===Most general Lorentz transformation=== The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:<ref name=ratcliffe>Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1</ref><ref>Naimark (1964), 2 in four dimensions</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} & =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\ -x_{0}y_{0}+\cdots+x_{n}y_{n} & =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime} \end{align} \\ \hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\ \downarrow\\ \begin{align}x_{0}^{\prime} & =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\ x_{1}^{\prime} & =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\ & \dots\\ x_{n}^{\prime} & =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn} \end{align} \\ \\ \mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\ \downarrow\\ \begin{align}x_{0} & =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\ x_{1} & =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\ & \dots\\ x_{n} & =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn} \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} & =\mathbf{g}^{-1}\\ \mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} & =\mathbf{A}\\ \mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} & =\mathbf{A}\\ \\ \end{align} \\ \begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \end{matrix} \end{matrix}</math>|{{equationRef|1a}}}} The invariance of the Lorentz interval with ''n''=1 between axes and conjugate diameters of hyperbolas was known for a long time since [[#Apo|Apollonius (ca. 200 BC)]]. Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.<ref>Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.</ref><ref>Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.</ref> They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881/82)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the "absolute" form <math>-x_{0}^{2}+\cdots+x_{n}^{2}=0</math> as discussed by [[#Klein|Klein (1871-73)]].<ref group=M>Killing (1885), p. 71</ref><ref>Müller (1910), p. 661, in particular footnote 247.</ref><ref>Sommerville (1911), p. 286, section K6.</ref> In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> by [[#Killing3|Killing (1888-1897)]]. ===Most general Lorentz transformation of velocity=== If <math>x_{i},\ x_{i}^{\prime}</math> in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates <math>u_{s},\ u_{s}^{\prime}</math> follow by :<math>\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)</math> defined by <math>u_{1}^{2}+u_{2}^{2}+\dots+u_{n}^{2}\le1</math> so that the Lorentz transformation becomes a [[w:homography]] inside the [[w:unit hypersphere]], which [[w:John Lighton Synge]] called "the most general formula for the composition of velocities" in terms of special relativity<ref>Synge (1955), p. 129 for ''n''=3</ref> (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})): {{NumBlk|:|<math>\begin{align}u_{s}^{\prime} & =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\ \\ u_{s} & =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}} \end{align} \left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \right.</math>|{{equationRef|1b}}}} Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere <math>-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0</math> were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]]. ==Historical notation== ==={{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas === [[w:Apollonius of Perga]] (c. 240–190 BC) defined the following proposition related to conjugate diameters of hyperbolas, which was translated and adapted to the modern reader by [[w:Thomas Heath (classicist)|w:Thomas Heath]] as follows:<ref group=M>Apollonius/Heath (1896), Proposition 129; (Apollonius, Book VII, Prop. 13).</ref> :[..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. <p style="background-color:Beige;border:1px solid black">Apollonius' result indicates the invariance of the Lorentz interval <math>-x_{0}^{\prime2}+x_{1}^{\prime2}=-x_{0}^{2}+x_{1}^{2}</math>, by which Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' can be interpreted as mapping from one pair of axes of a hyperbola to a pair of conjugate diameters.</p> This result was well known to modern authors, such as [[w:Guillaume de l'Hôpital]] (1707) who used [[w:analytic geomety]] in order to demonstrate the following proposition:<ref group=M>l'Hôpital (1707), Third book, Prop. XII, p. 76.</ref> :The difference of the squares of any two conjugate diameters ''Mm, Ss'' is equal to the difference of the squares of the two axes ''Aa, Bb.'' We have to prove that <math>\overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}</math>, or that <math>\overline{CM}^{2}-\overline{CS}^{2}=\overline{CA}^{2}-\overline{CB}^{2}</math>. ===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}=== After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows<ref group=M>Lagrange (1773/75), section 22</ref> :<math>\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\ \hline \begin{align}y & =Ms+Nx\\ z & =ms+nx \end{align} \left|\begin{matrix}\begin{align}P & =pM^{2}+2qMm+rm^{2}\\ Q & =pMN+q(Mn+Nm)+rmn\\ R & =pN^{2}+2qNn+rn^{2} \end{align} \\ \downarrow\\ PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2} \end{matrix}\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.</p> ==={{anchor|Gauss}} Gauss (1798–1818)=== ===={{anchor|Gauss1}} Binary quadratic forms==== The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:<ref group=M>Gauss (1798/1801), articles 157–158;</ref> :<math>\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\ F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\ \hline \begin{align}x & =\alpha x'+\beta y'\\ y & =\gamma x'+\delta y'\\ \\ x' & =\delta x-\beta y\\ y' & =-\gamma x+\alpha y \end{align} \left|\begin{matrix}\begin{align}a' & =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\ b' & =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\ c' & =a\beta^{2}+2b\beta\delta+c\delta^{2} \end{align} \\ \downarrow\\ b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2} \end{matrix}\right. \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called "proper equivalent" if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.<ref group=M>Gauss (1798/1801), section 159</ref> <p style="background-color:Beige;border:1px solid black">The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.</p> ===={{anchor|Gauss2}} Ternary quadratic forms==== Gauss (1798/1801)<ref group=M>Gauss (1798/1801), articles 266–285</ref> also discussed ternary quadratic forms with the general expression :<math>\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\\ g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)\\ \hline \begin{align}x & =\alpha y+\beta y'+\gamma y''\\ x' & =\alpha'y+\beta'y'+\gamma'y''\\ x'' & =\alpha''y+\beta''y'+\gamma''y'' \end{align} \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x²+y²+z²'', or indefinite in the case of different signs such as ''x²+y²-z²''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:<ref group=M>Gauss (1798/1801), article 277</ref> :<math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\\ \left(\begin{matrix}1, & -1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval <math>x^{2}+x^{\prime2}-x^{\prime\prime2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting: <br><math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)=\left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right)</math></p> ===={{anchor|Gauss4}} Homogeneous coordinates==== Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression :<math>(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E</math> into :<math>G+G'\cos T^{2}+G''\sin T^{2}</math> in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:<ref group=M>Gauss (1818), pp. 5–10</ref> :<math>\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\ k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\ \hline \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =k & \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =-k\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-k & \beta\beta-\beta'\beta'-\beta''\beta'' & =-k\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-k & \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =+k\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} }\right.\\ \hline k=1\\ \begin{align}t\cos E & =\alpha+\alpha'\cos T+\alpha''\sin T\\ t\sin E & =\beta+\beta'\cos T+\beta''\sin T\\ t & =\gamma+\gamma'\cos T+\gamma''\sin T \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}}) ''(n=2)''. Further setting <math>[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that :<math>aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz</math> can be transformed into :<math>Guu+G'u'u'+G''u''u''</math>, in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:<ref group=M>Gauss (1818), pp. 9–10</ref> :<math>\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =-\alpha x-\beta y+\gamma z\\ u' & =\alpha'x+\beta'y-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right.</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' satisfying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>, and can be related to Gauss' previous equations in terms of homogeneous coordinates <math>\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]</math>.</p> ==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation in 1827:<ref group=M>Jacobi (1827), p. 235, 239–240</ref> :<math>{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\ k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\ \hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P & =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\cos\vartheta & =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\sin\vartheta & =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \\ \cos\psi & =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\cos\varphi & =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\sin\varphi & =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta} \end{align} \\ \\ \hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z & =m\\ \alpha'\mu+\beta'x+\gamma'y+\delta'z & =m'\\ \alpha''\mu+\beta''x+\gamma''y+\delta''z & =m''\\ \alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z & =m'''\\ \\ Am+A'm'+A''m''+A'''m''' & =\mu\\ Bm+B'm'+B''m''+B'''m''' & =x\\ Cm+C'm'+C''m''+C'''m''' & =y\\ Dm+D'm'+D''m''+D'''m''' & =z\\ \\ \end{align} \\ \begin{align}\alpha & =-kA, & \beta & =-kB, & \gamma & =-kC, & \delta & =kD,\\ \alpha' & =kA', & \beta' & =kB', & \gamma' & =kC', & \delta' & =-kD',\\ \alpha'' & =kA'', & \beta'' & =kB'', & \gamma'' & =kC'', & \delta'' & =-kD'',\\ \alpha''' & =kA''', & \beta''' & =kB''', & \gamma''' & =kC''', & \delta''' & =-kD''', \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =-k\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' & =k\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' & =k\\ \alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' & =k\\ \alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' & =0\\ \alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' & =0\\ \alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' & =0\\ \alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' & =0\\ \alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' & =0\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' & =0\\ \\ -\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' & =k\\ -\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' & =k\\ -\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' & =k\\ -\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' & =-k\\ -\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' & =0\\ -\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' & =0\\ -\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' & =0\\ -\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' & =0\\ -\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' & =0\\ -\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' & =0 \end{align} \end{matrix}} \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] & =\left[u_{1},\ u_{2},\ u_{3}\right]\\{} [\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] & =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right] \end{align} }</math> and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing <math>m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}</math>.</p> Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):<ref group=M>The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.</ref> :<math>{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\ \mathbf{(1)}\begin{align}x & =\alpha s+\alpha's'+\alpha''s''\\ y & =\beta s+\beta's'+\beta''s''\\ z & =\gamma s+\gamma's'+\gamma''s''\\ \\ s & =\alpha x+\beta y+\gamma z\\ s' & =\alpha'x+\beta'y+\gamma'z\\ s'' & =\alpha''x+\beta''y+\gamma''z \end{align} \left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma & =1 & \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' & =1\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma' & =1 & \beta\beta+\beta'\beta'+\beta''\beta'' & =1\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' & =1 & \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' & =1\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma+\beta'\gamma'+\beta''\gamma'' & =0\\ \alpha''\alpha+\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' & =0\\ \alpha\alpha'+\beta\beta'+\gamma\gamma' & =0 & \alpha\beta+\alpha'\beta'+\alpha''\beta'' & =0 \end{align} \right.\\ \hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\ \left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\ \hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\ \left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\ \hline \begin{align}\cos\phi & =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \cos\eta & =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\ \sin\phi & =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \sin\eta & =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi} \end{align} \end{matrix}\\ \hline \\ \mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\ \hline \begin{align}y & =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, & s' & =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\ z & =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, & s'' & =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z}, \end{align} \end{matrix} \end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma & =1\\ \alpha'\alpha'-\beta'\beta'-\gamma'\gamma' & =-1\\ \alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' & =-1\\ \alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' & =0\\ \alpha''\alpha-\beta''\beta-\gamma''\gamma & =0\\ \alpha\alpha'-\beta\beta'-\gamma\gamma' & =0\\ \\ \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =1\\ \beta\beta-\beta'\beta'-\beta''\beta'' & =-1\\ \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =-1\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} \right. \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.</p> Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:<ref group =M>Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.</ref> :<math>{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\ \hline \mathbf{(1)\ }\begin{align}y_{\varkappa} & =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\ x_{\varkappa} & =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\ \\ \frac{y_{\varkappa}}{y_{n}} & =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\ \frac{x_{\varkappa}}{x_{n}} & =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}} \end{align} \left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} & =0\\ \alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} & =1\\ \\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} & =0\\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} & =1 \end{align} \right.\\ \hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\ \alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\ \hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} & =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\ \\ \xi_{\varkappa} & =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}} \end{align} \\ \hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1 \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.</p> He also stated the following transformation leaving invariant the Lorentz interval:<ref group=M>Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.</ref> :<math>\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\ \hline {\scriptstyle \begin{align}u & =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\ u_{1} & =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\ & \dots\\ u_{n-1} & =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\ \\ w & =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\ w_{1} & =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\ & \dots\\ w_{n-1} & =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1} \end{align} \left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =-1\\ \alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =0\\ \alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} & =0\\ \\ \alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} & =-1\\ \alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} & =0\\ \alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} & =0 \end{align} \text{ }\right.} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.</p> ==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids === [[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:<ref group=M>Chasles (1829), p. 139</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(a)\\ \\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(b)\\ \\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(c)\\ \\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(d) \end{matrix}</math> He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:<ref group=M>Chasles (1829), p. 141</ref> :<math>\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\ \gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\ \alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\ \beta''\quad\Rightarrow\quad\beta''\sqrt{-1} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:<ref group=M>Chasles (1829), pp. 143-144</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}x & =lx'+l'y'+l''z'\\ y & =mx'+m'y'+m''z'\\ z & =nx'+n'y'+n''z' \end{align} \\ \left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} & =0\\ \frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} & =0\\ \frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} & =0 \end{align} \right\} \\ \hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\ \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1 \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Chasles defined the conditional equations of ''l,m,n'' in the same way as those of <math>\alpha,\beta,\gamma</math> in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.</p> He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:<ref group=M>Chasles (1829), pp. 145-146</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =a^{2}\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =b^{2}\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(A)\\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(B)\\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =a^{2}\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =b^{2}\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(C)\\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(D) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates=== [[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation<ref group=M>Lebesgue (1837), pp. 338-341</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\ \hline {\scriptstyle \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\ \dots\\ y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} & =1 & (10)\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} & =0 & (11)\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} & =1 & (13)\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} & =0 & (14) \end{align} \right.} \end{matrix}</math> In order to achieve the invariance of the Lorentz interval<ref group=M>Lebesgue (1837), pp. 353–354</ref> :<math>x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}</math> he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign. <p style="background-color:Beige;border:1px solid black">These instructions give Lorentz transformation ({{equationNote|1a}}) in the form: <br><math>{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \hline \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\ \dots\\ -y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} & =1\\ a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} & =-1\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} & =0\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} & =1\\ a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} & =-1\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} & =0 \end{align} \right. \end{matrix}}</math></p> He went on to redefine the variables of the Lorentz interval and its transformation:<ref group=M>Lebesgue (1837), pp. 353–355</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \downarrow\\ \begin{align}x_{1} & =x_{n}\cos\theta_{1}, & x_{2} & =x_{n}\cos\theta_{2},\dots & x_{n-1} & =x_{n}\cos\theta_{n-1}\\ y_{1} & =y_{n}\cos\phi_{1}, & y_{2} & =y_{n}\cos\phi_{2},\dots & y_{n-1} & =y_{n}\cos\phi_{n-1} \end{align} \\ \downarrow\\ \cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\ \cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\ \hline \\ \cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\ (i=1,2,3\dots n) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Setting <math>[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]</math> it is equivalent to Lorentz transformation ({{equationNote|1b}}).</p> ==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids=== Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&alpha;), from which he derived equations (&beta;) and others:<ref group=M>Weddle (1847), p. 274</ref> :<math>\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1, & l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1, & l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} & =0\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1, & l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0 \end{align} \right\} & \dots(\alpha)\\ \\ \left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1, & l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1, & l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} & =0\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1, & m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0 \end{align} \right\} & \dots(\beta) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>, he defined three conjugate points <math>(x_{1}\dots,y_{1}\dots,z_{1}\dots)</math> on those two conjugate hyperboloids, related to each other in the same way as equations (&alpha;, &beta;) stated above:<ref group=M>Weddle (1847), pp. 275-276</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} & =0\\ \frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} & =0\\ \frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} & =0 \end{align} \quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} & =1\\ \frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} & =1\\ \frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} & =-1 \end{align} \\ \begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} & =a^{2}\\ y_{1}^{2}+y_{2}^{2}-y_{3}^{2} & =b^{2}\\ z_{1}^{2}+z_{2}^{2}-z_{3}^{2} & =-c^{2} \end{align} \quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} & =0\\ x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} & =0\\ y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} & =0 \end{align} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black"> These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:<ref group=M>Bour (1856), pp. 61; 64–65</ref> :<math>\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\ \hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \\ \hline \\ k=+1\\ t=\gamma+\gamma'\cos T+\gamma''\sin T,\\ 1=u,\ \cos T=u',\ \sin T=u',\\ t=z,\ t\cos E=x,\ t\sin E=y\\ \downarrow\\ \mathbf{(2)}\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =\gamma z-\alpha x-\beta y\\ u' & =\alpha'x+\beta'y'-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} & =k\\ -\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} & =-k\\ -\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} & =-k\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =k\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} } \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>. Furthermore, setting <math>[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math> in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> === {{anchor|Somov}} Somov (1863) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:<ref group=M>Somov (1863), pp. 12–14; p. 18 for differentials.</ref> :<math>\begin{matrix}\left.\begin{align}\cos\phi & =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\ \sin\phi & =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''} \end{align} \right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\ \cos^{2}\psi+\cos^{2}\psi=1 \end{matrix}\\ \hline \mathbf{(1)}\ \begin{align}\cos\phi & =x, & \cos\psi & =x'\\ \sin\phi & =y, & \sin\psi & =y' \end{align} \ \left|\begin{align}x & =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\ y & =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=1\\ x^{\prime2}+y^{\prime2}=1 \end{matrix}\\ \hline \begin{align}\cos\phi & =\frac{x}{z}, & \cos\psi & =\frac{x'}{z'}\\ \sin\phi & =\frac{y}{z}, & \sin\psi & =\frac{y'}{z'} \end{align} \ \left|\begin{align}\frac{x}{z} & =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\ \frac{y}{z} & =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\ x^{\prime2}+y^{\prime2}=z^{\prime2} \end{matrix}\\ \hline \mathbf{(2)}\ \left.\begin{align}x & =mx'+ny'+sz'\\ y & =m'x'+n'y'+s'z'\\ z & =m''x'+n''y'+s''z'\\ \\ x' & =mx+m'y-m''z\\ y' & =nx+n'y-n''z\\ z' & =-sx-s'y+s''z\\ \\ dx & =mdx'+ndy'+sdz'\\ dy & =m'dx'+n'dy'+s'dz'\\ dz & =m''dx'+n''dy'+s''dz' \end{align} \right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} & =1\\ n^{2}+n^{\prime2}-n^{\prime\prime2} & =1\\ -s^{2}-s^{\prime2}+s^{\prime\prime2} & =1\\ ns+n's'-n''s'' & =0\\ sm+s'm'-s''m'' & =0\\ mn+m'n'-m''n'' & =0\\ \\ m^{2}+n^{2}-s^{2} & =1\\ m^{\prime2}+n^{\prime2}-s^{\prime2} & =1\\ -m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} & =1\\ -m'm''-n'n''+s's'' & =0\\ -m''m-n''n+s''s & =0\\ mm'+nn'-ss' & =0 \end{align} }\\ dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Klein}} Klein (1871-73) – Cayley absolute and non-Euclidean geometry=== Elaborating on [[w:Arthur Cayley]]'s (1859) definition of an "absolute" ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a "fundamental [[w:conic section]]" in order to discuss motions such as rotation and translation in the non-Euclidean plane.<ref group=M>Klein (1871), pp. 601–602</ref> This was elaborated in (1873) when he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to a two-sheet [[w:hyperboloid]].<ref group=M>Klein (1873), pp. 127-128</ref> <p style="background-color:Beige;border:1px solid black">Klein's representation of hyperbolic space in terms of a two-sheet hyperboloid and its accompanied quadratic form suggests that Lorentz transformations can be geometrically interpreted as motions or isometries in hyperbolic space.</p> ==={{anchor|Killing}} Killing (1878–1893)=== ===={{anchor|Killing1}} Weierstrass coordinates==== [[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form :<math>k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}</math><ref group=M>Killing (1877/78), p. 74; Killing (1880), p. 279</ref> with <math>ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}</math><ref group=M>Killing (1880), eq. 25 on p. 283</ref> or<ref group=M>Killing (1880), p. 283</ref> :<math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> where ''k'' is the reciprocal measure of curvature, <math>k^{2}=\infty</math> denotes [[w:Euclidean geometry]], <math>k^{2}>0</math> [[w:elliptic geometry]], and <math>k^{2}<0</math> hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.<ref group=M>Killing (1877/78), eq. 25 on p. 283</ref> In (1879/80) he tried to formulate the corresponding transformations by plugging <math>k^{2}</math> into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:<ref group=M>Killing (1879/80), p. 274</ref> <math>\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\ \hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\ -k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\ k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, & -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}} \end{matrix}\\ \left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right) \end{matrix}</math> In (1885) he wrote the Weierstrass coordinates and their transformation as follows:<ref group=M>Killing (1885), pp. 18, 28–30, 53</ref> :<math>\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\ k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\ ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\ \hline \begin{align}k^{2}p' & =k^{2}wp+w'x+w''y\\ x' & =ap+a'x+a''y\\ y' & =bp+b'x+b''y\\ \\ k^{2}p & =k^{2}wp'+ax'+by'\\ x & =w'p'+a'x+b'y'\\ y & =w''p'+a''x'+b''y' \end{align} \left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} & =k^{2}\\ \frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} & =1\\ \frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} & =1\\ aw+a'w'+a''w'' & =0\\ bw+b'w'+b''w'' & =0\\ \frac{ab}{k^{2}}+a'b'+a''b'' & =0\\ \\ k^{2}w^{2}+a^{2}+b^{2} & =k^{2}\\ \frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} & =1\\ \frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} & =1\\ ww'+aa'+bb' & =0\\ ww''+aa''+bb'' & =0\\ \frac{w'w''}{k^{2}}+a'a''+b'b'' & =0 \end{align} }\right. \end{matrix}</math> In (1885) he also gave the transformation for ''n'' dimensions:<ref group=M>Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222</ref><ref>Ratcliffe (1994), § 3.6</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\ \hline \left.\begin{align}k^{2}\xi_{0} & =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\ \xi_{\varkappa} & =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\ \\ k^{2}x_{0} & =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\ x_{\varkappa} & =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n} \end{align} \right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ \frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} & =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa) \end{align} } \end{matrix}</math> In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.<ref group=M>Killing (1884/85), pp. 4–5</ref> Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting <math>k^{2}=-1</math> and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid <math>p^{2}-x^{2}-y^{2}=1</math> (known as [[w:hyperboloid model]]),<ref group=M>Killing (1885), Note 9 on p. 260</ref><ref name=rey /> by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space. <p style="background-color:Beige;border:1px solid black">All of Killing's transformations between 1879 and 1885 don't work when <math>k^{2}</math> is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> Finally, in (1893) he wrote:<ref group=M>Killing (1893), see pp. 144, 327–328</ref> :<math>\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\ \hline \begin{align}t' & =at+bu+cv\\ u' & =a't+b'u+c'v\\ v' & =a''t+b''u+c''v \end{align} \left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} & =k^{2}\\ k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} & =1\\ k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} & =1\\ k^{2}ab+a'b'+a''b'' & =0\\ k^{2}ac+a'c'+a''c'' & =0\\ k^{2}bc+b'c'+b''c'' & =0 \end{align} \right. \end{matrix}</math> and in ''n'' dimensions<ref group=M>Killing (1893), pp. 314–316, 216–217</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\ ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\ \hline \begin{align}y_{0} & =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\ y_{1} & =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\ & \,\,\,\vdots\\ y_{n} & =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n} \end{align} \left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} & =1\\ k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} & =0\\ (\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> ===={{anchor|Killing3}} Infinitesimal transformations and Lie group==== After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions, Killing (1887/88)<ref group=M>Killing (1887/88a), pp. 274–275</ref> defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere: :<math>\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\ \hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\ \text{where}\\ \left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\ \left[\iota\ne\varkappa\ne\lambda\ne\mu\right] \end{matrix}</math> and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:<ref group=M>Killing (1892), p. 177</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ \hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\ \text{where}\\ \left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f; \end{matrix}</math> In (1897/98) he showed the relation between Weierstrass coordinates <math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> and corrdinates <math>k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0</math> used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:<ref group=M>Killing (1897/98), pp. 255–256</ref> :<math>\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} & (a)\\ k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} & (b) \end{matrix}\\ \hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\ \text{where}\\ \left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\ \left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\ \left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\ \hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\ \downarrow\\ k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\ \hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota} \end{matrix} \end{matrix}</math> He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b). <p style="background-color:Beige;border:1px solid black">Setting <math>k^{2}=-1</math> denotes the group of hyperbolic motions and thus the Lorentz group.</p> === {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates === [[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).<ref>Gray (1997)</ref> He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:<ref group=M name=p1>Poincaré (1881a), pp. 133–134</ref><ref>Dickson (1923), pp. 220–221</ref> :<math>\begin{matrix}\begin{align}F & =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\ & =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\ F & =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\ & =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1 \end{align} \\ \hline \begin{align}\xi' & =\alpha\xi+\beta\eta+\gamma\zeta\\ \eta' & =\alpha'\xi+\beta'\eta+\gamma'\zeta\\ \zeta' & =\alpha''\xi+\beta''\eta+\gamma''\zeta \end{align} \left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1\\ \alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right. \end{matrix}</math> He went on to describe the properties of "hyperbolic coordinates".<ref group=M name=poinc>Poincaré (1881b), p. 333</ref><ref name=rey>Reynolds (1993)</ref> Poincaré mentioned the hyperboloid model also in (1887).<ref group=M>Poincaré (1887), p. 206</ref> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> === {{anchor|Cox}} Cox (1881–1883) – Weierstrass coordinates === [[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)<ref name=guder group=M>Gudermann (1830), §1–3, §18–19</ref> and [[w:George Salmon]] (1862)<ref group=M>Salmon (1862), section 212, p. 165</ref> on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)<ref group=M>Frischauf (1876), pp. 86–87</ref> in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:<ref group=M>Cox (1881), p. 186 for Weierstrass coordinates; (1881/82), pp. 193–194 for Lorentz transformation. On p. 193, the misprinted expression <math>x^{2}-y^{2}-z^{2}</math> should read <math>z^{2}-y^{2}-x^{2}</math></ref> :<math>\begin{matrix}z^{2}-x^{2}-y^{2}=1\\ z^{2}-y^{2}-x^{2}=Z^{2}-Y^{2}-X^{2}\\ \hline \begin{align}x & =l_{1}X+l_{2}Y+l_{3}Z\\ y & =m_{1}X+m_{2}Y+m_{3}Z\\ z & =n_{1}X+n_{2}Y+n_{3}Z\\ \\ X & =l_{1}x+m_{1}y-n_{1}z\\ Y & =l_{2}x+m_{2}y-n_{2}z\\ Z & =l_{3}x+m_{3}y-n_{3}z \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} & =0\\ \\ l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1\\ l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0\\ n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1 \end{align} }</math> with <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1 \end{align} }</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' up to a sign change in the inverse transformation.</p> Cox also gave the Weierstrass coordinates and their transformation in hyperbolic space:<ref group=M>Cox (1881), pp. 199, 206–207</ref> :<math>\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\ w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\ \hline \begin{align}x & =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\ y & =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\ z & =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\ w & =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\ \\ x' & =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\ y' & =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\ z' & =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\ w' & =l_{4}x+m_{4}y+n_{4}z-r_{4}w \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} & =1\\ l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} & =1\\ l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} & =0\\ l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} & =0\\ l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} & =0\\ l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}</math> with <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' up to a sign change in both the first as well as inverse transformation.</p> === {{anchor|Hill}} Hill (1882) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations<ref group=M>Hill (1882), pp. 323–325</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\ k\left(\sin^{2}E+\cos^{2}E-1\right)\\ \hline \begin{align} & & \cos E' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ & \mathbf{(1)} & \sin E' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \hline \\ & & x & =\alpha u+\alpha'u'+\alpha''u''\\ & & y & =\beta u+\beta'u'+\beta''u''\\ & & z & =\gamma u+\gamma'u'+\gamma''u''\\ & \mathbf{(2)}\\ & & u & =-\alpha x-\beta y+\gamma z\\ & & u' & =\alpha'x+\beta'y'-\gamma'z\\ & & u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =-1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =1\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ (k=-1)\\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =-k\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .</p> === {{anchor|Picard}} Picard (1882-1884) – Quadratic forms === [[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to "hyperfuchsian" functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:<ref group=M>Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system</ref><ref>Dickson (1923), pp. 280-281</ref> :<math>\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\ \\ \mathbf{(1)}\ \begin{align}x & =M_{1}X+P_{1}Y+R_{1}Z\\ y & =M_{2}X+P_{2}Y+R_{2}Z\\ z & =M_{3}X+P_{3}Y+R_{3}Z \end{align} \\ \\ \left[\begin{align}[][x,y,z]=\text{complex}\\ \left[x_{0},y_{0},z_{0}\right]=\text{conjugate} \end{align} \right]\\ \\ \hline \\ x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\ x=x'+ix'',\quad y=y'+iy''\\ \\ \mathbf{(2)}\ \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} & =1\\ P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0\\ P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ \\ M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} & =1\\ M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} & =1\\ M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} & =-1\\ \mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} & =0\\ \mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} & =0\\ \mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} & =0\\ \\ \left[\begin{align}[][M,P,R\dots]=\text{complex}\\ \left[\mu,\pi,\rho\dots\right]=\text{conjugate} \end{align} \right] \end{align} }\right.\end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x²+y²-z²=X²+Y²-Z²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:<ref group=M>Picard (1884a), p. 13</ref> :<math>\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\ \hline \begin{align}U & =\mathcal{A}u+\mathcal{B}v\\ V & =\mathcal{C}u+\mathcal{D}v \end{align} \left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} & =1\\ \mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} & =0\\ \mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} & =-1\\ \mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} & =1 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U²-V²=u²-v²''.</p> Or in (1884b):<ref group=M>Picard (1884b), p. 416</ref> :<math>\begin{matrix}xx_{0}+yy_{0}-1=0\\ \hline \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884c):<ref group=M>Picard (1884c), pp. 123–124; 163</ref> :<math>\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\ \hline \mathbf{(1)}\ \begin{align}U & =Mu+Pv+Rw\\ V & =M'u+P'v+R'w\\ W & =M''u+P''v+R''w\\ \\ u & =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\ v & =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\ w & =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W \end{align} \left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} & =1\\ PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} & =1\\ RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} & =-1\\ MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} & =0\\ MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} & =0\\ PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} & =0\\ \\ MM_{0}+PP_{0}-RR_{0} & =1\\ M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} & =1\\ M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} & =-1\\ M_{0}M'+P_{0}P'-R_{0}R' & =0\\ M_{0}M''+P_{0}P''-R_{0}R'' & =0\\ M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' & =0 \end{align} }\right.\\ \hline \text{Invariance of unit hypersphere:}\\ \mathbf{(2)}\ \begin{align}\xi' & =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\ \eta' & =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''} \end{align} \left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} & =1\\ BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} & =1\\ CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} & =-1\\ AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} & =0\\ AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} & =0\\ BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U²+V²-W²=u²+v²-w²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing <math>\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1</math>.</p> === {{anchor|Callandreau}} Callandreau (1885) – Homography === Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations<ref group=M>Callandreau (1885), pp. A.7; A.12</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\ {\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\ \hline \begin{align}\cos\varepsilon' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \sin\varepsilon' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T} \end{align} \left|{\scriptstyle \begin{align} & \left(k=1\right)\\ \alpha^{2}+\beta^{2}-\gamma^{2} & =-k & \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =+k & \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =+k & \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-1 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-1 & \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =+1 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>.</p> ==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations=== In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions.<ref group=M>Lie (1885/86), p. 411</ref> In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):<ref group=M>Werner (1889), pp. 4, 28</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\ \hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\ \text{where}\\ \left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\ \left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\ \left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right] \end{matrix}</math> More generally, Lie (1890)<ref group=M>Lie (1890a), p. 295;</ref> defined non-Euclidean motions in terms of two forms <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0</math> in which the imaginary form with <math>+1</math> denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:<ref group=M>Lie (1890a), p. 311</ref> :<math>\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\ \hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n) \end{matrix}</math> Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form: :<math>x^{2}+y^{2}-1=0</math><ref group=M>Lie (1893), p. 474</ref> or <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0</math><ref group=M>Lie (1893), p. 479</ref> or <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math>.<ref group=M>Lie (1893), p. 481</ref> <p style="background-color:Beige;border:1px solid black">The group of hyperbolic motions is isomorphic to the Lorentz group. The interval <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math> becomes the Lorentz interval <math>x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0</math> by setting <br><math>(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)</math></p> ==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates=== [[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':<ref group=M>Gérard (1892), pp. 40–41</ref> :<math>\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\ X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\ \hline \begin{align}X & =aX'+a'Y'+a''Z'\\ Y & =bX'+b'Y'+b''Z'\\ Z & =cX'+c'Y'+c''Z'\\ \\ X' & =aX+bY-cZ\\ Y' & =a'X+b'Y-c'Z\\ Z' & =-a''X-b''Y+c''Z \end{align} \left|\begin{align}a^{2}+b^{2}-c^{2} & =1\\ a^{\prime2}+b^{\prime2}-c^{\prime2} & =1\\ a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} & =-1\\ aa'+bb'-cc' & =0\\ a'a''+b'b''-c'c'' & =0\\ a''a+b''b-c''c & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> He gave the case of translation as follows:<ref group=M name=gerard>Gérard (1892), pp. 40–41</ref> :<math>\begin{align}X & =Z_{0}X'+X_{0}Z'\\ Y & =Y'\\ Z & =X_{0}X'+Z_{0}Z' \end{align} \ \text{with}\ \begin{align}X_{0} & =\operatorname{sh}OO'\\ Z_{0} & =\operatorname{ch}OO' \end{align} </math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz boost ({{equationNote|3b}}).</p> ==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates=== [[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:<ref group=M>Hausdorff (1899), p. 165, pp. 181-182</ref> :<math>\begin{matrix}p^{2}-x^{2}-y^{2}=1\\ \hline \begin{align}x & =a_{1}x'+a_{2}y'+x_{0}p'\\ y & =b_{1}x'+b{}_{2}y'+y_{0}p'\\ p & =e_{1}x'+e_{2}y'+p_{0}p'\\ \\ x' & =a_{1}x+b_{1}y-e_{1}p\\ y' & =a_{2}x+b_{2}y-e_{2}p\\ -p' & =x_{0}x+y_{0}y-p_{0}p \end{align} \left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} & =1\\ a_{2}^{2}+b_{2}^{2}-e_{2}^{2} & =1\\ -x_{0}^{2}-y_{0}^{2}+p_{0}^{2} & =1\\ a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} & =0\\ a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} & =0\\ a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} & =0\\ \\ a_{1}^{2}+a_{2}^{2}-x_{0}^{2} & =1\\ b_{1}^{2}+b_{2}^{2}-y_{0}^{2} & =1\\ -e_{1}^{2}-e_{2}^{2}+p_{0}^{2} & =1\\ b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} & =0\\ a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} & =0\\ a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates === In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting <math>k=\sqrt{-1}R</math> with ''R'' as real quantity):<ref group=M>Woods (1901/02), p. 98, 104</ref> :<math>\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\ \hline \begin{align}u' & =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ v' & =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ w' & =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}} \end{align} \left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} & =k^{2}\\ (i=1,2,3)\\ k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} & =1\\ \alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} & =0\\ (i,h=1,2,3,4;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''²=-1.</p> Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with <math>k=\sqrt{-1}k</math> for hyperbolic space):<ref group=M>Woods (1903/05), pp. 45–46; p. 48)</ref> :<math>\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\ ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\ \hline \begin{align}x_{1}^{\prime} & =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\ x_{2}^{\prime} & =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\ x_{3}^{\prime} & =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\ x_{0}^{\prime} & =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0} \end{align} \left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) & =1\\ \delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) & =k^{2}\\ (i=1,2,3)\\ \delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) & =0\\ (i,h=0,1,2,3;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''²=-1.</p> ==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates=== [[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''<ref group=M>Liebmann (1904/05), p. 168; pp. 175–176</ref> :<math>\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\ \hline \begin{align}x_{1} & =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\ y_{1} & =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\ x_{1} & =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\ \\ x & =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\ y & =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\ p & =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1} \end{align} \left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} & =1\\ -\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} & =1\\ -\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} & =1\\ \alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} & =0\\ \alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} & =0\\ \alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==References== ===Historical mathematical sources=== {{reflist|3|group=M}} *{{#section:History of Topics in Special Relativity/mathsource|apo}} *{{#section:History of Topics in Special Relativity/mathsource|bour56att}} *{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} *{{#section:History of Topics in Special Relativity/mathsource|chas29}} *{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} *{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} *{{#section:History of Topics in Special Relativity/mathsource|fris76}} *{{#section:History of Topics in Special Relativity/mathsource|gau98}} *{{#section:History of Topics in Special Relativity/mathsource|gau18}} *{{#section:History of Topics in Special Relativity/mathsource|ger92}} *{{#section:History of Topics in Special Relativity/mathsource|gud30}} *{{#section:History of Topics in Special Relativity/mathsource|haus99}} *{{#section:History of Topics in Special Relativity/mathsource|hill82}} *{{#section:History of Topics in Special Relativity/mathsource|jac27}} *{{#section:History of Topics in Special Relativity/mathsource|jac32a}} *{{#section:History of Topics in Special Relativity/mathsource|jac32b}} *{{#section:History of Topics in Special Relativity/mathsource|jac33}} *{{#section:History of Topics in Special Relativity/mathsource|kil77}} *{{#section:History of Topics in Special Relativity/mathsource|kil79}} *{{#section:History of Topics in Special Relativity/mathsource|kil84}} *{{#section:History of Topics in Special Relativity/mathsource|kil85}} *{{#section:History of Topics in Special Relativity/mathsource|kil93}} *{{#section:History of Topics in Special Relativity/mathsource|kil97}} *{{#section:History of Topics in Special Relativity/mathsource|klei71}} *{{#section:History of Topics in Special Relativity/mathsource|klei73}} *{{#section:History of Topics in Special Relativity/mathsource|lag73}} *{{#section:History of Topics in Special Relativity/mathsource|leb37}} *{{#section:History of Topics in Special Relativity/mathsource|lie85}} *{{#section:History of Topics in Special Relativity/mathsource|lie90}} *{{#section:History of Topics in Special Relativity/mathsource|lie93}} *{{#section:History of Topics in Special Relativity/mathsource|lieb04}} *{{#section:History of Topics in Special Relativity/mathsource|lop}} *{{#section:History of Topics in Special Relativity/mathsource|pic82}} *{{#section:History of Topics in Special Relativity/mathsource|pic84a}} *{{#section:History of Topics in Special Relativity/mathsource|pic84b}} *{{#section:History of Topics in Special Relativity/mathsource|pic84c}} *{{#section:History of Topics in Special Relativity/mathsource|poin81a}} *{{#section:History of Topics in Special Relativity/mathsource|poin81b}} *{{#section:History of Topics in Special Relativity/mathsource|poin87}} *{{#section:History of Topics in Special Relativity/mathsource|sal62}} *{{#section:History of Topics in Special Relativity/mathsource|som63}} *{{#section:History of Topics in Special Relativity/mathsource|wedd47}} *{{#section:History of Topics in Special Relativity/mathsource|wern89}} *{{#section:History of Topics in Special Relativity/mathsource|woo01}} *{{#section:History of Topics in Special Relativity/mathsource|woo03}} ===Secondary sources=== {{reflist|3}} {{#section:History of Topics in Special Relativity/secsource|L1}} [[Category:Special Relativity]] [[Category:History of Physics]] aj82exm30z0d79vkoidb5eqr5zr5p9s 2416266 2416152 2022-08-18T19:22:46Z D.H 52339 /* {{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas */ wikitext text/x-wiki {{../Lorentz transformation (header)}} ==Most general Lorentz transformations== ===General quadratic form=== The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as<ref>Bôcher (1907), chapter X</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}\begin{align}q=\sum_{0}^{n}A_{ij}x_{i}x_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align} & =q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\ b=\sum_{0}^{n}A_{ij}x_{i}y_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y} & =b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}' \end{align} \quad\left(A_{ij}=A_{ji}\right)\\ \hline \left.\begin{align}x_{i}^{\prime} & =\sum_{j=0}^{n}g_{ij}x_{j}=\mathbf{g}\cdot\mathbf{x}\\ x_{i} & =\sum_{j=0}^{n}g_{ij}^{(-1)}x_{j}^{\prime}=\mathbf{g}^{-1}\cdot\mathbf{x}' \end{align} \right|\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}' \end{matrix}</math>|{{equationRef|Q1}}}} The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc. ===Most general Lorentz transformation=== The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:<ref name=ratcliffe>Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1</ref><ref>Naimark (1964), 2 in four dimensions</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} & =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\ -x_{0}y_{0}+\cdots+x_{n}y_{n} & =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime} \end{align} \\ \hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\ \downarrow\\ \begin{align}x_{0}^{\prime} & =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\ x_{1}^{\prime} & =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\ & \dots\\ x_{n}^{\prime} & =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn} \end{align} \\ \\ \mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\ \downarrow\\ \begin{align}x_{0} & =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\ x_{1} & =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\ & \dots\\ x_{n} & =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn} \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} & =\mathbf{g}^{-1}\\ \mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} & =\mathbf{A}\\ \mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} & =\mathbf{A}\\ \\ \end{align} \\ \begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \end{matrix} \end{matrix}</math>|{{equationRef|1a}}}} The invariance of the Lorentz interval with ''n''=1 between axes and conjugate diameters of hyperbolas was known for a long time since [[#Apo|Apollonius (ca. 200 BC)]]. Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.<ref>Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.</ref><ref>Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.</ref> They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881/82)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the "absolute" form <math>-x_{0}^{2}+\cdots+x_{n}^{2}=0</math> as discussed by [[#Klein|Klein (1871-73)]].<ref group=M>Killing (1885), p. 71</ref><ref>Müller (1910), p. 661, in particular footnote 247.</ref><ref>Sommerville (1911), p. 286, section K6.</ref> In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> by [[#Killing3|Killing (1888-1897)]]. ===Most general Lorentz transformation of velocity=== If <math>x_{i},\ x_{i}^{\prime}</math> in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates <math>u_{s},\ u_{s}^{\prime}</math> follow by :<math>\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)</math> defined by <math>u_{1}^{2}+u_{2}^{2}+\dots+u_{n}^{2}\le1</math> so that the Lorentz transformation becomes a [[w:homography]] inside the [[w:unit hypersphere]], which [[w:John Lighton Synge]] called "the most general formula for the composition of velocities" in terms of special relativity<ref>Synge (1955), p. 129 for ''n''=3</ref> (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})): {{NumBlk|:|<math>\begin{align}u_{s}^{\prime} & =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\ \\ u_{s} & =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}} \end{align} \left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \right.</math>|{{equationRef|1b}}}} Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere <math>-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0</math> were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]]. ==Historical notation== ==={{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas === [[w:Apollonius of Perga]] (c. 240–190 BC) defined the following proposition related to conjugate diameters of hyperbolas, which was translated and adapted to the modern reader by [[w:Thomas Heath (classicist)|w:Thomas Heath]] as follows:<ref group=M>Apollonius/Heath (1896), Proposition 129; (Apollonius, Book VII, Prop. 13).</ref> :[..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. <p style="background-color:Beige;border:1px solid black">Apollonius' result produces the relation <math>-x_{0}^{\prime2}+x_{1}^{\prime2}=-x_{0}^{2}+x_{1}^{2}</math> in agreement with the invariance of the Lorentz interval, so that the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' can be interpreted as mapping from one pair of axes of a hyperbola to a pair of conjugate diameters.</p> This well known result was demonstrated in terms of [[w:analytic geomety]] by [[w:Guillaume de l'Hôpital]] (1707), who summarized Apollonius' proposition as follows:<ref group=M>l'Hôpital (1707), Third book, Prop. XII, p. 76.</ref> :The difference of the squares of any two conjugate diameters ''Mm, Ss'' is equal to the difference of the squares of the two axes ''Aa, Bb.'' We have to prove that <math>\overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}</math>, or that <math>\overline{CM}^{2}-\overline{CS}^{2}=\overline{CA}^{2}-\overline{CB}^{2}</math>. ===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}=== After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows<ref group=M>Lagrange (1773/75), section 22</ref> :<math>\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\ \hline \begin{align}y & =Ms+Nx\\ z & =ms+nx \end{align} \left|\begin{matrix}\begin{align}P & =pM^{2}+2qMm+rm^{2}\\ Q & =pMN+q(Mn+Nm)+rmn\\ R & =pN^{2}+2qNn+rn^{2} \end{align} \\ \downarrow\\ PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2} \end{matrix}\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.</p> ==={{anchor|Gauss}} Gauss (1798–1818)=== ===={{anchor|Gauss1}} Binary quadratic forms==== The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:<ref group=M>Gauss (1798/1801), articles 157–158;</ref> :<math>\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\ F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\ \hline \begin{align}x & =\alpha x'+\beta y'\\ y & =\gamma x'+\delta y'\\ \\ x' & =\delta x-\beta y\\ y' & =-\gamma x+\alpha y \end{align} \left|\begin{matrix}\begin{align}a' & =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\ b' & =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\ c' & =a\beta^{2}+2b\beta\delta+c\delta^{2} \end{align} \\ \downarrow\\ b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2} \end{matrix}\right. \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called "proper equivalent" if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.<ref group=M>Gauss (1798/1801), section 159</ref> <p style="background-color:Beige;border:1px solid black">The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.</p> ===={{anchor|Gauss2}} Ternary quadratic forms==== Gauss (1798/1801)<ref group=M>Gauss (1798/1801), articles 266–285</ref> also discussed ternary quadratic forms with the general expression :<math>\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\\ g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)\\ \hline \begin{align}x & =\alpha y+\beta y'+\gamma y''\\ x' & =\alpha'y+\beta'y'+\gamma'y''\\ x'' & =\alpha''y+\beta''y'+\gamma''y'' \end{align} \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x²+y²+z²'', or indefinite in the case of different signs such as ''x²+y²-z²''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:<ref group=M>Gauss (1798/1801), article 277</ref> :<math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\\ \left(\begin{matrix}1, & -1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval <math>x^{2}+x^{\prime2}-x^{\prime\prime2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting: <br><math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)=\left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right)</math></p> ===={{anchor|Gauss4}} Homogeneous coordinates==== Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression :<math>(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E</math> into :<math>G+G'\cos T^{2}+G''\sin T^{2}</math> in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:<ref group=M>Gauss (1818), pp. 5–10</ref> :<math>\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\ k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\ \hline \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =k & \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =-k\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-k & \beta\beta-\beta'\beta'-\beta''\beta'' & =-k\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-k & \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =+k\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} }\right.\\ \hline k=1\\ \begin{align}t\cos E & =\alpha+\alpha'\cos T+\alpha''\sin T\\ t\sin E & =\beta+\beta'\cos T+\beta''\sin T\\ t & =\gamma+\gamma'\cos T+\gamma''\sin T \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}}) ''(n=2)''. Further setting <math>[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that :<math>aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz</math> can be transformed into :<math>Guu+G'u'u'+G''u''u''</math>, in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:<ref group=M>Gauss (1818), pp. 9–10</ref> :<math>\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =-\alpha x-\beta y+\gamma z\\ u' & =\alpha'x+\beta'y-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right.</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' satisfying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>, and can be related to Gauss' previous equations in terms of homogeneous coordinates <math>\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]</math>.</p> ==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation in 1827:<ref group=M>Jacobi (1827), p. 235, 239–240</ref> :<math>{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\ k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\ \hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P & =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\cos\vartheta & =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\sin\vartheta & =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \\ \cos\psi & =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\cos\varphi & =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\sin\varphi & =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta} \end{align} \\ \\ \hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z & =m\\ \alpha'\mu+\beta'x+\gamma'y+\delta'z & =m'\\ \alpha''\mu+\beta''x+\gamma''y+\delta''z & =m''\\ \alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z & =m'''\\ \\ Am+A'm'+A''m''+A'''m''' & =\mu\\ Bm+B'm'+B''m''+B'''m''' & =x\\ Cm+C'm'+C''m''+C'''m''' & =y\\ Dm+D'm'+D''m''+D'''m''' & =z\\ \\ \end{align} \\ \begin{align}\alpha & =-kA, & \beta & =-kB, & \gamma & =-kC, & \delta & =kD,\\ \alpha' & =kA', & \beta' & =kB', & \gamma' & =kC', & \delta' & =-kD',\\ \alpha'' & =kA'', & \beta'' & =kB'', & \gamma'' & =kC'', & \delta'' & =-kD'',\\ \alpha''' & =kA''', & \beta''' & =kB''', & \gamma''' & =kC''', & \delta''' & =-kD''', \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =-k\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' & =k\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' & =k\\ \alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' & =k\\ \alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' & =0\\ \alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' & =0\\ \alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' & =0\\ \alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' & =0\\ \alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' & =0\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' & =0\\ \\ -\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' & =k\\ -\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' & =k\\ -\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' & =k\\ -\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' & =-k\\ -\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' & =0\\ -\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' & =0\\ -\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' & =0\\ -\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' & =0\\ -\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' & =0\\ -\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' & =0 \end{align} \end{matrix}} \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] & =\left[u_{1},\ u_{2},\ u_{3}\right]\\{} [\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] & =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right] \end{align} }</math> and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing <math>m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}</math>.</p> Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):<ref group=M>The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.</ref> :<math>{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\ \mathbf{(1)}\begin{align}x & =\alpha s+\alpha's'+\alpha''s''\\ y & =\beta s+\beta's'+\beta''s''\\ z & =\gamma s+\gamma's'+\gamma''s''\\ \\ s & =\alpha x+\beta y+\gamma z\\ s' & =\alpha'x+\beta'y+\gamma'z\\ s'' & =\alpha''x+\beta''y+\gamma''z \end{align} \left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma & =1 & \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' & =1\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma' & =1 & \beta\beta+\beta'\beta'+\beta''\beta'' & =1\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' & =1 & \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' & =1\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma+\beta'\gamma'+\beta''\gamma'' & =0\\ \alpha''\alpha+\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' & =0\\ \alpha\alpha'+\beta\beta'+\gamma\gamma' & =0 & \alpha\beta+\alpha'\beta'+\alpha''\beta'' & =0 \end{align} \right.\\ \hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\ \left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\ \hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\ \left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\ \hline \begin{align}\cos\phi & =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \cos\eta & =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\ \sin\phi & =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \sin\eta & =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi} \end{align} \end{matrix}\\ \hline \\ \mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\ \hline \begin{align}y & =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, & s' & =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\ z & =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, & s'' & =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z}, \end{align} \end{matrix} \end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma & =1\\ \alpha'\alpha'-\beta'\beta'-\gamma'\gamma' & =-1\\ \alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' & =-1\\ \alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' & =0\\ \alpha''\alpha-\beta''\beta-\gamma''\gamma & =0\\ \alpha\alpha'-\beta\beta'-\gamma\gamma' & =0\\ \\ \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =1\\ \beta\beta-\beta'\beta'-\beta''\beta'' & =-1\\ \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =-1\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} \right. \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.</p> Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:<ref group =M>Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.</ref> :<math>{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\ \hline \mathbf{(1)\ }\begin{align}y_{\varkappa} & =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\ x_{\varkappa} & =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\ \\ \frac{y_{\varkappa}}{y_{n}} & =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\ \frac{x_{\varkappa}}{x_{n}} & =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}} \end{align} \left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} & =0\\ \alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} & =1\\ \\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} & =0\\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} & =1 \end{align} \right.\\ \hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\ \alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\ \hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} & =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\ \\ \xi_{\varkappa} & =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}} \end{align} \\ \hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1 \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.</p> He also stated the following transformation leaving invariant the Lorentz interval:<ref group=M>Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.</ref> :<math>\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\ \hline {\scriptstyle \begin{align}u & =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\ u_{1} & =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\ & \dots\\ u_{n-1} & =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\ \\ w & =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\ w_{1} & =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\ & \dots\\ w_{n-1} & =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1} \end{align} \left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =-1\\ \alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =0\\ \alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} & =0\\ \\ \alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} & =-1\\ \alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} & =0\\ \alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} & =0 \end{align} \text{ }\right.} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.</p> ==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids === [[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:<ref group=M>Chasles (1829), p. 139</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(a)\\ \\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(b)\\ \\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(c)\\ \\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(d) \end{matrix}</math> He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:<ref group=M>Chasles (1829), p. 141</ref> :<math>\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\ \gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\ \alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\ \beta''\quad\Rightarrow\quad\beta''\sqrt{-1} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:<ref group=M>Chasles (1829), pp. 143-144</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}x & =lx'+l'y'+l''z'\\ y & =mx'+m'y'+m''z'\\ z & =nx'+n'y'+n''z' \end{align} \\ \left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} & =0\\ \frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} & =0\\ \frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} & =0 \end{align} \right\} \\ \hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\ \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1 \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Chasles defined the conditional equations of ''l,m,n'' in the same way as those of <math>\alpha,\beta,\gamma</math> in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.</p> He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:<ref group=M>Chasles (1829), pp. 145-146</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =a^{2}\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =b^{2}\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(A)\\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(B)\\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =a^{2}\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =b^{2}\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(C)\\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(D) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates=== [[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation<ref group=M>Lebesgue (1837), pp. 338-341</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\ \hline {\scriptstyle \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\ \dots\\ y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} & =1 & (10)\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} & =0 & (11)\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} & =1 & (13)\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} & =0 & (14) \end{align} \right.} \end{matrix}</math> In order to achieve the invariance of the Lorentz interval<ref group=M>Lebesgue (1837), pp. 353–354</ref> :<math>x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}</math> he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign. <p style="background-color:Beige;border:1px solid black">These instructions give Lorentz transformation ({{equationNote|1a}}) in the form: <br><math>{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \hline \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\ \dots\\ -y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} & =1\\ a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} & =-1\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} & =0\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} & =1\\ a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} & =-1\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} & =0 \end{align} \right. \end{matrix}}</math></p> He went on to redefine the variables of the Lorentz interval and its transformation:<ref group=M>Lebesgue (1837), pp. 353–355</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \downarrow\\ \begin{align}x_{1} & =x_{n}\cos\theta_{1}, & x_{2} & =x_{n}\cos\theta_{2},\dots & x_{n-1} & =x_{n}\cos\theta_{n-1}\\ y_{1} & =y_{n}\cos\phi_{1}, & y_{2} & =y_{n}\cos\phi_{2},\dots & y_{n-1} & =y_{n}\cos\phi_{n-1} \end{align} \\ \downarrow\\ \cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\ \cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\ \hline \\ \cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\ (i=1,2,3\dots n) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Setting <math>[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]</math> it is equivalent to Lorentz transformation ({{equationNote|1b}}).</p> ==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids=== Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&alpha;), from which he derived equations (&beta;) and others:<ref group=M>Weddle (1847), p. 274</ref> :<math>\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1, & l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1, & l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} & =0\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1, & l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0 \end{align} \right\} & \dots(\alpha)\\ \\ \left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1, & l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1, & l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} & =0\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1, & m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0 \end{align} \right\} & \dots(\beta) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>, he defined three conjugate points <math>(x_{1}\dots,y_{1}\dots,z_{1}\dots)</math> on those two conjugate hyperboloids, related to each other in the same way as equations (&alpha;, &beta;) stated above:<ref group=M>Weddle (1847), pp. 275-276</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} & =0\\ \frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} & =0\\ \frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} & =0 \end{align} \quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} & =1\\ \frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} & =1\\ \frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} & =-1 \end{align} \\ \begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} & =a^{2}\\ y_{1}^{2}+y_{2}^{2}-y_{3}^{2} & =b^{2}\\ z_{1}^{2}+z_{2}^{2}-z_{3}^{2} & =-c^{2} \end{align} \quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} & =0\\ x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} & =0\\ y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} & =0 \end{align} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black"> These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:<ref group=M>Bour (1856), pp. 61; 64–65</ref> :<math>\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\ \hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \\ \hline \\ k=+1\\ t=\gamma+\gamma'\cos T+\gamma''\sin T,\\ 1=u,\ \cos T=u',\ \sin T=u',\\ t=z,\ t\cos E=x,\ t\sin E=y\\ \downarrow\\ \mathbf{(2)}\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =\gamma z-\alpha x-\beta y\\ u' & =\alpha'x+\beta'y'-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} & =k\\ -\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} & =-k\\ -\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} & =-k\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =k\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} } \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>. Furthermore, setting <math>[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math> in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> === {{anchor|Somov}} Somov (1863) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:<ref group=M>Somov (1863), pp. 12–14; p. 18 for differentials.</ref> :<math>\begin{matrix}\left.\begin{align}\cos\phi & =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\ \sin\phi & =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''} \end{align} \right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\ \cos^{2}\psi+\cos^{2}\psi=1 \end{matrix}\\ \hline \mathbf{(1)}\ \begin{align}\cos\phi & =x, & \cos\psi & =x'\\ \sin\phi & =y, & \sin\psi & =y' \end{align} \ \left|\begin{align}x & =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\ y & =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=1\\ x^{\prime2}+y^{\prime2}=1 \end{matrix}\\ \hline \begin{align}\cos\phi & =\frac{x}{z}, & \cos\psi & =\frac{x'}{z'}\\ \sin\phi & =\frac{y}{z}, & \sin\psi & =\frac{y'}{z'} \end{align} \ \left|\begin{align}\frac{x}{z} & =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\ \frac{y}{z} & =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\ x^{\prime2}+y^{\prime2}=z^{\prime2} \end{matrix}\\ \hline \mathbf{(2)}\ \left.\begin{align}x & =mx'+ny'+sz'\\ y & =m'x'+n'y'+s'z'\\ z & =m''x'+n''y'+s''z'\\ \\ x' & =mx+m'y-m''z\\ y' & =nx+n'y-n''z\\ z' & =-sx-s'y+s''z\\ \\ dx & =mdx'+ndy'+sdz'\\ dy & =m'dx'+n'dy'+s'dz'\\ dz & =m''dx'+n''dy'+s''dz' \end{align} \right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} & =1\\ n^{2}+n^{\prime2}-n^{\prime\prime2} & =1\\ -s^{2}-s^{\prime2}+s^{\prime\prime2} & =1\\ ns+n's'-n''s'' & =0\\ sm+s'm'-s''m'' & =0\\ mn+m'n'-m''n'' & =0\\ \\ m^{2}+n^{2}-s^{2} & =1\\ m^{\prime2}+n^{\prime2}-s^{\prime2} & =1\\ -m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} & =1\\ -m'm''-n'n''+s's'' & =0\\ -m''m-n''n+s''s & =0\\ mm'+nn'-ss' & =0 \end{align} }\\ dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Klein}} Klein (1871-73) – Cayley absolute and non-Euclidean geometry=== Elaborating on [[w:Arthur Cayley]]'s (1859) definition of an "absolute" ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a "fundamental [[w:conic section]]" in order to discuss motions such as rotation and translation in the non-Euclidean plane.<ref group=M>Klein (1871), pp. 601–602</ref> This was elaborated in (1873) when he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to a two-sheet [[w:hyperboloid]].<ref group=M>Klein (1873), pp. 127-128</ref> <p style="background-color:Beige;border:1px solid black">Klein's representation of hyperbolic space in terms of a two-sheet hyperboloid and its accompanied quadratic form suggests that Lorentz transformations can be geometrically interpreted as motions or isometries in hyperbolic space.</p> ==={{anchor|Killing}} Killing (1878–1893)=== ===={{anchor|Killing1}} Weierstrass coordinates==== [[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form :<math>k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}</math><ref group=M>Killing (1877/78), p. 74; Killing (1880), p. 279</ref> with <math>ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}</math><ref group=M>Killing (1880), eq. 25 on p. 283</ref> or<ref group=M>Killing (1880), p. 283</ref> :<math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> where ''k'' is the reciprocal measure of curvature, <math>k^{2}=\infty</math> denotes [[w:Euclidean geometry]], <math>k^{2}>0</math> [[w:elliptic geometry]], and <math>k^{2}<0</math> hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.<ref group=M>Killing (1877/78), eq. 25 on p. 283</ref> In (1879/80) he tried to formulate the corresponding transformations by plugging <math>k^{2}</math> into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:<ref group=M>Killing (1879/80), p. 274</ref> <math>\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\ \hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\ -k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\ k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, & -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}} \end{matrix}\\ \left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right) \end{matrix}</math> In (1885) he wrote the Weierstrass coordinates and their transformation as follows:<ref group=M>Killing (1885), pp. 18, 28–30, 53</ref> :<math>\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\ k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\ ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\ \hline \begin{align}k^{2}p' & =k^{2}wp+w'x+w''y\\ x' & =ap+a'x+a''y\\ y' & =bp+b'x+b''y\\ \\ k^{2}p & =k^{2}wp'+ax'+by'\\ x & =w'p'+a'x+b'y'\\ y & =w''p'+a''x'+b''y' \end{align} \left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} & =k^{2}\\ \frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} & =1\\ \frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} & =1\\ aw+a'w'+a''w'' & =0\\ bw+b'w'+b''w'' & =0\\ \frac{ab}{k^{2}}+a'b'+a''b'' & =0\\ \\ k^{2}w^{2}+a^{2}+b^{2} & =k^{2}\\ \frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} & =1\\ \frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} & =1\\ ww'+aa'+bb' & =0\\ ww''+aa''+bb'' & =0\\ \frac{w'w''}{k^{2}}+a'a''+b'b'' & =0 \end{align} }\right. \end{matrix}</math> In (1885) he also gave the transformation for ''n'' dimensions:<ref group=M>Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222</ref><ref>Ratcliffe (1994), § 3.6</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\ \hline \left.\begin{align}k^{2}\xi_{0} & =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\ \xi_{\varkappa} & =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\ \\ k^{2}x_{0} & =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\ x_{\varkappa} & =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n} \end{align} \right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ \frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} & =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa) \end{align} } \end{matrix}</math> In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.<ref group=M>Killing (1884/85), pp. 4–5</ref> Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting <math>k^{2}=-1</math> and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid <math>p^{2}-x^{2}-y^{2}=1</math> (known as [[w:hyperboloid model]]),<ref group=M>Killing (1885), Note 9 on p. 260</ref><ref name=rey /> by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space. <p style="background-color:Beige;border:1px solid black">All of Killing's transformations between 1879 and 1885 don't work when <math>k^{2}</math> is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> Finally, in (1893) he wrote:<ref group=M>Killing (1893), see pp. 144, 327–328</ref> :<math>\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\ \hline \begin{align}t' & =at+bu+cv\\ u' & =a't+b'u+c'v\\ v' & =a''t+b''u+c''v \end{align} \left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} & =k^{2}\\ k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} & =1\\ k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} & =1\\ k^{2}ab+a'b'+a''b'' & =0\\ k^{2}ac+a'c'+a''c'' & =0\\ k^{2}bc+b'c'+b''c'' & =0 \end{align} \right. \end{matrix}</math> and in ''n'' dimensions<ref group=M>Killing (1893), pp. 314–316, 216–217</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\ ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\ \hline \begin{align}y_{0} & =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\ y_{1} & =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\ & \,\,\,\vdots\\ y_{n} & =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n} \end{align} \left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} & =1\\ k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} & =0\\ (\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> ===={{anchor|Killing3}} Infinitesimal transformations and Lie group==== After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions, Killing (1887/88)<ref group=M>Killing (1887/88a), pp. 274–275</ref> defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere: :<math>\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\ \hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\ \text{where}\\ \left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\ \left[\iota\ne\varkappa\ne\lambda\ne\mu\right] \end{matrix}</math> and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:<ref group=M>Killing (1892), p. 177</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ \hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\ \text{where}\\ \left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f; \end{matrix}</math> In (1897/98) he showed the relation between Weierstrass coordinates <math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> and corrdinates <math>k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0</math> used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:<ref group=M>Killing (1897/98), pp. 255–256</ref> :<math>\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} & (a)\\ k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} & (b) \end{matrix}\\ \hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\ \text{where}\\ \left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\ \left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\ \left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\ \hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\ \downarrow\\ k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\ \hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota} \end{matrix} \end{matrix}</math> He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b). <p style="background-color:Beige;border:1px solid black">Setting <math>k^{2}=-1</math> denotes the group of hyperbolic motions and thus the Lorentz group.</p> === {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates === [[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).<ref>Gray (1997)</ref> He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:<ref group=M name=p1>Poincaré (1881a), pp. 133–134</ref><ref>Dickson (1923), pp. 220–221</ref> :<math>\begin{matrix}\begin{align}F & =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\ & =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\ F & =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\ & =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1 \end{align} \\ \hline \begin{align}\xi' & =\alpha\xi+\beta\eta+\gamma\zeta\\ \eta' & =\alpha'\xi+\beta'\eta+\gamma'\zeta\\ \zeta' & =\alpha''\xi+\beta''\eta+\gamma''\zeta \end{align} \left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1\\ \alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right. \end{matrix}</math> He went on to describe the properties of "hyperbolic coordinates".<ref group=M name=poinc>Poincaré (1881b), p. 333</ref><ref name=rey>Reynolds (1993)</ref> Poincaré mentioned the hyperboloid model also in (1887).<ref group=M>Poincaré (1887), p. 206</ref> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> === {{anchor|Cox}} Cox (1881–1883) – Weierstrass coordinates === [[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)<ref name=guder group=M>Gudermann (1830), §1–3, §18–19</ref> and [[w:George Salmon]] (1862)<ref group=M>Salmon (1862), section 212, p. 165</ref> on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)<ref group=M>Frischauf (1876), pp. 86–87</ref> in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:<ref group=M>Cox (1881), p. 186 for Weierstrass coordinates; (1881/82), pp. 193–194 for Lorentz transformation. On p. 193, the misprinted expression <math>x^{2}-y^{2}-z^{2}</math> should read <math>z^{2}-y^{2}-x^{2}</math></ref> :<math>\begin{matrix}z^{2}-x^{2}-y^{2}=1\\ z^{2}-y^{2}-x^{2}=Z^{2}-Y^{2}-X^{2}\\ \hline \begin{align}x & =l_{1}X+l_{2}Y+l_{3}Z\\ y & =m_{1}X+m_{2}Y+m_{3}Z\\ z & =n_{1}X+n_{2}Y+n_{3}Z\\ \\ X & =l_{1}x+m_{1}y-n_{1}z\\ Y & =l_{2}x+m_{2}y-n_{2}z\\ Z & =l_{3}x+m_{3}y-n_{3}z \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} & =0\\ \\ l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1\\ l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0\\ n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1 \end{align} }</math> with <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1 \end{align} }</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' up to a sign change in the inverse transformation.</p> Cox also gave the Weierstrass coordinates and their transformation in hyperbolic space:<ref group=M>Cox (1881), pp. 199, 206–207</ref> :<math>\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\ w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\ \hline \begin{align}x & =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\ y & =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\ z & =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\ w & =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\ \\ x' & =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\ y' & =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\ z' & =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\ w' & =l_{4}x+m_{4}y+n_{4}z-r_{4}w \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} & =1\\ l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} & =1\\ l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} & =0\\ l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} & =0\\ l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} & =0\\ l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}</math> with <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' up to a sign change in both the first as well as inverse transformation.</p> === {{anchor|Hill}} Hill (1882) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations<ref group=M>Hill (1882), pp. 323–325</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\ k\left(\sin^{2}E+\cos^{2}E-1\right)\\ \hline \begin{align} & & \cos E' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ & \mathbf{(1)} & \sin E' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \hline \\ & & x & =\alpha u+\alpha'u'+\alpha''u''\\ & & y & =\beta u+\beta'u'+\beta''u''\\ & & z & =\gamma u+\gamma'u'+\gamma''u''\\ & \mathbf{(2)}\\ & & u & =-\alpha x-\beta y+\gamma z\\ & & u' & =\alpha'x+\beta'y'-\gamma'z\\ & & u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =-1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =1\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ (k=-1)\\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =-k\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .</p> === {{anchor|Picard}} Picard (1882-1884) – Quadratic forms === [[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to "hyperfuchsian" functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:<ref group=M>Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system</ref><ref>Dickson (1923), pp. 280-281</ref> :<math>\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\ \\ \mathbf{(1)}\ \begin{align}x & =M_{1}X+P_{1}Y+R_{1}Z\\ y & =M_{2}X+P_{2}Y+R_{2}Z\\ z & =M_{3}X+P_{3}Y+R_{3}Z \end{align} \\ \\ \left[\begin{align}[][x,y,z]=\text{complex}\\ \left[x_{0},y_{0},z_{0}\right]=\text{conjugate} \end{align} \right]\\ \\ \hline \\ x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\ x=x'+ix'',\quad y=y'+iy''\\ \\ \mathbf{(2)}\ \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} & =1\\ P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0\\ P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ \\ M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} & =1\\ M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} & =1\\ M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} & =-1\\ \mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} & =0\\ \mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} & =0\\ \mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} & =0\\ \\ \left[\begin{align}[][M,P,R\dots]=\text{complex}\\ \left[\mu,\pi,\rho\dots\right]=\text{conjugate} \end{align} \right] \end{align} }\right.\end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x²+y²-z²=X²+Y²-Z²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:<ref group=M>Picard (1884a), p. 13</ref> :<math>\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\ \hline \begin{align}U & =\mathcal{A}u+\mathcal{B}v\\ V & =\mathcal{C}u+\mathcal{D}v \end{align} \left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} & =1\\ \mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} & =0\\ \mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} & =-1\\ \mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} & =1 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U²-V²=u²-v²''.</p> Or in (1884b):<ref group=M>Picard (1884b), p. 416</ref> :<math>\begin{matrix}xx_{0}+yy_{0}-1=0\\ \hline \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884c):<ref group=M>Picard (1884c), pp. 123–124; 163</ref> :<math>\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\ \hline \mathbf{(1)}\ \begin{align}U & =Mu+Pv+Rw\\ V & =M'u+P'v+R'w\\ W & =M''u+P''v+R''w\\ \\ u & =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\ v & =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\ w & =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W \end{align} \left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} & =1\\ PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} & =1\\ RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} & =-1\\ MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} & =0\\ MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} & =0\\ PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} & =0\\ \\ MM_{0}+PP_{0}-RR_{0} & =1\\ M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} & =1\\ M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} & =-1\\ M_{0}M'+P_{0}P'-R_{0}R' & =0\\ M_{0}M''+P_{0}P''-R_{0}R'' & =0\\ M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' & =0 \end{align} }\right.\\ \hline \text{Invariance of unit hypersphere:}\\ \mathbf{(2)}\ \begin{align}\xi' & =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\ \eta' & =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''} \end{align} \left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} & =1\\ BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} & =1\\ CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} & =-1\\ AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} & =0\\ AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} & =0\\ BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U²+V²-W²=u²+v²-w²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing <math>\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1</math>.</p> === {{anchor|Callandreau}} Callandreau (1885) – Homography === Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations<ref group=M>Callandreau (1885), pp. A.7; A.12</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\ {\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\ \hline \begin{align}\cos\varepsilon' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \sin\varepsilon' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T} \end{align} \left|{\scriptstyle \begin{align} & \left(k=1\right)\\ \alpha^{2}+\beta^{2}-\gamma^{2} & =-k & \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =+k & \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =+k & \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-1 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-1 & \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =+1 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>.</p> ==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations=== In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions.<ref group=M>Lie (1885/86), p. 411</ref> In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):<ref group=M>Werner (1889), pp. 4, 28</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\ \hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\ \text{where}\\ \left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\ \left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\ \left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right] \end{matrix}</math> More generally, Lie (1890)<ref group=M>Lie (1890a), p. 295;</ref> defined non-Euclidean motions in terms of two forms <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0</math> in which the imaginary form with <math>+1</math> denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:<ref group=M>Lie (1890a), p. 311</ref> :<math>\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\ \hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n) \end{matrix}</math> Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form: :<math>x^{2}+y^{2}-1=0</math><ref group=M>Lie (1893), p. 474</ref> or <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0</math><ref group=M>Lie (1893), p. 479</ref> or <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math>.<ref group=M>Lie (1893), p. 481</ref> <p style="background-color:Beige;border:1px solid black">The group of hyperbolic motions is isomorphic to the Lorentz group. The interval <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math> becomes the Lorentz interval <math>x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0</math> by setting <br><math>(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)</math></p> ==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates=== [[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':<ref group=M>Gérard (1892), pp. 40–41</ref> :<math>\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\ X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\ \hline \begin{align}X & =aX'+a'Y'+a''Z'\\ Y & =bX'+b'Y'+b''Z'\\ Z & =cX'+c'Y'+c''Z'\\ \\ X' & =aX+bY-cZ\\ Y' & =a'X+b'Y-c'Z\\ Z' & =-a''X-b''Y+c''Z \end{align} \left|\begin{align}a^{2}+b^{2}-c^{2} & =1\\ a^{\prime2}+b^{\prime2}-c^{\prime2} & =1\\ a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} & =-1\\ aa'+bb'-cc' & =0\\ a'a''+b'b''-c'c'' & =0\\ a''a+b''b-c''c & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> He gave the case of translation as follows:<ref group=M name=gerard>Gérard (1892), pp. 40–41</ref> :<math>\begin{align}X & =Z_{0}X'+X_{0}Z'\\ Y & =Y'\\ Z & =X_{0}X'+Z_{0}Z' \end{align} \ \text{with}\ \begin{align}X_{0} & =\operatorname{sh}OO'\\ Z_{0} & =\operatorname{ch}OO' \end{align} </math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz boost ({{equationNote|3b}}).</p> ==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates=== [[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:<ref group=M>Hausdorff (1899), p. 165, pp. 181-182</ref> :<math>\begin{matrix}p^{2}-x^{2}-y^{2}=1\\ \hline \begin{align}x & =a_{1}x'+a_{2}y'+x_{0}p'\\ y & =b_{1}x'+b{}_{2}y'+y_{0}p'\\ p & =e_{1}x'+e_{2}y'+p_{0}p'\\ \\ x' & =a_{1}x+b_{1}y-e_{1}p\\ y' & =a_{2}x+b_{2}y-e_{2}p\\ -p' & =x_{0}x+y_{0}y-p_{0}p \end{align} \left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} & =1\\ a_{2}^{2}+b_{2}^{2}-e_{2}^{2} & =1\\ -x_{0}^{2}-y_{0}^{2}+p_{0}^{2} & =1\\ a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} & =0\\ a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} & =0\\ a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} & =0\\ \\ a_{1}^{2}+a_{2}^{2}-x_{0}^{2} & =1\\ b_{1}^{2}+b_{2}^{2}-y_{0}^{2} & =1\\ -e_{1}^{2}-e_{2}^{2}+p_{0}^{2} & =1\\ b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} & =0\\ a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} & =0\\ a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates === In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting <math>k=\sqrt{-1}R</math> with ''R'' as real quantity):<ref group=M>Woods (1901/02), p. 98, 104</ref> :<math>\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\ \hline \begin{align}u' & =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ v' & =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ w' & =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}} \end{align} \left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} & =k^{2}\\ (i=1,2,3)\\ k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} & =1\\ \alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} & =0\\ (i,h=1,2,3,4;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''²=-1.</p> Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with <math>k=\sqrt{-1}k</math> for hyperbolic space):<ref group=M>Woods (1903/05), pp. 45–46; p. 48)</ref> :<math>\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\ ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\ \hline \begin{align}x_{1}^{\prime} & =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\ x_{2}^{\prime} & =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\ x_{3}^{\prime} & =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\ x_{0}^{\prime} & =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0} \end{align} \left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) & =1\\ \delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) & =k^{2}\\ (i=1,2,3)\\ \delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) & =0\\ (i,h=0,1,2,3;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''²=-1.</p> ==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates=== [[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''<ref group=M>Liebmann (1904/05), p. 168; pp. 175–176</ref> :<math>\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\ \hline \begin{align}x_{1} & =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\ y_{1} & =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\ x_{1} & =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\ \\ x & =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\ y & =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\ p & =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1} \end{align} \left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} & =1\\ -\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} & =1\\ -\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} & =1\\ \alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} & =0\\ \alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} & =0\\ \alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==References== ===Historical mathematical sources=== {{reflist|3|group=M}} *{{#section:History of Topics in Special Relativity/mathsource|apo}} *{{#section:History of Topics in Special Relativity/mathsource|bour56att}} *{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} *{{#section:History of Topics in Special Relativity/mathsource|chas29}} *{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} *{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} *{{#section:History of Topics in Special Relativity/mathsource|fris76}} *{{#section:History of Topics in Special Relativity/mathsource|gau98}} *{{#section:History of Topics in Special Relativity/mathsource|gau18}} *{{#section:History of Topics in Special Relativity/mathsource|ger92}} *{{#section:History of Topics in Special Relativity/mathsource|gud30}} *{{#section:History of Topics in Special Relativity/mathsource|haus99}} *{{#section:History of Topics in Special Relativity/mathsource|hill82}} *{{#section:History of Topics in Special Relativity/mathsource|jac27}} *{{#section:History of Topics in Special Relativity/mathsource|jac32a}} *{{#section:History of Topics in Special Relativity/mathsource|jac32b}} *{{#section:History of Topics in Special Relativity/mathsource|jac33}} *{{#section:History of Topics in Special Relativity/mathsource|kil77}} *{{#section:History of Topics in Special Relativity/mathsource|kil79}} *{{#section:History of Topics in Special Relativity/mathsource|kil84}} *{{#section:History of Topics in Special Relativity/mathsource|kil85}} *{{#section:History of Topics in Special Relativity/mathsource|kil93}} *{{#section:History of Topics in Special Relativity/mathsource|kil97}} *{{#section:History of Topics in Special Relativity/mathsource|klei71}} *{{#section:History of Topics in Special Relativity/mathsource|klei73}} *{{#section:History of Topics in Special Relativity/mathsource|lag73}} *{{#section:History of Topics in Special Relativity/mathsource|leb37}} *{{#section:History of Topics in Special Relativity/mathsource|lie85}} *{{#section:History of Topics in Special Relativity/mathsource|lie90}} *{{#section:History of Topics in Special Relativity/mathsource|lie93}} *{{#section:History of Topics in Special Relativity/mathsource|lieb04}} *{{#section:History of Topics in Special Relativity/mathsource|lop}} *{{#section:History of Topics in Special Relativity/mathsource|pic82}} *{{#section:History of Topics in Special Relativity/mathsource|pic84a}} *{{#section:History of Topics in Special Relativity/mathsource|pic84b}} *{{#section:History of Topics in Special Relativity/mathsource|pic84c}} *{{#section:History of Topics in Special Relativity/mathsource|poin81a}} *{{#section:History of Topics in Special Relativity/mathsource|poin81b}} *{{#section:History of Topics in Special Relativity/mathsource|poin87}} *{{#section:History of Topics in Special Relativity/mathsource|sal62}} *{{#section:History of Topics in Special Relativity/mathsource|som63}} *{{#section:History of Topics in Special Relativity/mathsource|wedd47}} *{{#section:History of Topics in Special Relativity/mathsource|wern89}} *{{#section:History of Topics in Special Relativity/mathsource|woo01}} *{{#section:History of Topics in Special Relativity/mathsource|woo03}} ===Secondary sources=== {{reflist|3}} {{#section:History of Topics in Special Relativity/secsource|L1}} [[Category:Special Relativity]] [[Category:History of Physics]] gggqtztoil52yddkpkmhg7vfpxg1q0v 2416453 2416266 2022-08-19T07:09:06Z D.H 52339 /* {{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas */ wikitext text/x-wiki {{../Lorentz transformation (header)}} ==Most general Lorentz transformations== ===General quadratic form=== The general [[w:quadratic form]] ''q(x)'' with coefficients of a [[w:symmetric matrix]] '''A''', the associated [[w:bilinear form]] ''b(x,y)'', and the [[w:linear transformation]]s of ''q(x)'' and ''b(x,y)'' into ''q(x′)'' and ''b(x′,y′)'' using the [[w:transformation matrix]] '''g''', can be written as<ref>Bôcher (1907), chapter X</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}\begin{align}q=\sum_{0}^{n}A_{ij}x_{i}x_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{x}\end{align} & =q'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{x}'\\ b=\sum_{0}^{n}A_{ij}x_{i}y_{j}=\mathbf{x}^{\mathrm{T}}\cdot\mathbf{A}\cdot\mathbf{y} & =b'=\mathbf{x}^{\mathrm{\prime T}}\cdot\mathbf{A}'\cdot\mathbf{y}' \end{align} \quad\left(A_{ij}=A_{ji}\right)\\ \hline \left.\begin{align}x_{i}^{\prime} & =\sum_{j=0}^{n}g_{ij}x_{j}=\mathbf{g}\cdot\mathbf{x}\\ x_{i} & =\sum_{j=0}^{n}g_{ij}^{(-1)}x_{j}^{\prime}=\mathbf{g}^{-1}\cdot\mathbf{x}' \end{align} \right|\mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g}=\mathbf{A}' \end{matrix}</math>|{{equationRef|Q1}}}} The case ''n=1'' is the [[w:binary quadratic form]] introduced by [[#Lagrange|Lagrange (1773)]] and [[#Gauss|Gauss (1798/1801)]], ''n=2'' is the ternary quadratic form introduced by [[#Gauss2|Gauss (1798/1801)]], ''n=3'' is the quaternary quadratic form etc. ===Most general Lorentz transformation=== The general Lorentz transformation follows from ({{equationNote|Q1}}) by setting '''A'''='''A′'''=diag(-1,1,...,1) and det '''g'''=±1. It forms an [[w:indefinite orthogonal group]] called the [[w:Lorentz group]] O(1,n), while the case det '''g'''=+1 forms the restricted [[w:Lorentz group]] SO(1,n). The quadratic form ''q(x)'' becomes the [[w:Lorentz interval]] in terms of an [[w:indefinite quadratic form]] of [[w:Minkowski space]] (being a special case of [[w:pseudo-Euclidean space]]), and the associated bilinear form ''b(x)'' becomes the [[w:Minkowski inner product]]:<ref name=ratcliffe>Ratcliffe (1994), 3.1 and Theorem 3.1.4 and Exercise 3.1</ref><ref>Naimark (1964), 2 in four dimensions</ref> {{NumBlk|:|<math>\begin{matrix}\begin{align}-x_{0}^{2}+\cdots+x_{n}^{2} & =-x_{0}^{\prime2}+\dots+x_{n}^{\prime2}\\ -x_{0}y_{0}+\cdots+x_{n}y_{n} & =-x_{0}^{\prime}y_{0}^{\prime}+\cdots+x_{n}^{\prime}y_{n}^{\prime} \end{align} \\ \hline \left.\begin{matrix}\mathbf{x}'=\mathbf{g}\cdot\mathbf{x}\\ \downarrow\\ \begin{align}x_{0}^{\prime} & =x_{0}g_{00}+x_{1}g_{01}+\dots+x_{n}g_{0n}\\ x_{1}^{\prime} & =x_{0}g_{10}+x_{1}g_{11}+\dots+x_{n}g_{1n}\\ & \dots\\ x_{n}^{\prime} & =x_{0}g_{n0}+x_{1}g_{n1}+\dots+x_{n}g_{nn} \end{align} \\ \\ \mathbf{x}=\mathbf{g}^{-1}\cdot\mathbf{x}'\\ \downarrow\\ \begin{align}x_{0} & =x_{0}^{\prime}g_{00}-x_{1}^{\prime}g_{10}-\dots-x_{n}^{\prime}g_{n0}\\ x_{1} & =-x_{0}^{\prime}g_{01}+x_{1}^{\prime}g_{11}+\dots+x_{n}^{\prime}g_{n1}\\ & \dots\\ x_{n} & =-x_{0}^{\prime}g_{0n}+x_{1}^{\prime}g_{1n}+\dots+x_{n}^{\prime}g_{nn} \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}}\cdot\mathbf{A} & =\mathbf{g}^{-1}\\ \mathbf{g}^{{\rm T}}\cdot\mathbf{A}\cdot\mathbf{g} & =\mathbf{A}\\ \mathbf{g}\cdot\mathbf{A}\cdot\mathbf{g}^{\mathrm{T}} & =\mathbf{A}\\ \\ \end{align} \\ \begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \end{matrix} \end{matrix}</math>|{{equationRef|1a}}}} The invariance of the Lorentz interval with ''n''=1 between axes and conjugate diameters of hyperbolas was known for a long time since [[#Apo|Apollonius (ca. 200 BC)]]. Lorentz transformations ({{equationNote|1a}}) for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]] in order to simplify computations of [[w:elliptic function]]s and integrals.<ref>Musen (1970) pointed out the intimate connection of Hill's scalar development and Minkowski's pseudo-Euclidean 3D space.</ref><ref>Touma et al. (2009) showed the analogy between Gauss and Hill's equations and Lorentz transformations, see eq. 22-29.</ref> They were also used by [[#Chasles|Chasles (1829)]] and [[#Weddle|Weddle (1847)]] to describe relations on hyperboloids, as well as by [[#Poincare|Poincaré (1881)]], [[#Cox|Cox (1881/82)]], [[#Picard|Picard (1882, 1884)]], [[#Killing|Killing (1885, 1893)]], [[#Gerard|Gérard (1892)]], [[#Hausdorff|Hausdorff (1899)]], [[#Woods2|Woods (1901, 1903)]], [[#Liebmann|Liebmann (1904/05)]] to describe [[w:hyperbolic motion]]s (i.e. rigid motions in the [[w:hyperbolic plane]] or [[w:hyperbolic space]]), which were expressed in terms of Weierstrass coordinates of the [[w:hyperboloid model]] satisfying the relation <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> or in terms of the [[w:Cayley–Klein metric]] of [[w:projective geometry]] using the "absolute" form <math>-x_{0}^{2}+\cdots+x_{n}^{2}=0</math> as discussed by [[#Klein|Klein (1871-73)]].<ref group=M>Killing (1885), p. 71</ref><ref>Müller (1910), p. 661, in particular footnote 247.</ref><ref>Sommerville (1911), p. 286, section K6.</ref> In addition, [[w:infinitesimal transformation]]s related to the [[w:Lie algebra]] of the group of hyperbolic motions were given in terms of Weierstrass coordinates <math>-x_{0}^{2}+\cdots+x_{n}^{2}=-1</math> by [[#Killing3|Killing (1888-1897)]]. ===Most general Lorentz transformation of velocity=== If <math>x_{i},\ x_{i}^{\prime}</math> in ({{equationNote|1a}}) are interpreted as [[w:homogeneous coordinates]], then the corresponding inhomogenous coordinates <math>u_{s},\ u_{s}^{\prime}</math> follow by :<math>\frac{x_{s}}{x_{0}}=u_{s},\ \frac{x_{s}^{\prime}}{x_{0}^{\prime}}=u_{s}^{\prime}\ (s=1,2\dots n)</math> defined by <math>u_{1}^{2}+u_{2}^{2}+\dots+u_{n}^{2}\le1</math> so that the Lorentz transformation becomes a [[w:homography]] inside the [[w:unit hypersphere]], which [[w:John Lighton Synge]] called "the most general formula for the composition of velocities" in terms of special relativity<ref>Synge (1955), p. 129 for ''n''=3</ref> (the transformation matrix '''g''' stays the same as in ({{equationNote|1a}})): {{NumBlk|:|<math>\begin{align}u_{s}^{\prime} & =\frac{g_{s0}+g_{s1}u_{1}+\dots+g_{sn}u_{n}}{g_{00}+g_{01}u_{1}+\dots+g_{0n}u_{n}}\\ \\ u_{s} & =\frac{-g_{0s}+g_{1s}u_{1}^{\prime}+\dots+g_{ns}u_{n}^{\prime}}{g_{00}-g_{10}u_{1}^{\prime}-\dots-g_{n0}u_{n}^{\prime}} \end{align} \left|\begin{align}\sum_{i=1}^{n}g_{ij}g_{ik}-g_{0j}g_{0k} & =\left\{ \begin{align}-1\quad & (j=k=0)\\ 1\quad & (j=k>0)\\ 0\quad & (j\ne k) \end{align} \right.\\ \sum_{j=1}^{n}g_{ij}g_{kj}-g_{i0}g_{k0} & =\left\{ \begin{align}-1\quad & (i=k=0)\\ 1\quad & (i=k>0)\\ 0\quad & (i\ne k) \end{align} \right. \end{align} \right.</math>|{{equationRef|1b}}}} Such Lorentz transformations for various dimensions were used by [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827–1833)]], [[#Lebesgue|Lebesgue (1837)]], [[#Bour|Bour (1856)]], [[#Somov|Somov (1863)]], [[#Hill|Hill (1882)]], [[#Callandreau|Callandreau (1885)]] in order to simplify computations of elliptic functions and integrals, by [[#Picard|Picard (1882-1884)]] in relation to [[w:Hermitian form|Hermitian quadratic form]]s, or by [[#Woods2|Woods (1901, 1903)]] in terms of the [[w:Beltrami–Klein model]] of hyperbolic geometry. In addition, infinitesimal transformations in terms of the [[w:Lie algebra]] of the group of hyperbolic motions leaving invariant the unit sphere <math>-1+u_{1}^{\prime2}+\cdots+u_{n}^{\prime2}=0</math> were given by [[#Lie3|Lie (1885-1893) and Werner (1889)]] and [[#Killing3|Killing (1888-1897)]]. ==Historical notation== ==={{anchor|Apo}} Apollonius (BC) – Conjugate diameters of hyperbolas === [[w:Apollonius of Perga]] (c. 240–190 BC) defined the following proposition in his 7th book on conics related to conjugate diameters of hyperbolas, which was translated and adapted to the modern reader by [[w:Thomas Heath (classicist)|w:Thomas Heath]] as follows:<ref group=M>Apollonius/Heath (1896), Proposition 129; (Apollonius, Book VII, Prop. 13).</ref> :[..] in every hyperbola the difference of the squares on any two conjugate diameters is equal to the [..] difference [..] of the squares on the axes. <p style="background-color:Beige;border:1px solid black">Apollonius' result produces the relation <math>-x_{0}^{\prime2}+x_{1}^{\prime2}=-x_{0}^{2}+x_{1}^{2}</math> in agreement with the invariance of the Lorentz interval, so that the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' can be interpreted as mapping from one pair of axes of a hyperbola to a pair of conjugate diameters.</p> [[File:Lhopital Conjugate Diameters.png|thumb|250px|Original illustration of <math>\scriptstyle \overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}</math> by l'Hôpital.]] This well known result (Apollonius' 7th book on conics survived in Arabian translation, and was translated into Latin in 1661 and 1710) was demonstrated in terms of [[w:analytic geomety]] by [[w:Guillaume de l'Hôpital]] (1707), who summarized Apollonius' proposition as follows:<ref group=M>l'Hôpital (1707), Third book, Prop. XII, p. 76.</ref> :The difference of the squares of any two conjugate diameters ''Mm, Ss'' is equal to the difference of the squares of the two axes ''Aa, Bb.'' We have to prove that <math>\overline{CS}^{2}-\overline{CM}^{2}=\overline{CB}^{2}-\overline{CA}^{2}</math>, or that <math>\overline{CM}^{2}-\overline{CS}^{2}=\overline{CA}^{2}-\overline{CB}^{2}</math>. ===Lagrange (1773) – Binary quadratic forms {{anchor|Lagrange}}=== After the invariance of the sum of squares under linear substitutions was discussed by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]], the general expressions of a [[w:binary quadratic form]] and its transformation was formulated by [[w:Joseph-Louis Lagrange]] (1773/75) as follows<ref group=M>Lagrange (1773/75), section 22</ref> :<math>\begin{matrix}py^{2}+2qyz+rz^{2}=Ps^{2}+2Qsx+Rx^{2}\\ \hline \begin{align}y & =Ms+Nx\\ z & =ms+nx \end{align} \left|\begin{matrix}\begin{align}P & =pM^{2}+2qMm+rm^{2}\\ Q & =pMN+q(Mn+Nm)+rmn\\ R & =pN^{2}+2qNn+rn^{2} \end{align} \\ \downarrow\\ PR-Q^{2}=\left(pr-q^{2}\right)(Mn-Nm)^{2} \end{matrix}\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to ({{equationNote|Q1}}) ''(n=1)''. The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(p,q,r)=(P,Q,R)=(1,0,-1)''.</p> ==={{anchor|Gauss}} Gauss (1798–1818)=== ===={{anchor|Gauss1}} Binary quadratic forms==== The theory of binary quadratic forms was considerably expanded by [[w:Carl Friedrich Gauss]] (1798, published 1801) in his [[w:Disquisitiones Arithmeticae]]. He rewrote Lagrange's formalism as follows using integer coefficients α,β,γ,δ:<ref group=M>Gauss (1798/1801), articles 157–158;</ref> :<math>\begin{matrix}F=ax^{2}+2bxy+cy^{2}=(a,b,c)\\ F'=a'x^{\prime2}+2b'x'y'+c'y^{\prime2}=(a',b',c')\\ \hline \begin{align}x & =\alpha x'+\beta y'\\ y & =\gamma x'+\delta y'\\ \\ x' & =\delta x-\beta y\\ y' & =-\gamma x+\alpha y \end{align} \left|\begin{matrix}\begin{align}a' & =a\alpha^{2}+2b\alpha\gamma+c\gamma^{2}\\ b' & =a\alpha\beta+b(\alpha\delta+\beta\gamma)+c\gamma\delta\\ c' & =a\beta^{2}+2b\beta\delta+c\delta^{2} \end{align} \\ \downarrow\\ b^{2}-a'c'=\left(b^{2}-ac\right)(\alpha\delta-\beta\gamma)^{2} \end{matrix}\right. \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=1)''. As pointed out by Gauss, ''F'' and ''F′'' are called "proper equivalent" if αδ-βγ=1, so that ''F'' is contained in ''F′'' as well as ''F′'' is contained in ''F''. In addition, if another form ''F″'' is contained by the same procedure in ''F′'' it is also contained in ''F'' and so forth.<ref group=M>Gauss (1798/1801), section 159</ref> <p style="background-color:Beige;border:1px solid black">The Lorentz interval <math>-x_{0}^{2}+x_{1}^{2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' are a special case of the binary quadratic form by setting ''(a,b,c)=(a',b',c')=(1,0,-1)''.</p> ===={{anchor|Gauss2}} Ternary quadratic forms==== Gauss (1798/1801)<ref group=M>Gauss (1798/1801), articles 266–285</ref> also discussed ternary quadratic forms with the general expression :<math>\begin{matrix}f=ax^{2}+a'x^{\prime2}+a''x^{\prime\prime2}+2bx'x''+2b'xx''+2b''xx'=\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\\ g=my^{2}+m'y^{\prime2}+m''y^{\prime\prime2}+2ny'y''+2n'yy''+2n''yy'=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)\\ \hline \begin{align}x & =\alpha y+\beta y'+\gamma y''\\ x' & =\alpha'y+\beta'y'+\gamma'y''\\ x'' & =\alpha''y+\beta''y'+\gamma''y'' \end{align} \end{matrix}</math> which is equivalent to ({{equationNote|Q1}}) ''(n=2)''. Gauss called these forms definite when they have the same sign such as ''x²+y²+z²'', or indefinite in the case of different signs such as ''x²+y²-z²''. While discussing the classification of ternary quadratic forms, Gauss (1801) presented twenty special cases, among them these six variants:<ref group=M>Gauss (1798/1801), article 277</ref> :<math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)\Rightarrow\begin{matrix}\left(\begin{matrix}1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & 1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\\ \left(\begin{matrix}1, & -1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right),\ \left(\begin{matrix}-1, & -1, & 1\\ 0, & 0, & 0 \end{matrix}\right) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are all six types of Lorentz interval in 2+1 dimensions that can be produced as special cases of a ternary quadratic form. In general: The Lorentz interval <math>x^{2}+x^{\prime2}-x^{\prime\prime2}</math> and the Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' is an indefinite ternary quadratic form, which follows from the general ternary form by setting: <br><math>\left(\begin{matrix}a, & a', & a''\\ b, & b', & b'' \end{matrix}\right)=\left(\begin{matrix}m, & m', & m''\\ n, & n', & n'' \end{matrix}\right)=\left(\begin{matrix}1, & 1, & -1\\ 0, & 0, & 0 \end{matrix}\right)</math></p> ===={{anchor|Gauss4}} Homogeneous coordinates==== Gauss (1818) discussed planetary motions together with formulating [[w:elliptic function]]s. In order to simplify the integration, he transformed the expression :<math>(AA+BB+CC)tt+aa(t\cos E)^{2}+bb(t\sin E)^{2}-2aAt\cdot t\cos E-2bBt\cdot t\sin E</math> into :<math>G+G'\cos T^{2}+G''\sin T^{2}</math> in which the [[w:eccentric anomaly]] ''E'' is connected to the new variable ''T'' by the following transformation including an arbitrary constant ''k'', which Gauss then rewrote by setting ''k''=1:<ref group=M>Gauss (1818), pp. 5–10</ref> :<math>\begin{matrix}{\scriptstyle \left(\alpha+\alpha'\cos T+\alpha''\sin T\right)^{2}+\left(\beta+\beta'\cos T+\beta''\sin T\right)^{2}-\left(\gamma+\gamma'\cos T+\gamma''\sin T\right)^{2}}=0\\ k\left(\cos^{2}T+\sin^{2}T-1\right)=0\\ \hline \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =k & \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =-k\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-k & \beta\beta-\beta'\beta'-\beta''\beta'' & =-k\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-k & \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =+k\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} }\right.\\ \hline k=1\\ \begin{align}t\cos E & =\alpha+\alpha'\cos T+\alpha''\sin T\\ t\sin E & =\beta+\beta'\cos T+\beta''\sin T\\ t & =\gamma+\gamma'\cos T+\gamma''\sin T \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The coefficients α,β,γ,... of Gauss' case ''k''=1 are equivalent to the coefficient system in Lorentz transformations ({{equationNote|1a}}) and ({{equationNote|1b}}) ''(n=2)''. Further setting <math>[\cos T,\sin T,\cos E,\sin E]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, Gauss' transformation becomes Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> Subsequently, he showed that these relations can be reformulated using three variables ''x,y,z'' and ''u,u′,u″'', so that :<math>aaxx+bbyy+(AA+BB+CC)zz-2aAxz-2bByz</math> can be transformed into :<math>Guu+G'u'u'+G''u''u''</math>, in which ''x,y,z'' and ''u,u′,u″'' are related by the transformation:<ref group=M>Gauss (1818), pp. 9–10</ref> :<math>\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =-\alpha x-\beta y+\gamma z\\ u' & =\alpha'x+\beta'y-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}-\alpha\alpha-\beta\beta+\gamma\gamma & =1\\ -\alpha'\alpha'-\beta'\beta'+\gamma'\gamma' & =-1\\ -\alpha''\alpha''-\beta''\beta''+\gamma''\gamma'' & =-1\\ -\alpha'\alpha''-\beta'\beta''+\gamma'\gamma'' & =0\\ -\alpha''\alpha-\beta''\beta+\gamma''\gamma & =0\\ -\alpha\alpha'-\beta\beta'+\gamma\gamma' & =0 \end{align} }\right.</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' satisfying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>, and can be related to Gauss' previous equations in terms of homogeneous coordinates <math>\left[\cos T,\sin T,\cos E,\sin E\right]=\left[\tfrac{x}{z},\ \tfrac{y}{z},\ \tfrac{u'}{u},\ \tfrac{u''}{u}\right]</math>.</p> ==={{anchor|Jacobi}} Jacobi (1827, 1833/34) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Carl Gustav Jacob Jacobi]] extended Gauss' transformation in 1827:<ref group=M>Jacobi (1827), p. 235, 239–240</ref> :<math>{\scriptstyle \begin{matrix}\cos P^{2}+\sin P^{2}\cos\vartheta^{2}+\sin P^{2}\sin\vartheta^{2}=1\\ k\left(\cos\psi^{2}+\sin\psi^{2}\cos\varphi^{2}+\sin\psi^{2}\sin\varphi^{2}-1\right)=0\\ \hline {\left.\begin{matrix}\mathbf{(1)}\begin{align}\cos P & =\frac{\alpha+\alpha'\cos\psi+\alpha''\sin\psi\cos\varphi+\alpha'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\cos\vartheta & =\frac{\beta+\beta'\cos\psi+\beta''\sin\psi\cos\varphi+\beta'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \sin P\sin\vartheta & =\frac{\gamma+\beta'\cos\psi+\gamma''\sin\psi\cos\varphi+\gamma'''\sin\psi\sin\varphi}{\delta+\delta'\cos\psi+\delta''\sin\psi\cos\varphi+\delta'''\sin\psi\sin\varphi}\\ \\ \cos\psi & =\frac{-\delta'+\alpha'\cos P+\beta'\sin P\cos\vartheta+\gamma'\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\cos\varphi & =\frac{-\delta''+\alpha''\cos P+\beta''\sin P\cos\vartheta+\gamma''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta}\\ \sin\psi\sin\varphi & =\frac{-\delta'''+\alpha'''\cos P+\beta'''\sin P\cos\vartheta+\gamma'''\sin P\sin\vartheta}{\delta-\alpha\cos P-\beta\sin P\cos\vartheta-\gamma\sin P\sin\vartheta} \end{align} \\ \\ \hline \mathbf{(2)}\begin{align}\alpha\mu+\beta x+\gamma y+\delta z & =m\\ \alpha'\mu+\beta'x+\gamma'y+\delta'z & =m'\\ \alpha''\mu+\beta''x+\gamma''y+\delta''z & =m''\\ \alpha'''\mu+\beta'''x+\gamma'''y+\delta'''z & =m'''\\ \\ Am+A'm'+A''m''+A'''m''' & =\mu\\ Bm+B'm'+B''m''+B'''m''' & =x\\ Cm+C'm'+C''m''+C'''m''' & =y\\ Dm+D'm'+D''m''+D'''m''' & =z\\ \\ \end{align} \\ \begin{align}\alpha & =-kA, & \beta & =-kB, & \gamma & =-kC, & \delta & =kD,\\ \alpha' & =kA', & \beta' & =kB', & \gamma' & =kC', & \delta' & =-kD',\\ \alpha'' & =kA'', & \beta'' & =kB'', & \gamma'' & =kC'', & \delta'' & =-kD'',\\ \alpha''' & =kA''', & \beta''' & =kB''', & \gamma''' & =kC''', & \delta''' & =-kD''', \end{align} \end{matrix}\right|\begin{matrix}\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma-\delta\delta & =-k\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma'-\delta'\delta' & =k\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma''-\delta''\delta'' & =k\\ \alpha'''\alpha'''+\beta'''\beta'''+\gamma'''\gamma'''-\delta'''\delta''' & =k\\ \alpha\alpha'+\beta\beta'+\gamma\gamma'-\delta\delta' & =0\\ \alpha\alpha''+\beta\beta''+\gamma\gamma''-\delta\delta'' & =0\\ \alpha\alpha'''+\beta\beta'''+\gamma\gamma'''-\delta\delta''' & =0\\ \alpha''\alpha'''+\beta''\beta'''+\gamma''\gamma'''-\delta''\delta''' & =0\\ \alpha'''\alpha'+\beta'''\beta'+\gamma'''\gamma'-\delta'''\delta' & =0\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma''-\delta'\delta'' & =0\\ \\ -\alpha\alpha+\alpha'\alpha'+\alpha''\alpha''+\alpha'''\alpha''' & =k\\ -\beta\beta+\beta'\beta'+\beta''\beta''+\beta'''\beta''' & =k\\ -\gamma\gamma+\gamma'\gamma'+\gamma''\gamma''+\gamma'''\gamma''' & =k\\ -\delta\delta+\delta'\delta'+\delta''\delta''+\delta'''\delta''' & =-k\\ -\alpha\beta+\alpha'\beta'+\alpha''\beta''+\alpha'''\beta''' & =0\\ -\alpha\gamma+\alpha'\gamma'+\alpha''\gamma''+\alpha'''\gamma''' & =0\\ -\alpha\delta+\alpha'\delta'+\alpha''\delta''+\alpha'''\delta''' & =0\\ -\beta\gamma+\beta'\gamma'+\beta''\gamma''+\beta'''\gamma''' & =0\\ -\gamma\delta+\gamma'\delta'+\gamma''\delta''+\gamma'''\delta''' & =0\\ -\delta\beta+\delta'\beta'+\delta''\beta''+\delta'''\beta''' & =0 \end{align} \end{matrix}} \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>{\scriptstyle \begin{align}\left[\cos P,\ \sin P\cos\varphi,\ \sin P\sin\varphi\right] & =\left[u_{1},\ u_{2},\ u_{3}\right]\\{} [\cos\psi,\ \sin\psi\cos\vartheta,\ \sin\psi\sin\vartheta] & =\left[u_{1}^{\prime},\ u_{2}^{\prime},\ u_{3}^{\prime}\right] \end{align} }</math> and ''k''=1 in the (1827) formulas, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'', and by setting ''k''=1 in transformation system (2) it becomes equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' producing <math>m^{2}+m^{\prime2}+m^{\prime\prime2}-m^{\prime\prime\prime2}=\mu^{2}+x^{2}+y^{2}-z^{2}</math>.</p> Alternatively, in two papers from 1832 Jacobi started with an ordinary orthogonal transformation, and by using an imaginary substitution he arrived at Gauss' transformation (up to a sign change):<ref group=M>The orthogonal substitution and the imaginary transformation was defined in Jacobi (1832a), pp. 257, 265–267; Transformation system (2) and (3) and coefficients in Jacobi (1832b), pp. 321-325.</ref> :<math>{\scriptstyle \begin{matrix}xx+yy+zz=ss+s's'+s''s''=0\\ \mathbf{(1)}\begin{align}x & =\alpha s+\alpha's'+\alpha''s''\\ y & =\beta s+\beta's'+\beta''s''\\ z & =\gamma s+\gamma's'+\gamma''s''\\ \\ s & =\alpha x+\beta y+\gamma z\\ s' & =\alpha'x+\beta'y+\gamma'z\\ s'' & =\alpha''x+\beta''y+\gamma''z \end{align} \left|\begin{align}\alpha\alpha+\beta\beta+\gamma\gamma & =1 & \alpha\alpha+\alpha'\alpha'+\alpha''\alpha'' & =1\\ \alpha'\alpha'+\beta'\beta'+\gamma'\gamma' & =1 & \beta\beta+\beta'\beta'+\beta''\beta'' & =1\\ \alpha''\alpha''+\beta''\beta''+\gamma''\gamma'' & =1 & \gamma\gamma+\gamma'\gamma'+\gamma''\gamma'' & =1\\ \alpha'\alpha''+\beta'\beta''+\gamma'\gamma'' & =0 & \beta\gamma+\beta'\gamma'+\beta''\gamma'' & =0\\ \alpha''\alpha+\beta''\beta+\gamma''\gamma & =0 & \gamma\alpha+\gamma'\alpha'+\gamma''\alpha'' & =0\\ \alpha\alpha'+\beta\beta'+\gamma\gamma' & =0 & \alpha\beta+\alpha'\beta'+\alpha''\beta'' & =0 \end{align} \right.\\ \hline \left[\frac{y}{x},\ \frac{z}{x},\ \frac{s'}{s},\ \frac{s''}{s}\right]=\left[-i\cos\varphi,\ -i\sin\varphi,\ i\cos\eta,\ i\sin\eta\right]\\ \left[\alpha',\ \alpha'',\ \beta,\ \gamma\right]=\left[i\alpha',\ i\alpha'',\ -i\beta,\ -i\gamma\right]\\ \hline \begin{matrix}\mathbf{(2)}\begin{matrix}\left(\alpha-\alpha'\cos\eta-\alpha''\sin\eta\right)^{2}=\left(\beta-\beta'\cos\eta-\beta''\sin\eta\right)^{2}+\left(\gamma-\gamma'\cos\eta-\gamma''\sin\eta\right)^{2}\\ \left(\alpha-\beta\cos\phi-\gamma\sin\phi\right)^{2}=\left(\alpha'-\beta'\cos\phi-\gamma'\sin\phi\right)^{2}+\left(\alpha''-\beta''\cos\phi-\gamma''\sin\phi\right)^{2}\\ \hline \begin{align}\cos\phi & =\frac{\beta-\beta'\cos\eta-\beta''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \cos\eta & =\frac{\alpha'-\beta'\cos\phi-\gamma'\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi}\\ \sin\phi & =\frac{\gamma-\gamma'\cos\eta-\gamma''\sin\eta}{\alpha-\alpha'\cos\eta-\alpha''\sin\eta}, & \sin\eta & =\frac{\alpha''-\beta''\cos\phi-\gamma''\sin\phi}{\alpha-\beta\cos\phi-\gamma\sin\phi} \end{align} \end{matrix}\\ \hline \\ \mathbf{(3)}\begin{matrix}1-zz-yy=\frac{1-s's'-s''s''}{\left(\alpha-\alpha's'-\alpha''s''\right)^{2}}\\ \hline \begin{align}y & =\frac{\beta-\beta's'-\beta''s''}{\alpha-\alpha's'-\alpha''s''}, & s' & =\frac{\alpha'-\beta'y-\gamma'z}{\alpha-\beta y-\gamma z},\\ z & =\frac{\gamma-\gamma's'-\gamma''s''}{\alpha-\alpha's'-\alpha''s'''}, & s'' & =\frac{\alpha''-\beta''y-\gamma''z}{\alpha-\beta y-\gamma z}, \end{align} \end{matrix} \end{matrix}\left|\begin{align}\alpha\alpha-\beta\beta-\gamma\gamma & =1\\ \alpha'\alpha'-\beta'\beta'-\gamma'\gamma' & =-1\\ \alpha''\alpha''-\beta''\beta''-\gamma''\gamma'' & =-1\\ \alpha'\alpha''-\beta'\beta''-\gamma'\gamma'' & =0\\ \alpha''\alpha-\beta''\beta-\gamma''\gamma & =0\\ \alpha\alpha'-\beta\beta'-\gamma\gamma' & =0\\ \\ \alpha\alpha-\alpha'\alpha'-\alpha''\alpha'' & =1\\ \beta\beta-\beta'\beta'-\beta''\beta'' & =-1\\ \gamma\gamma-\gamma'\gamma'-\gamma''\gamma'' & =-1\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \gamma\alpha-\gamma'\alpha'-\gamma''\alpha'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} \right. \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">By setting <math>[\cos\phi,\ \sin\phi,\ \cos\eta,\ \sin\eta]=\left[u_{1},\ u_{2},\ u_{1}^{\prime},\ u_{2}^{\prime}\right]</math>, transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Also transformation system (3) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' up to a sign change.</p> Extending his previous result, Jacobi (1833) started with [[#Cauchy|Cauchy's (1829)]] orthogonal transformation for ''n'' dimensions, and by using an imaginary substitution he formulated Gauss' transformation (up to a sign change) in the case of ''n'' dimensions:<ref group =M>Jacobi (1833/34), pp. 7–8, 34–35, 41; Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 229–230.</ref> :<math>{\scriptstyle \begin{matrix}x_{1}x_{1}+x_{2}x_{2}+\dots+x_{n}x_{n}=y_{1}y_{1}+y_{2}y_{2}+\dots+y_{n}y_{n}\\ \hline \mathbf{(1)\ }\begin{align}y_{\varkappa} & =\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}\\ x_{\varkappa} & =\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}\\ \\ \frac{y_{\varkappa}}{y_{n}} & =\frac{\alpha_{1}^{(\varkappa)}x_{1}+\alpha_{2}^{(\varkappa)}x_{2}+\dots+\alpha_{n}^{(\varkappa)}x_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}}\\ \frac{x_{\varkappa}}{x_{n}} & =\frac{\alpha_{\varkappa}^{\prime}y_{1}+\alpha_{\varkappa}^{\prime\prime}y_{2}+\dots+\alpha_{\varkappa}^{(n)}y_{n}}{\alpha_{1}^{(n)}x_{1}+\alpha_{2}^{(n)}x_{2}+\dots+\alpha_{n}^{(n)}x_{n}} \end{align} \left|\begin{align}\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\lambda}^{(n)} & =0\\ \alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}+\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}+\dots+\alpha_{\varkappa}^{(n)}\alpha_{\varkappa}^{(n)} & =1\\ \\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\lambda)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\lambda)} & =0\\ \alpha_{1}^{(\varkappa)}\alpha_{1}^{(\varkappa)}+\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\varkappa)}+\dots+\alpha_{n}^{(\varkappa)}\alpha_{n}^{(\varkappa)} & =1 \end{align} \right.\\ \hline \frac{x_{\varkappa}}{x_{n}}=-i\xi_{\varkappa},\ \frac{y_{\varkappa}}{y_{n}}=i\nu_{\varkappa}\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{y_{n}y_{n}}{x_{n}x_{n}}\left(1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}\right)\\ \alpha_{n}^{(\varkappa)}=i\alpha^{(\varkappa)},\ \alpha_{\varkappa}^{(n)}=-i\alpha_{\varkappa},\ \alpha_{n}^{(n)}=\alpha\\ 1-\xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=\frac{1-\nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}}{\left[\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}\right]^{2}}\\ \hline \mathbf{(2)\ }\begin{align}\nu_{\varkappa} & =\frac{\alpha^{(\varkappa)}-\alpha_{1}^{(\varkappa)}\xi_{1}-\alpha_{2}^{(\varkappa)}\xi_{2}\dots-\alpha_{n-1}^{(\varkappa)}\xi_{n-1}}{\alpha-\alpha_{1}\xi_{1}-\alpha_{2}\xi_{2}\dots-\alpha_{n-1}\xi_{n-1}}\\ \\ \xi_{\varkappa} & =\frac{\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\nu_{1}-\alpha_{2}^{\prime\prime}\nu_{2}\dots-\alpha_{\varkappa}^{(n-1)}\nu_{n-1}}{\alpha-\alpha^{\prime}\nu_{1}-\alpha^{\prime\prime}\nu_{2}\dots-\alpha^{(n-1)}\nu_{n-1}} \end{align} \\ \hline \xi_{1}\xi_{1}-\xi_{2}\xi_{2}-\dots-\xi_{n-1}\xi_{n-1}=1\ \Rightarrow\ \nu_{1}\nu_{1}-\nu_{2}\nu_{2}-\dots-\nu_{n-1}\nu_{n-1}=1 \end{matrix}}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) up to a sign change.</p> He also stated the following transformation leaving invariant the Lorentz interval:<ref group=M>Jacobi (1833/34), p. 37. Some misprints were corrected in Jacobi's collected papers, vol 3, pp. 232–233.</ref> :<math>\begin{matrix}uu-u_{1}u_{1}-u_{2}u_{2}-\dots-u_{n-1}u_{n-1}=ww-w_{1}w_{1}-w_{2}w_{2}-\dots-w_{n-1}w_{n-1}\\ \hline {\scriptstyle \begin{align}u & =\alpha w-\alpha'w_{1}-\alpha''w_{2}-\dots-\alpha^{(n-1)}w_{n-1}\\ u_{1} & =\alpha_{1}w-\alpha_{1}^{\prime}w_{1}-\alpha_{1}^{\prime\prime}w_{2}-\dots-\alpha_{1}^{(n-1)}w_{n-1}\\ & \dots\\ u_{n-1} & =\alpha_{n-1}w-\alpha_{n-1}^{\prime}w_{1}-\alpha_{n-1}^{\prime\prime}w_{2}-\dots-\alpha_{n-1}^{(n-1)}w_{n-1}\\ \\ w & =\alpha u-\alpha_{1}u_{1}-\alpha_{2}^{\prime\prime}u_{2}-\dots-\alpha_{n-1}u_{n-1}\\ w_{1} & =\alpha'u-\alpha_{1}^{\prime}u_{1}-\alpha_{2}^{\prime}u_{2}-\dots-\alpha_{n-1}^{\prime}u_{n-1}\\ & \dots\\ w_{n-1} & =\alpha^{(n-1)}u-\alpha_{1}^{(n-1)}u_{1}-\alpha_{2}^{(n-1)}u_{2}-\dots-\alpha_{n-1}^{(n-1)}u_{n-1} \end{align} \left|\begin{align}\alpha\alpha-\alpha'\alpha'-\alpha''\alpha''\dots-\alpha^{(n-1)}\alpha^{(n-1)} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{\varkappa}^{\prime}\alpha_{\varkappa}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =-1\\ \alpha\alpha_{\varkappa}-\alpha^{\prime}\alpha_{\varkappa}^{\prime}-\alpha^{\prime\prime}\alpha_{\varkappa}^{\prime\prime}\dots-\alpha^{(n-1)}\alpha_{\varkappa}^{(n-1)} & =0\\ \alpha_{\varkappa}\alpha_{\lambda}-\alpha_{\varkappa}^{\prime}\alpha_{\lambda}^{\prime}-\alpha_{\varkappa}^{\prime\prime}\alpha_{\lambda}^{\prime\prime}\dots-\alpha_{\varkappa}^{(n-1)}\alpha_{\lambda}^{(n-1)} & =0\\ \\ \alpha\alpha-\alpha_{1}\alpha_{1}-\alpha_{2}\alpha_{2}\dots-\alpha_{n-1}\alpha_{n-1} & =+1\\ \alpha_{\varkappa}\alpha_{\varkappa}-\alpha_{1}^{\varkappa}\alpha_{1}^{\varkappa}-\alpha_{2}^{\prime\prime}\alpha_{2}^{\prime\prime}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\varkappa)} & =-1\\ \alpha\alpha^{(\varkappa)}-\alpha_{1}\alpha_{1}^{(\varkappa)}-\alpha_{2}\alpha_{2}^{(\varkappa)}\dots-\alpha_{n-1}\alpha_{n-1}^{(\varkappa)} & =0\\ \alpha^{(\varkappa)}\alpha^{(\lambda)}-\alpha_{1}^{(\varkappa)}\alpha_{1}^{\lambda l)}-\alpha_{2}^{(\varkappa)}\alpha_{2}^{(\lambda)}\dots-\alpha_{n-1}^{(\varkappa)}\alpha_{n-1}^{(\lambda)} & =0 \end{align} \text{ }\right.} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) up to a sign change.</p> ==={{anchor|Chasles}} Chasles (1829) – Conjugate hyperboloids === [[w:Michel Chasles]] (1829) independently introduced the same equation systems as [[#Gauss4|Gauss (1818)]] and [[#Jacobi|Jacobi (1827)]], albeit in the different context of conjugate hyperboloids. He started with two equation systems (a) and (b) from which he derived systems (c), (d) and others:<ref group=M>Chasles (1829), p. 139</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(a)\\ \\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(b)\\ \\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1 \end{align} \right\} & \dots(c)\\ \\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(d) \end{matrix}</math> He noted that those quantities become the “frequently employed” formulas of Lagrange [i.e. the coefficients of the Euclidean orthogonal transformation first given by [[../Lorentz transformation (imaginary)#Euler|E:Euler (1771)]]] by setting:<ref group=M>Chasles (1829), p. 141</ref> :<math>\begin{matrix}\gamma\quad\Rightarrow\quad-\gamma\sqrt{-1}\\ \gamma'\quad\Rightarrow\quad-\gamma'\sqrt{-1}\\ \alpha''\quad\Rightarrow\quad\alpha''\sqrt{-1}\\ \beta''\quad\Rightarrow\quad\beta''\sqrt{-1} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equations (a,b,c,d) are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Chasles now showed that equation systems (a,b,c,d) are of importance when discussing the relations between conjugate diameters of hyperboloids. He used the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid having the same principal axes (x,y,z), thus sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>. He then transformed those two hyperboloids to new axes (x',y',z') sharing the property of conjugacy:<ref group=M>Chasles (1829), pp. 143-144</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}x & =lx'+l'y'+l''z'\\ y & =mx'+m'y'+m''z'\\ z & =nx'+n'y'+n''z' \end{align} \\ \left\{ \begin{align}\frac{ll'}{a{{}^2}}+\frac{mm'}{b^{2}}-\frac{nn'}{c^{2}} & =0\\ \frac{ll''}{a{{}^2}}+\frac{mm''}{b^{2}}-\frac{nn''}{c^{2}} & =0\\ \frac{l'l''}{a{{}^2}}+\frac{m'm''}{b^{2}}-\frac{n'n''}{c^{2}} & =0 \end{align} \right\} \\ \hline \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=1\\ \left(\frac{l^{2}}{a{{}^2}}+\frac{m^{2}}{b^{2}}-\frac{n^{2}}{c^{2}}\right)x^{\prime2}+\left(\frac{l^{\prime2}}{a{{}^2}}+\frac{m^{\prime2}}{b^{2}}-\frac{n^{\prime2}}{c^{2}}\right)y^{\prime2}+\left(\frac{l^{\prime\prime2}}{a{{}^2}}+\frac{m^{\prime\prime2}}{b^{2}}-\frac{n^{\prime\prime2}}{c^{2}}\right)z^{\prime2}=-1 \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Chasles defined the conditional equations of ''l,m,n'' in the same way as those of <math>\alpha,\beta,\gamma</math> in equation system (b) above, so his transformation of x,y,z into x',y',z' represents Lorentz transformation ({{equationNote|1a}}, n=2) by applying equation system (a) as well.</p> He went on to use two semi-diameters of the one-sheet hyperboloid and one semi-diameter of the two-sheet hyperboloid in order to define equation system (A), and went on to suggest that the other equations related to this system can be obtained using the above transformation from oblique coordinates to other oblique ones, but he deemed it more simple to use a geometric argument to obtain system (B), which together with (A) then allowed him to algebraically determine systems (C), (D) and additional ones, leading Chasles to announce that “''from these formulas one can very easily conclude the various properties of conjugated diameters of hyperboloids''”:<ref group=M>Chasles (1829), pp. 145-146</ref> :<math>\begin{matrix}\left.\begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =a^{2}\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =b^{2}\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(A)\\ \left.\begin{align}\alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma' & =0 \end{align} \right\} & \dots(B)\\ \left.\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =a^{2}\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =b^{2}\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-c^{2} \end{align} \right\} & \dots(C)\\ \left.\begin{align}\alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right\} & \dots(D) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Equation systems (A,B,C,D), being equivalent to systems (a,b,c,d) above, are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Lebesgue}} Lebesgue (1837) – Homogeneous coordinates=== [[w:Victor-Amédée Lebesgue]] (1837) summarized the previous work of [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], [[#Cauchy|Cauchy (1829)]]. He started with the orthogonal transformation<ref group=M>Lebesgue (1837), pp. 338-341</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}\ (9)\\ \hline {\scriptstyle \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n,2}x_{n}\ (12)\ \\ \dots\\ y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n,\alpha}^{2} & =1 & (10)\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n,\alpha}a_{n,\beta} & =0 & (11)\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n}^{2} & =1 & (13)\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n}a_{\beta,n} & =0 & (14) \end{align} \right.} \end{matrix}</math> In order to achieve the invariance of the Lorentz interval<ref group=M>Lebesgue (1837), pp. 353–354</ref> :<math>x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}</math> he gave the following instructions as to how the previous equations shall be modified: In equation (9) change the sign of the last term of each member. In the first ''n-1'' equations of (10) change the sign of the last term of the left-hand side, and in the one which satisfies α=''n'' change the sign of the last term of the left-hand side as well as the sign of the right-hand side. In all equations (11) the last term will change sign. In equations (12) the last terms of the right-hand side will change sign, and so will the left-hand side of the ''n''-th equation. In equations (13) the signs of the last terms of the left-hand side will change, moreover in the ''n''-th equation change the sign of the right-hand side. In equations (14) the last terms will change sign. <p style="background-color:Beige;border:1px solid black">These instructions give Lorentz transformation ({{equationNote|1a}}) in the form: <br><math>{\scriptstyle \begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \hline \begin{align}x_{1} & =a_{1,1}y_{1}+a_{1,2}y_{2}+\dots+a_{1,n}y_{n}\\ x_{2} & =a_{2,1}y_{1}+a_{2,2}y_{2}+\dots+a_{2,n}y_{n}\\ \dots\\ x_{n} & =a_{n,1}x_{1}+a_{n,2}x_{2}+\dots+a_{n,n}x_{n}\\ \\ y_{1} & =a_{1,1}x_{1}+a_{2,1}x_{2}+\dots+a_{n-1,1}x_{n-1}-a_{n,1}x_{n}\\ y_{2} & =a_{1,2}x_{1}+a_{2,2}x_{2}+\dots+a_{n-1,2}x_{n-1}-a_{n,2}x_{n}\\ \dots\\ -y_{n} & =a_{1,n}x_{1}+a_{2,n}x_{2}+\dots+a_{n-1,n}x_{n-1}-a_{n,n}x_{n} \end{align} \left|\begin{align}a_{1,\alpha}^{2}+a_{2,\alpha}^{2}+\dots+a_{n-1,\alpha}^{2}-a_{n,\alpha}^{2} & =1\\ a_{1,n}^{2}+a_{2,n}^{2}+\dots+a_{n-1,n}^{2}-a_{n,n}^{2} & =-1\\ a_{1,\alpha}a_{1,\beta}+a_{2,\alpha}a_{2,\beta}+\dots+a_{n-1,\alpha}a_{n-1,\beta}-a_{n,\alpha}a_{n,\beta} & =0\\ a_{\alpha,1}^{2}+a_{\alpha,2}^{2}+\dots+a_{\alpha,n-1}^{2}-a_{\alpha,n}^{2} & =1\\ a_{n,1}^{2}+a_{n,2}^{2}+\dots+a_{n,n-1}^{2}-a_{n,n}^{2} & =-1\\ a_{\alpha,1}a_{\beta,1}+a_{\alpha,2}a_{\beta,2}+\dots+a_{\alpha,n-1}a_{\beta,n-1}-a_{\alpha,n}a_{\beta,n} & =0 \end{align} \right. \end{matrix}}</math></p> He went on to redefine the variables of the Lorentz interval and its transformation:<ref group=M>Lebesgue (1837), pp. 353–355</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n-1}^{2}-x_{n}^{2}=y_{1}^{2}+y_{2}^{2}+\dots+y_{n-1}^{2}-y_{n}^{2}\\ \downarrow\\ \begin{align}x_{1} & =x_{n}\cos\theta_{1}, & x_{2} & =x_{n}\cos\theta_{2},\dots & x_{n-1} & =x_{n}\cos\theta_{n-1}\\ y_{1} & =y_{n}\cos\phi_{1}, & y_{2} & =y_{n}\cos\phi_{2},\dots & y_{n-1} & =y_{n}\cos\phi_{n-1} \end{align} \\ \downarrow\\ \cos^{2}\theta_{1}+\cos^{2}\theta_{2}+\dots+\cos^{2}\theta_{n-1}=1\\ \cos^{2}\phi_{1}+\cos^{2}\phi_{2}+\dots+\cos^{2}\phi_{n-1}=1\\ \hline \\ \cos\theta_{i}=\frac{a_{i,1}\cos\phi_{1}+a_{i,2}\cos\phi_{2}+\dots+a_{i,n-1}\cos\phi_{n-1}+a_{i,n}}{a_{n,1}\cos\phi_{1}+a_{n,2}\cos\phi_{2}+\dots+a_{n,n-1}\cos\phi_{n-1}+a_{n,n}}\\ (i=1,2,3\dots n) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Setting <math>[\cos\theta_{i},\ \cos\phi_{i}]=\left[u_{s},\ u_{s}^{\prime}\right]</math> it is equivalent to Lorentz transformation ({{equationNote|1b}}).</p> ==={{anchor|Weddle}} Weddle (1847) – Conjugate hyperboloids=== Very similar to [[#Chasles|Chasles (1829)]], though without reference to him, [[w:Thomas Weddle]] discussed conjugate hyperboloids using the following equation system (&alpha;), from which he derived equations (&beta;) and others:<ref group=M>Weddle (1847), p. 274</ref> :<math>\begin{matrix}\left.\begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1, & l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1, & l_{1}l_{3}+m_{1}m_{3}-n_{1}n_{3} & =0\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1, & l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0 \end{align} \right\} & \dots(\alpha)\\ \\ \left.\begin{align}l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1, & l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1, & l_{1}n_{1}+l_{2}n_{2}-l_{3}n_{3} & =0\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1, & m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0 \end{align} \right\} & \dots(\beta) \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2).</p> Using the equations of a one-sheet hyperboloid and of a two-sheet hyperboloid sharing the same conjugate axes, and having the common asymptotic cone <math>\tfrac{x^{2}}{a{{}^2}}+\tfrac{y^{2}}{b^{2}}-\tfrac{z^{2}}{c^{2}}=0</math>, he defined three conjugate points <math>(x_{1}\dots,y_{1}\dots,z_{1}\dots)</math> on those two conjugate hyperboloids, related to each other in the same way as equations (&alpha;, &beta;) stated above:<ref group=M>Weddle (1847), pp. 275-276</ref> :<math>\begin{matrix}\frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1\\ \frac{x^{2}}{a{{}^2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=-1\\ \hline \begin{align}\frac{x_{1}x_{2}}{a{{}^2}}+\frac{y_{1}y_{2}}{b^{2}}-\frac{z_{1}z_{2}}{c^{2}} & =0\\ \frac{x_{1}x_{3}}{a{{}^2}}+\frac{y_{1}y_{3}}{b^{2}}-\frac{z_{1}z_{3}}{c^{2}} & =0\\ \frac{x_{2}x_{3}}{a{{}^2}}+\frac{y_{2}y_{3}}{b^{2}}-\frac{z_{2}z_{3}}{c^{2}} & =0 \end{align} \quad\begin{align}\frac{x_{1}^{2}}{a{{}^2}}+\frac{y_{1}^{2}}{b^{2}}-\frac{z_{1}^{2}}{c^{2}} & =1\\ \frac{x_{2}^{2}}{a{{}^2}}+\frac{y_{2}^{2}}{b^{2}}-\frac{z_{2}^{2}}{c^{2}} & =1\\ \frac{x_{3}^{2}}{a{{}^2}}+\frac{y_{3}^{2}}{b^{2}}-\frac{z_{3}^{2}}{c^{2}} & =-1 \end{align} \\ \begin{align}x_{1}^{2}+x_{2}^{2}-x_{3}^{2} & =a^{2}\\ y_{1}^{2}+y_{2}^{2}-y_{3}^{2} & =b^{2}\\ z_{1}^{2}+z_{2}^{2}-z_{3}^{2} & =-c^{2} \end{align} \quad\begin{align}x_{1}y_{1}+x_{2}y_{2}-x_{3}y_{3} & =0\\ x_{1}z_{1}+x_{2}z_{2}-x_{3}z_{3} & =0\\ y_{1}z_{1}+y_{2}z_{2}-y_{3}z_{3} & =0 \end{align} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black"> These are the coefficients of Lorentz transformation ({{equationNote|1a}}, n=2) by setting ''a=b=c=1''.</p> ==={{anchor|Bour}} Bour (1856) – Homogeneous coordinates=== Following [[#Gauss4|Gauss (1818)]], [[w:Edmond Bour]] (1856) wrote the transformations:<ref group=M>Bour (1856), pp. 61; 64–65</ref> :<math>\begin{matrix}\cos^{2}E+\sin^{2}E-1=k\left(\cos^{2}T+\sin^{2}T-1\right)\\ \hline \left.\begin{matrix}\mathbf{(1)}\ \begin{align}\cos E & =\frac{\alpha+\alpha'\cos T+\alpha''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T}\\ \sin E & =\frac{\beta+\beta'\cos T+\beta''\sin T}{\gamma+\gamma'\cos T+\gamma''\sin T} \end{align} \\ \hline \\ k=+1\\ t=\gamma+\gamma'\cos T+\gamma''\sin T,\\ 1=u,\ \cos T=u',\ \sin T=u',\\ t=z,\ t\cos E=x,\ t\sin E=y\\ \downarrow\\ \mathbf{(2)}\begin{align}x & =\alpha u+\alpha'u'+\alpha''u''\\ y & =\beta u+\beta'u'+\beta''u''\\ z & =\gamma u+\gamma'u'+\gamma''u''\\ \\ u & =\gamma z-\alpha x-\beta y\\ u' & =\alpha'x+\beta'y'-\gamma'z\\ u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \end{matrix}\right|{\scriptstyle \begin{align}-\alpha^{2}-\beta^{2}+\gamma^{2} & =k\\ -\alpha^{\prime2}-\beta^{\prime2}+\gamma^{\prime2} & =-k\\ -\alpha^{\prime\prime2}-\beta^{\prime\prime2}+\gamma^{\prime\prime2} & =-k\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =k\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0 \end{align} } \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'', implying <math>x^{2}+y^{2}-z^{2}=u^{\prime2}+u^{\prime\prime2}-u^{2}</math>. Furthermore, setting <math>[k,\cos T,\sin T,\cos E,\sin E]=\left[1,u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math> in transformation system (1) produces Lorentz transformation ({{equationNote|1b}}) ''(n=2)''.</p> === {{anchor|Somov}} Somov (1863) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[#Jacobi|Jacobi (1827, 1833)]], and [[#Bour|Bour (1856)]], [[w:Osip Ivanovich Somov]] (1863) wrote the transformation systems:<ref group=M>Somov (1863), pp. 12–14; p. 18 for differentials.</ref> :<math>\begin{matrix}\left.\begin{align}\cos\phi & =\frac{m\cos\psi+n\sin\psi+s}{m''\cos\psi+n''\sin\psi+s''}\\ \sin\phi & =\frac{m'\cos\psi+n'\sin\psi+s'}{m''\cos\psi+n''\sin\psi+s''} \end{align} \right|\begin{matrix}\cos^{2}\phi+\cos^{2}\phi=1\\ \cos^{2}\psi+\cos^{2}\psi=1 \end{matrix}\\ \hline \mathbf{(1)}\ \begin{align}\cos\phi & =x, & \cos\psi & =x'\\ \sin\phi & =y, & \sin\psi & =y' \end{align} \ \left|\begin{align}x & =\frac{mx'+ny'+s}{m''x'+n''y'+s''}\\ y & =\frac{m'x'+n'y'+s'}{m''x'+n''y'+s''} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=1\\ x^{\prime2}+y^{\prime2}=1 \end{matrix}\\ \hline \begin{align}\cos\phi & =\frac{x}{z}, & \cos\psi & =\frac{x'}{z'}\\ \sin\phi & =\frac{y}{z}, & \sin\psi & =\frac{y'}{z'} \end{align} \ \left|\begin{align}\frac{x}{z} & =\frac{mx'+ny'+sz'}{m''x'+n''y'+s''z'}\\ \frac{y}{z} & =\frac{m'x'+n'y'+s'z'}{m''x'+n''y'+s''z'} \end{align} \right|\ \begin{matrix}x^{2}+y^{2}=z^{2}\\ x^{\prime2}+y^{\prime2}=z^{\prime2} \end{matrix}\\ \hline \mathbf{(2)}\ \left.\begin{align}x & =mx'+ny'+sz'\\ y & =m'x'+n'y'+s'z'\\ z & =m''x'+n''y'+s''z'\\ \\ x' & =mx+m'y-m''z\\ y' & =nx+n'y-n''z\\ z' & =-sx-s'y+s''z\\ \\ dx & =mdx'+ndy'+sdz'\\ dy & =m'dx'+n'dy'+s'dz'\\ dz & =m''dx'+n''dy'+s''dz' \end{align} \right|{\scriptstyle \begin{align}m^{2}+m^{\prime2}-m^{\prime\prime2} & =1\\ n^{2}+n^{\prime2}-n^{\prime\prime2} & =1\\ -s^{2}-s^{\prime2}+s^{\prime\prime2} & =1\\ ns+n's'-n''s'' & =0\\ sm+s'm'-s''m'' & =0\\ mn+m'n'-m''n'' & =0\\ \\ m^{2}+n^{2}-s^{2} & =1\\ m^{\prime2}+n^{\prime2}-s^{\prime2} & =1\\ -m^{\prime\prime2}-n^{\prime\prime2}+s^{\prime\prime2} & =1\\ -m'm''-n'n''+s's'' & =0\\ -m''m-n''n+s''s & =0\\ mm'+nn'-ss' & =0 \end{align} }\\ dx^{2}+dy^{2}-dz^{2}=dx^{\prime2}+dy^{\prime2}-dz^{\prime2} \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)''. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Klein}} Klein (1871-73) – Cayley absolute and non-Euclidean geometry=== Elaborating on [[w:Arthur Cayley]]'s (1859) definition of an "absolute" ([[w:Cayley–Klein metric]]), [[w:Felix Klein]] (1871) defined a "fundamental [[w:conic section]]" in order to discuss motions such as rotation and translation in the non-Euclidean plane.<ref group=M>Klein (1871), pp. 601–602</ref> This was elaborated in (1873) when he pointed out that hyperbolic geometry in terms of a surface of constant negative curvature can be related to a quadratic equation, which can be transformed into a sum of squares of which one square has a different sign, and can also be related to the interior of a surface of second degree corresponding to a two-sheet [[w:hyperboloid]].<ref group=M>Klein (1873), pp. 127-128</ref> <p style="background-color:Beige;border:1px solid black">Klein's representation of hyperbolic space in terms of a two-sheet hyperboloid and its accompanied quadratic form suggests that Lorentz transformations can be geometrically interpreted as motions or isometries in hyperbolic space.</p> ==={{anchor|Killing}} Killing (1878–1893)=== ===={{anchor|Killing1}} Weierstrass coordinates==== [[w:Wilhelm Killing]] (1878–1880) described non-Euclidean geometry by using [[w:hyperboloid model|Weierstrass coordinates]] (named after [[w:Karl Weierstrass]] who described them in lectures in 1872 which Killing attended) obeying the form :<math>k^{2}t^{2}+u^{2}+v^{2}+w^{2}=k^{2}</math><ref group=M>Killing (1877/78), p. 74; Killing (1880), p. 279</ref> with <math>ds^{2}=k^{2}dt^{2}+du^{2}+dv^{2}+dw^{2}</math><ref group=M>Killing (1880), eq. 25 on p. 283</ref> or<ref group=M>Killing (1880), p. 283</ref> :<math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> where ''k'' is the reciprocal measure of curvature, <math>k^{2}=\infty</math> denotes [[w:Euclidean geometry]], <math>k^{2}>0</math> [[w:elliptic geometry]], and <math>k^{2}<0</math> hyperbolic geometry. In (1877/78) he pointed out the possibility and some characteristics of a transformation (indicating rigid motions) preserving the above form.<ref group=M>Killing (1877/78), eq. 25 on p. 283</ref> In (1879/80) he tried to formulate the corresponding transformations by plugging <math>k^{2}</math> into a [[w:Rotation matrix#Rotation matrix from axis and angle|general rotation matrix]]:<ref group=M>Killing (1879/80), p. 274</ref> <math>\begin{matrix}k^{2}u^{2}+v^{2}+w^{2}=k^{2}\\ \hline \begin{matrix}\cos\eta\tau+\lambda^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \nu\frac{\sin\eta\tau}{\eta}+\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & -\mu\frac{\sin\eta\tau}{\eta}+\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}\\ -k^{2}\nu\frac{\sin\eta\tau}{\eta}+k^{2}\lambda\mu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\mu^{2}\frac{1-\cos\eta\tau}{\eta^{2}}, & \lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}\\ k^{2}\mu\frac{\sin\eta\tau}{\eta}+k^{2}\nu\lambda\frac{1-\cos\eta\tau}{\eta^{2}}, & -\lambda\frac{\sin\eta\tau}{\eta}+k^{2}\mu\nu\frac{1-\cos\eta\tau}{\eta^{2}}, & \cos\eta\tau+\nu^{2}\frac{1-\cos\eta\tau}{\eta^{2}} \end{matrix}\\ \left(\lambda^{2}+k^{2}\mu^{2}+k^{2}\nu^{2}=\eta^{2}\right) \end{matrix}</math> In (1885) he wrote the Weierstrass coordinates and their transformation as follows:<ref group=M>Killing (1885), pp. 18, 28–30, 53</ref> :<math>\begin{matrix}k^{2}p^{2}+x^{2}+y^{2}=k^{2}\\ k^{2}p^{2}+x^{2}+y^{2}=k^{2}p^{\prime2}+x^{\prime2}+y^{\prime2}\\ ds^{2}=k^{2}dp^{2}+dx^{2}+dy^{2}\\ \hline \begin{align}k^{2}p' & =k^{2}wp+w'x+w''y\\ x' & =ap+a'x+a''y\\ y' & =bp+b'x+b''y\\ \\ k^{2}p & =k^{2}wp'+ax'+by'\\ x & =w'p'+a'x+b'y'\\ y & =w''p'+a''x'+b''y' \end{align} \left|{\scriptstyle \begin{align}k^{2}w^{2}+w^{\prime2}+w^{\prime\prime2} & =k^{2}\\ \frac{a^{2}}{k^{2}}+a^{\prime2}+a^{\prime\prime2} & =1\\ \frac{b^{2}}{k^{2}}+b^{\prime2}+b^{\prime\prime2} & =1\\ aw+a'w'+a''w'' & =0\\ bw+b'w'+b''w'' & =0\\ \frac{ab}{k^{2}}+a'b'+a''b'' & =0\\ \\ k^{2}w^{2}+a^{2}+b^{2} & =k^{2}\\ \frac{w^{\prime2}}{k^{2}}+a^{\prime2}+b^{\prime2} & =1\\ \frac{w^{\prime\prime2}}{k^{2}}+a^{\prime\prime2}+b^{\prime\prime2} & =1\\ ww'+aa'+bb' & =0\\ ww''+aa''+bb'' & =0\\ \frac{w'w''}{k^{2}}+a'a''+b'b'' & =0 \end{align} }\right. \end{matrix}</math> In (1885) he also gave the transformation for ''n'' dimensions:<ref group=M>Killing (1884/85), pp. 42–43; Killing (1885), pp. 73–74, 222</ref><ref>Ratcliffe (1994), § 3.6</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ ds^{2}=k^{2}dx_{0}^{2}+dx_{1}^{2}+\dots+dx_{n}^{2}\\ \hline \left.\begin{align}k^{2}\xi_{0} & =k^{2}a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{0}\\ \xi_{\varkappa} & =a_{\varkappa0}x_{0}+a_{\varkappa1}x_{1}+\dots+a_{\varkappa n}x_{n}\\ \\ k^{2}x_{0} & =a_{00}k^{2}\xi_{0}+a_{10}\xi_{1}+\dots+a_{n0}\xi_{n}\\ x_{\varkappa} & =a_{0\varkappa}\xi_{0}+a_{1\varkappa}\xi_{1}+\dots+a_{n\varkappa}\xi_{n} \end{align} \right|{\scriptstyle \begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ \frac{a_{0\iota}a_{0\varkappa}}{k^{2}}+a_{0\iota}a_{1\varkappa}+\dots+a_{n\iota}a_{n\varkappa}=\delta_{\iota\kappa} & =1\ (\iota=\kappa)\ \text{or}\ 0\ (\iota\ne\kappa) \end{align} } \end{matrix}</math> In (1885) he applied his transformations to mechanics and defined four-dimensional vectors of velocity and force.<ref group=M>Killing (1884/85), pp. 4–5</ref> Regarding the geometrical interpretation of his transformations, Killing argued in (1885) that by setting <math>k^{2}=-1</math> and using ''p,x,y'' as rectangular space coordinates, the hyperbolic plane is mapped on one side of a two-sheet hyperboloid <math>p^{2}-x^{2}-y^{2}=1</math> (known as [[w:hyperboloid model]]),<ref group=M>Killing (1885), Note 9 on p. 260</ref><ref name=rey /> by which the previous formulas become equivalent to Lorentz transformations and the geometry becomes that of Minkowski space. <p style="background-color:Beige;border:1px solid black">All of Killing's transformations between 1879 and 1885 don't work when <math>k^{2}</math> is negative, thus they fail to produce Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> Finally, in (1893) he wrote:<ref group=M>Killing (1893), see pp. 144, 327–328</ref> :<math>\begin{matrix}k^{2}t^{2}+u^{2}+v^{2}=k^{2}\\ \hline \begin{align}t' & =at+bu+cv\\ u' & =a't+b'u+c'v\\ v' & =a''t+b''u+c''v \end{align} \left|\begin{align}k^{2}a^{2}+a^{\prime2}+a^{\prime\prime2} & =k^{2}\\ k^{2}b^{2}+b^{\prime2}+b^{\prime\prime2} & =1\\ k^{2}c^{2}+b^{\prime2}+c^{\prime\prime2} & =1\\ k^{2}ab+a'b'+a''b'' & =0\\ k^{2}ac+a'c'+a''c'' & =0\\ k^{2}bc+b'c'+b''c'' & =0 \end{align} \right. \end{matrix}</math> and in ''n'' dimensions<ref group=M>Killing (1893), pp. 314–316, 216–217</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ k^{2}y_{0}y_{0}^{\prime}+y_{1}y_{1}^{\prime}+\cdots+y_{n}y_{n}^{\prime}=k^{2}x_{0}x_{0}^{\prime}+x_{1}x_{1}^{\prime}+\cdots+x_{n}x_{n}^{\prime}\\ ds^{2}=k^{2}dx_{0}^{2}+\dots+dx_{n}^{2}\\ \hline \begin{align}y_{0} & =a_{00}x_{0}+a_{01}x_{1}+\dots+a_{0n}x_{n}\\ y_{1} & =a_{10}x_{0}+a_{11}x_{1}+\dots+a_{1n}x_{n}\\ & \,\,\,\vdots\\ y_{n} & =a_{n0}x_{0}+a_{n1}x_{1}+\dots+a_{nn}x_{n} \end{align} \left|\begin{align}k^{2}a_{00}^{2}+a_{10}^{2}+\dots+a_{n0}^{2} & =k^{2}\\ k^{2}a_{0\varkappa}^{2}+a_{1\varkappa}^{2}+\dots+a_{n\varkappa}^{2} & =1\\ k^{2}a_{00}a_{0\varkappa}+a_{10}a_{1\varkappa}+\dots+a_{n0}a_{n\varkappa} & =0\\ k^{2}a_{0\varkappa}a_{0\lambda}+a_{1\varkappa}a_{1\lambda}+\dots+a_{n\varkappa}a_{n\lambda} & =0\\ (\varkappa,\lambda=1,\dots, n,\ \lambda\lessgtr\varkappa) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) with <math>k^{2}=-1</math>.</p> ===={{anchor|Killing3}} Infinitesimal transformations and Lie group==== After [[#Lie3|Lie (1885/86)]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions, Killing (1887/88)<ref group=M>Killing (1887/88a), pp. 274–275</ref> defined the infinitesimal projective transformations (Lie algebra) in relation to the unit hypersphere: :<math>\begin{matrix}x_{1}^{2}+\dots+x_{m+1}^{2}=1\\ \hline X_{\iota\varkappa}f=x_{i}\frac{\partial f}{\partial x_{\varkappa}}-x_{\varkappa}\frac{\partial f}{\partial x_{\iota}}\\ \text{where}\\ \left(X_{\iota\varkappa},X_{\iota\lambda}\right)=X_{\varkappa\lambda};\ \left(X_{\iota\varkappa},X_{\lambda\mu}\right)=0;\\ \left[\iota\ne\varkappa\ne\lambda\ne\mu\right] \end{matrix}</math> and in (1892) he defined the infinitesimal transformation for non-Euclidean motions in terms of Weierstrass coordinates:<ref group=M>Killing (1892), p. 177</ref> :<math>\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}\\ \hline X_{\iota\varkappa}=x_{\iota}p_{\varkappa}-x_{\varkappa}p_{\iota},\quad X_{\iota}=x_{0}p_{\iota}-\frac{x_{\iota}p_{0}}{k^{2}}\\ \text{where}\\ \left(X_{\iota}X_{\iota\varkappa}\right)=X_{\varkappa}f;\ \left(X_{\iota}X_{\varkappa\lambda}\right)=0;\ \left(X_{\iota}X_{\varkappa}\right)=-\frac{1}{k^{2}}X_{\iota\varkappa}f; \end{matrix}</math> In (1897/98) he showed the relation between Weierstrass coordinates <math>k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2}</math> and corrdinates <math>k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0</math> used by himself in (1887/88) and by [[#Lie3|Werner (1889), Lie (1890)]]:<ref group=M>Killing (1897/98), pp. 255–256</ref> :<math>\begin{matrix}\begin{matrix}k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2} & (a)\\ k^{2}x_{0}^{2}+x_{1}^{2}+\dots+x_{n}^{2}=k^{2} & (b) \end{matrix}\\ \hline V_{\varkappa}=k^{2}x_{0}p_{\varkappa}-x_{\varkappa}p_{0},\quad U_{\iota\varkappa}=p_{\iota}x_{\varkappa}-p_{\varkappa}x_{\iota}\\ \text{where}\\ \left(V_{\iota},V_{\varkappa}\right)=k^{2}U_{\iota\varkappa},\ \left(V_{\iota},U_{\iota\varkappa}\right)=-V_{\varkappa},\ \left(V_{\iota},U_{\varkappa\lambda}\right)=0,\\ \left(U_{\iota\varkappa},U_{\iota\lambda}\right)=U_{\varkappa\lambda},\ \left(U_{\iota\varkappa},U_{\lambda\mu}\right)=0\\ \left[\iota,\varkappa,\lambda,\mu=1,2,\dots n\right]\\ \hline \begin{matrix}y_{1}=\frac{x_{1}}{x_{0}},\ y_{2}=\frac{x_{2}}{x_{0}},\dots y_{n}=\frac{x_{n}}{x_{0}}\\ \downarrow\\ k^{2}+y_{1}^{2}+y_{2}^{2}+\dots+y_{n}^{2}=0\\ \hline q_{\varkappa}+\frac{y_{\varkappa}}{k^{2}}\sum_{\varrho}y_{y}q_{\varrho},\quad q_{\iota}y_{\varkappa}-q_{\varkappa}y_{\iota} \end{matrix} \end{matrix}</math> He pointed out that the corresponding group of non-Euclidean motions in terms of Weierstrass coordinates is intransitive when related to quadratic form (a) and [[w:Group action (mathematics)|transitive]] when related to quadratic form (b). <p style="background-color:Beige;border:1px solid black">Setting <math>k^{2}=-1</math> denotes the group of hyperbolic motions and thus the Lorentz group.</p> === {{anchor|Poincare}} Poincaré (1881) – Weierstrass coordinates === [[w:Henri Poincaré]] (1881) connected the work of [[../Lorentz transformation (Cayley-Hermite)#Hermite|E:Hermite (1853)]] and [[../Lorentz transformation (Möbius)#Selling|E:Selling (1873)]] on indefinite quadratic forms with non-Euclidean geometry (Poincaré already discussed such relations in an unpublished manuscript in 1880).<ref>Gray (1997)</ref> He used two indefinite ternary forms in terms of three squares and then defined them in terms of Weierstrass coordinates (without using that expression) connected by a transformation with integer coefficients:<ref group=M name=p1>Poincaré (1881a), pp. 133–134</ref><ref>Dickson (1923), pp. 220–221</ref> :<math>\begin{matrix}\begin{align}F & =(ax+by+cz)^{2}+(a'x+b'y+c'z)^{2}-(a''x+b''y+c''z)^{2}\\ & =\xi^{2}+\eta^{2}-\zeta^{2}=-1\\ F & =(ax'+by'+cz')^{2}+(a'x'+b'y'+c'z')^{2}-(a''x'+b''y'+c''z')^{2}\\ & =\xi^{\prime2}+\eta^{\prime2}-\zeta^{\prime2}=-1 \end{align} \\ \hline \begin{align}\xi' & =\alpha\xi+\beta\eta+\gamma\zeta\\ \eta' & =\alpha'\xi+\beta'\eta+\gamma'\zeta\\ \zeta' & =\alpha''\xi+\beta''\eta+\gamma''\zeta \end{align} \left|\begin{align}\alpha^{2}+\alpha^{\prime2}-\alpha^{\prime\prime2} & =1\\ \beta^{2}+\beta^{\prime2}-\beta^{\prime\prime2} & =1\\ \gamma^{2}+\gamma^{\prime2}-\gamma^{\prime\prime2} & =-1\\ \alpha\beta+\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma+\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma+\beta'\gamma'-\beta''\gamma'' & =0 \end{align} \right. \end{matrix}</math> He went on to describe the properties of "hyperbolic coordinates".<ref group=M name=poinc>Poincaré (1881b), p. 333</ref><ref name=rey>Reynolds (1993)</ref> Poincaré mentioned the hyperboloid model also in (1887).<ref group=M>Poincaré (1887), p. 206</ref> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> === {{anchor|Cox}} Cox (1881–1883) – Weierstrass coordinates === [[w:Homersham Cox (mathematician)|Homersham Cox]] (1881/82) – referring to similar rectangular coordinates used by [[w:Christoph Gudermann|Gudermann]] (1830)<ref name=guder group=M>Gudermann (1830), §1–3, §18–19</ref> and [[w:George Salmon]] (1862)<ref group=M>Salmon (1862), section 212, p. 165</ref> on a sphere, and to [[#Escherich|Escherich (1874)]] as reported by [[w:Johannes Frischauf]] (1876)<ref group=M>Frischauf (1876), pp. 86–87</ref> in the hyperbolic plane – defined the Weierstrass coordinates (without using that expression) and their transformation:<ref group=M>Cox (1881), p. 186 for Weierstrass coordinates; (1881/82), pp. 193–194 for Lorentz transformation. On p. 193, the misprinted expression <math>x^{2}-y^{2}-z^{2}</math> should read <math>z^{2}-y^{2}-x^{2}</math></ref> :<math>\begin{matrix}z^{2}-x^{2}-y^{2}=1\\ z^{2}-y^{2}-x^{2}=Z^{2}-Y^{2}-X^{2}\\ \hline \begin{align}x & =l_{1}X+l_{2}Y+l_{3}Z\\ y & =m_{1}X+m_{2}Y+m_{3}Z\\ z & =n_{1}X+n_{2}Y+n_{3}Z\\ \\ X & =l_{1}x+m_{1}y-n_{1}z\\ Y & =l_{2}x+m_{2}y-n_{2}z\\ Z & =l_{3}x+m_{3}y-n_{3}z \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}-n_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}-n_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ l_{1}l_{2}+m_{1}m_{2}-n_{1}n_{2} & =0\\ l_{2}l_{3}+m_{2}m_{3}-n_{2}n_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}-n_{3}n_{1} & =0\\ \\ l_{1}^{2}+l_{2}^{2}-l_{3}^{2} & =1\\ m_{1}^{2}+m_{2}^{2}-m_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1\\ l_{1}m_{1}+l_{2}m_{2}-l_{3}m_{3} & =0\\ m_{1}n_{1}+m_{2}n_{2}-m_{3}n_{3} & =0\\ n_{1}l_{1}+n_{2}l_{2}-n_{3}l_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =1 \end{align} }</math> with <math>{\scriptstyle \begin{align}l_{3}^{2}+m_{3}^{2}-n_{3}^{2} & =-1\\ n_{1}^{2}+n_{2}^{2}-n_{3}^{2} & =-1 \end{align} }</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' up to a sign change in the inverse transformation.</p> Cox also gave the Weierstrass coordinates and their transformation in hyperbolic space:<ref group=M>Cox (1881), pp. 199, 206–207</ref> :<math>\begin{matrix}w^{2}-x^{2}-y^{2}-z^{2}=1\\ w^{2}-x^{2}-y^{2}-z^{2}=w^{\prime2}-x^{\prime2}-y^{\prime2}-z^{\prime2}\\ \hline \begin{align}x & =l_{1}x'+l_{2}y'+l_{3}z'-l_{4}w'\\ y & =m_{1}x'+m_{2}y'+m_{3}z'-m_{4}w'\\ z & =n_{1}x'+n_{2}y'+n_{3}z'-n_{4}w'\\ w & =r_{1}x'+r_{2}y'+r_{3}z'-r_{4}w'\\ \\ x' & =l_{1}x+m_{1}y+n_{1}z-r_{1}w\\ y' & =l_{2}x+m_{2}y+n_{2}z-r_{2}w\\ z' & =l_{3}x+m_{3}y+n_{3}z-r_{3}w\\ w' & =l_{4}x+m_{4}y+n_{4}z-r_{4}w \end{align} \left|{\scriptstyle \begin{align}l_{1}^{2}+m_{1}^{2}+n_{1}^{2}-r_{1}^{2} & =1\\ l_{2}^{2}+m_{2}^{2}+n_{2}^{2}-r_{2}^{2} & =1\\ l_{3}^{2}+m_{3}^{2}+n_{3}^{2}-r_{3}^{2} & =1\\ l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2} & =1\\ l_{2}l_{3}+m_{2}m_{3}+n_{2}n_{3}-r_{2}r_{3} & =0\\ l_{3}l_{1}+m_{3}m_{1}+n_{3}n_{1}-r_{3}r_{1} & =0\\ l_{1}l_{4}+m_{1}m_{4}+n_{1}n_{4}-r_{1}r_{4} & =0\\ l_{2}l_{4}+m_{2}m_{4}+n_{2}n_{4}-r_{2}r_{4} & =0\\ l_{3}l_{4}+m_{3}m_{4}+n_{3}n_{4}-r_{3}r_{4} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=1}</math> with <math>{\scriptstyle l_{4}^{2}+m_{4}^{2}+n_{4}^{2}-r_{4}^{2}=-1}</math>, this becomes Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' up to a sign change in both the first as well as inverse transformation.</p> === {{anchor|Hill}} Hill (1882) – Homogeneous coordinates === Following [[#Gauss4|Gauss (1818)]], [[w:George William Hill]] (1882) formulated the equations<ref group=M>Hill (1882), pp. 323–325</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)\\ k\left(\sin^{2}E+\cos^{2}E-1\right)\\ \hline \begin{align} & & \cos E' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ & \mathbf{(1)} & \sin E' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \hline \\ & & x & =\alpha u+\alpha'u'+\alpha''u''\\ & & y & =\beta u+\beta'u'+\beta''u''\\ & & z & =\gamma u+\gamma'u'+\gamma''u''\\ & \mathbf{(2)}\\ & & u & =-\alpha x-\beta y+\gamma z\\ & & u' & =\alpha'x+\beta'y'-\gamma'z\\ & & u'' & =\alpha''x+\beta''y-\gamma''z \end{align} \left|{\scriptstyle \begin{align}\alpha^{2}+\beta^{2}-\gamma^{2} & =-1\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =1\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =1\\ \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ (k=-1)\\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =k\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =k\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =-k\\ \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos E',\sin E']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>. Transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' .</p> === {{anchor|Picard}} Picard (1882-1884) – Quadratic forms === [[w:Émile Picard]] (1882) analyzed the invariance of indefinite ternary [[w:Hermitian form|Hermitian quadratic forms]] with integer coefficients and their relation to [[w:Group action (mathematics)|discontinuous groups]], extending Poincaré's Fuchsian functions of one complex variable related to a circle, to "hyperfuchsian" functions of two complex variables related to a [[w:hypersphere]]. He formulated the following special case of an Hermitian form:<ref group=M>Picard (1882), pp. 307–308 first transformation system; pp. 315-317 second transformation system</ref><ref>Dickson (1923), pp. 280-281</ref> :<math>\begin{matrix}\begin{matrix}xx_{0}+yy_{0}-zz_{0}\\ \\ \mathbf{(1)}\ \begin{align}x & =M_{1}X+P_{1}Y+R_{1}Z\\ y & =M_{2}X+P_{2}Y+R_{2}Z\\ z & =M_{3}X+P_{3}Y+R_{3}Z \end{align} \\ \\ \left[\begin{align}[][x,y,z]=\text{complex}\\ \left[x_{0},y_{0},z_{0}\right]=\text{conjugate} \end{align} \right]\\ \\ \hline \\ x^{\prime2}+x^{\prime\prime2}+y^{\prime2}+y^{\prime\prime2}=1\\ x=x'+ix'',\quad y=y'+iy''\\ \\ \mathbf{(2)}\ \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \end{matrix}\left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3} & =1\\ P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0\\ P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ \\ M_{1}\mu_{1}+P_{1}\pi_{1}-R_{1}\rho_{1} & =1\\ M_{2}\mu_{2}+P_{2}\pi_{2}-R_{2}\rho_{2} & =1\\ M_{3}\mu_{3}+P_{3}\pi_{3}-R_{3}\rho_{3} & =-1\\ \mu_{2}M_{1}+\pi_{2}P_{1}-R_{1}\rho_{2} & =0\\ \mu_{2}M_{3}+\pi_{2}P_{3}-R_{3}\rho_{2} & =0\\ \mu_{3}M_{1}+\pi_{3}P_{1}-R_{1}\rho_{3} & =0\\ \\ \left[\begin{align}[][M,P,R\dots]=\text{complex}\\ \left[\mu,\pi,\rho\dots\right]=\text{conjugate} \end{align} \right] \end{align} }\right.\end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''x²+y²-z²=X²+Y²-Z²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884a) in relation to indefinite binary Hermitian quadratic forms:<ref group=M>Picard (1884a), p. 13</ref> :<math>\begin{matrix}UU_{0}-VV_{0}=uu_{0}-vv_{0}\\ \hline \begin{align}U & =\mathcal{A}u+\mathcal{B}v\\ V & =\mathcal{C}u+\mathcal{D}v \end{align} \left|\begin{align}\mathcal{A}\mathcal{A}_{0}-\mathcal{C}\mathcal{C}_{0} & =1\\ \mathcal{A}\mathcal{B}_{0}-\mathcal{C}\mathcal{D}_{0} & =0\\ \mathcal{B}\mathcal{B}_{0}-\mathcal{D}\mathcal{D}_{0} & =-1\\ \mathcal{D}\mathcal{D}_{0}-\mathcal{C}\mathcal{C}_{0} & =1 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=1)'' producing ''U²-V²=u²-v²''.</p> Or in (1884b):<ref group=M>Picard (1884b), p. 416</ref> :<math>\begin{matrix}xx_{0}+yy_{0}-1=0\\ \hline \begin{align}X & =\frac{M_{1}x+P_{1}y+R_{1}}{M_{3}x+P_{3}y+R_{3}}\\ Y & =\frac{M_{2}x+P_{2}y+R_{2}}{M_{3}x+P_{3}y+R_{3}} \end{align} \left|{\scriptstyle \begin{align}M_{1}\mu_{1}+M_{2}\mu_{2}-M_{3}\mu_{3}=P_{1}\pi_{1}+P_{2}\pi_{2}-P_{3}\pi_{3} & =1\\ R_{1}\rho_{1}+R_{2}\rho_{2}-R_{3}\rho_{3} & =-1\\ P_{1}\mu_{1}+P_{2}\mu_{2}-P_{3}\mu_{3}=M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3}=P_{1}\rho_{1}+P_{2}\rho_{2}-P_{3}\rho_{3} & =0\\ M_{1}\rho_{1}+M_{2}\rho_{2}-M_{3}\rho_{3} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, this is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing ''x²+y²=X²+Y²=1''.</p> Or in (1884c):<ref group=M>Picard (1884c), pp. 123–124; 163</ref> :<math>\begin{matrix}UU_{0}+VV_{0}-WW_{0}=uu_{0}+vv_{0}-ww_{0}\\ \hline \mathbf{(1)}\ \begin{align}U & =Mu+Pv+Rw\\ V & =M'u+P'v+R'w\\ W & =M''u+P''v+R''w\\ \\ u & =M_{0}U+M_{0}^{\prime}V-M_{0}^{\prime\prime}W\\ v & =P_{0}U+P_{0}^{\prime}V-P_{0}^{\prime\prime}W\\ w & =-R_{0}U-R_{0}^{\prime}V+R_{0}^{\prime\prime}W \end{align} \left|{\scriptstyle \begin{align}MM_{0}+M'M_{0}^{\prime}-M''M_{0}^{\prime\prime} & =1\\ PP_{0}+P'P_{0}^{\prime}-P''P_{0}^{\prime\prime} & =1\\ RR_{0}+R'R_{0}^{\prime}-R''R_{0}^{\prime\prime} & =-1\\ MP_{0}+M'P_{0}^{\prime}-M''P_{0}^{\prime\prime} & =0\\ MR_{0}+M'R_{0}^{\prime}-M''R_{0}^{\prime\prime} & =0\\ PR_{0}+P'R_{0}^{\prime}-P''R_{0}^{\prime\prime} & =0\\ \\ MM_{0}+PP_{0}-RR_{0} & =1\\ M'M_{0}^{\prime}+P'P_{0}^{\prime}-R'R_{0}^{\prime} & =1\\ M''M_{0}^{\prime\prime}+P''P_{0}^{\prime\prime}-R''R_{0}^{\prime\prime} & =-1\\ M_{0}M'+P_{0}P'-R_{0}R' & =0\\ M_{0}M''+P_{0}P''-R_{0}R'' & =0\\ M_{0}^{\prime}M''+P_{0}^{\prime}P''-R_{0}^{\prime}R'' & =0 \end{align} }\right.\\ \hline \text{Invariance of unit hypersphere:}\\ \mathbf{(2)}\ \begin{align}\xi' & =\frac{A\xi+A'\eta+A''}{C\xi+C'\eta+C''}\\ \eta' & =\frac{B\xi+B'\eta+B''}{C\xi+C'\eta+C''} \end{align} \left|{\scriptstyle \begin{align}AA_{0}+A'A_{0}^{\prime}-A''A_{0}^{\prime\prime} & =1\\ BB_{0}+B'B_{0}^{\prime}-B''B_{0}^{\prime\prime} & =1\\ CC_{0}+C'C_{0}^{\prime}-C''C_{0}^{\prime\prime} & =-1\\ AB_{0}+A'B_{0}^{\prime}-A''B_{0}^{\prime\prime} & =0\\ AC_{0}+A'C_{0}^{\prime}-A''C_{0}^{\prime\prime} & =0\\ BC_{0}+B'C_{0}^{\prime}-B''C_{0}^{\prime\prime} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">Replacing the imaginary variables and coefficients with real ones, transformation system (1) is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)'' producing ''U²+V²-W²=u²+v²-w²'' and transformation system (2) is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' producing <math>\xi^{\prime2}+\eta^{\prime2}=\xi^{2}+\eta^{2}=1</math>.</p> === {{anchor|Callandreau}} Callandreau (1885) – Homography === Following [[#Gauss4|Gauss (1818)]] and [[#Hill|Hill (1882)]], [[w:Octave Callandreau]] (1885) formulated the equations<ref group=M>Callandreau (1885), pp. A.7; A.12</ref> :<math>\begin{matrix}k\left(\sin^{2}T+\cos^{2}T-1\right)=\\ {\scriptstyle (\alpha+\alpha'\sin T+\alpha''\cos T)^{2}+(\beta+\beta'\sin T+\beta''\cos T)^{2}-(\gamma+\gamma'\sin T+\gamma''\cos T)^{2}}\\ \hline \begin{align}\cos\varepsilon' & =\frac{\alpha+\alpha'\sin T+\alpha''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T}\\ \sin\varepsilon' & =\frac{\beta+\beta'\sin T+\beta''\cos T}{\gamma+\gamma'\sin T+\gamma''\cos T} \end{align} \left|{\scriptstyle \begin{align} & \left(k=1\right)\\ \alpha^{2}+\beta^{2}-\gamma^{2} & =-k & \alpha\alpha'+\beta\beta'-\gamma\gamma' & =0\\ \alpha^{\prime2}+\beta^{\prime2}-\gamma^{\prime2} & =+k & \alpha\alpha''+\beta\beta''-\gamma\gamma'' & =0\\ \alpha^{\prime\prime2}+\beta^{\prime\prime2}-\gamma^{\prime\prime2} & =+k & \alpha'\alpha''+\beta'\beta''-\gamma'\gamma'' & =0\\ \\ \alpha^{2}-\alpha^{\prime2}-\alpha^{\prime\prime2} & =-1 & \alpha\beta-\alpha'\beta'-\alpha''\beta'' & =0\\ \beta^{2}-\beta^{\prime2}-\beta^{\prime\prime2} & =-1 & \alpha\gamma-\alpha'\gamma'-\alpha''\gamma'' & =0\\ \gamma^{2}-\gamma^{\prime2}-\gamma^{\prime\prime2} & =+1 & \beta\gamma-\beta'\gamma'-\beta''\gamma'' & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">The transformation system is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=2)'' with <math>[\cos T,\sin T,\cos\varepsilon',\sin\varepsilon']=\left[u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}\right]</math>.</p> ==={{anchor|Lie3}} Lie (1885-1890) – Lie group, hyperbolic motions, and infinitesimal transformations=== In (1885/86), [[w:Sophus Lie]] identified the projective group of a general surface of second degree <math>\sum f_{ik}x_{i}'x_{k}'=0</math> with the group of non-Euclidean motions.<ref group=M>Lie (1885/86), p. 411</ref> In a thesis guided by Lie, [[w:Hermann Werner]] (1889) discussed this projective group by using the equation of a unit hypersphere as the surface of second degree (which was already given before by [[#Killing3|Killing (1887)]]), and also gave the corresponding infinitesimal projective transformations (Lie algebra):<ref group=M>Werner (1889), pp. 4, 28</ref> :<math>\begin{matrix}x_{1}^{2}+x_{2}^{2}+\dots+x_{n}^{2}=1\\ \hline x_{i}p_{\varkappa}-x_{\varkappa}p_{i},\quad p_{i}-x_{i}\sum_{1}^{n}{\scriptstyle j}\ x_{j}p_{j}\quad(i,\varkappa=1,\dots, n)\\ \text{where}\\ \left(Q_{i},Q_{\varkappa}\right)=R_{i,\varkappa};\ \left(Q_{i},Q_{j,\varkappa}\right)=\varepsilon_{i,j}Q_{\varkappa}-\varepsilon_{i,\varkappa}Q_{j};\\ \left(R_{i,\varkappa},R_{\mu,\nu}\right)=\varepsilon_{\varkappa,\mu}R_{i,\nu}-\varepsilon_{\varkappa,\nu}R_{i,\mu}-\varepsilon_{,\mu}R_{\varkappa,\nu}+\varepsilon_{i,\nu}R_{\varkappa,\mu}\\ \left[\varepsilon_{i,\varkappa}\equiv0\ \text{for}\ i\ne\varkappa;\ \varepsilon_{i,i}=1\right] \end{matrix}</math> More generally, Lie (1890)<ref group=M>Lie (1890a), p. 295;</ref> defined non-Euclidean motions in terms of two forms <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\pm1=0</math> in which the imaginary form with <math>+1</math> denotes the group of elliptic motions (in Klein's terminology), the real form with −1 the group of hyperbolic motions, with the latter having the same form as Werner's transformation:<ref group=M>Lie (1890a), p. 311</ref> :<math>\begin{matrix}x_{1}^{2}+\dots+x_{n}^{2}-1=0\\ \hline p_{k}-x_{k}\sum j_{1}^{0}x_{j}p_{j},\quad x_{i}p_{k}-x_{k}p_{i}\quad(i,k=1\dots n) \end{matrix}</math> Summarizing, Lie (1893) discussed the real continuous groups of the conic sections representing non-Euclidean motions, which in the case of hyperbolic motions have the form: :<math>x^{2}+y^{2}-1=0</math><ref group=M>Lie (1893), p. 474</ref> or <math>x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-1=0</math><ref group=M>Lie (1893), p. 479</ref> or <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math>.<ref group=M>Lie (1893), p. 481</ref> <p style="background-color:Beige;border:1px solid black">The group of hyperbolic motions is isomorphic to the Lorentz group. The interval <math>x_{1}^{2}+\dots+x_{n}^{2}-1=0</math> becomes the Lorentz interval <math>x_{1}^{2}+\dots+x_{n}^{2}-x_{0}^{2}=0</math> by setting <br><math>(x_{1},\dots,\ x_{n},\ 1)=\left(\frac{x_{1}}{x_{0}},\dots,\ \frac{x_{n}}{x_{0}},\ \frac{x_{0}}{x_{0}}\right)</math></p> ==={{anchor|Gerard}} Gérard (1892) – Weierstrass coordinates=== [[w:Louis Gérard]] (1892) – in a thesis examined by Poincaré – discussed Weierstrass coordinates (without using that name) in the plane using the following invariant and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)'':<ref group=M>Gérard (1892), pp. 40–41</ref> :<math>\begin{matrix}X^{2}+Y^{2}-Z^{2}=1\\ X^{2}+Y^{2}-Z^{2}=X^{\prime2}+Y^{\prime2}-Z^{\prime2}\\ \hline \begin{align}X & =aX'+a'Y'+a''Z'\\ Y & =bX'+b'Y'+b''Z'\\ Z & =cX'+c'Y'+c''Z'\\ \\ X' & =aX+bY-cZ\\ Y' & =a'X+b'Y-c'Z\\ Z' & =-a''X-b''Y+c''Z \end{align} \left|\begin{align}a^{2}+b^{2}-c^{2} & =1\\ a^{\prime2}+b^{\prime2}-c^{\prime2} & =1\\ a^{\prime\prime2}+b^{\prime\prime2}-c^{\prime\prime2} & =-1\\ aa'+bb'-cc' & =0\\ a'a''+b'b''-c'c'' & =0\\ a''a+b''b-c''c & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> He gave the case of translation as follows:<ref group=M name=gerard>Gérard (1892), pp. 40–41</ref> :<math>\begin{align}X & =Z_{0}X'+X_{0}Z'\\ Y & =Y'\\ Z & =X_{0}X'+Z_{0}Z' \end{align} \ \text{with}\ \begin{align}X_{0} & =\operatorname{sh}OO'\\ Z_{0} & =\operatorname{ch}OO' \end{align} </math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz boost ({{equationNote|3b}}).</p> ==={{anchor|Hausdorff}} Hausdorff (1899) – Weierstrass coordinates=== [[w:Felix Hausdorff]] (1899) – citing Killing (1885) – discussed Weierstrass coordinates in the plane using the following invariant and its transformation:<ref group=M>Hausdorff (1899), p. 165, pp. 181-182</ref> :<math>\begin{matrix}p^{2}-x^{2}-y^{2}=1\\ \hline \begin{align}x & =a_{1}x'+a_{2}y'+x_{0}p'\\ y & =b_{1}x'+b{}_{2}y'+y_{0}p'\\ p & =e_{1}x'+e_{2}y'+p_{0}p'\\ \\ x' & =a_{1}x+b_{1}y-e_{1}p\\ y' & =a_{2}x+b_{2}y-e_{2}p\\ -p' & =x_{0}x+y_{0}y-p_{0}p \end{align} \left|{\scriptstyle \begin{align}a_{1}^{2}+b_{1}^{2}-e_{1}^{2} & =1\\ a_{2}^{2}+b_{2}^{2}-e_{2}^{2} & =1\\ -x_{0}^{2}-y_{0}^{2}+p_{0}^{2} & =1\\ a_{2}x_{0}+b_{2}y_{0}-e_{2}p_{0} & =0\\ a_{1}x_{0}+b_{1}y_{0}-e_{1}p_{0} & =0\\ a_{1}a_{2}+b_{1}b_{2}-e_{1}e_{2} & =0\\ \\ a_{1}^{2}+a_{2}^{2}-x_{0}^{2} & =1\\ b_{1}^{2}+b_{2}^{2}-y_{0}^{2} & =1\\ -e_{1}^{2}-e_{2}^{2}+p_{0}^{2} & =1\\ b_{1}e_{1}+b_{2}e_{2}-y_{0}p_{0} & =0\\ a_{1}e_{1}+a_{2}e_{2}-x_{0}p_{0} & =0\\ a_{1}b_{1}+a_{2}b_{2}-x_{0}y_{0} & =0 \end{align} }\right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==={{anchor|Woods2}} Woods (1901-05) – Beltrami and Weierstrass coordinates === In (1901/02) [[w:Frederick S. Woods]] defined the following invariant quadratic form and its [[w:projective transformation]] in terms of Beltrami coordinates (he pointed out that this can be connected to hyperbolic geometry by setting <math>k=\sqrt{-1}R</math> with ''R'' as real quantity):<ref group=M>Woods (1901/02), p. 98, 104</ref> :<math>\begin{matrix}k^{2}\left(u^{2}+v^{2}+w^{2}\right)+1=0\\ \hline \begin{align}u' & =\frac{\alpha_{1}u+\alpha_{2}v+\alpha_{3}w+\alpha_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ v' & =\frac{\beta_{1}u+\beta_{2}v+\beta_{3}w+\beta_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}}\\ w' & =\frac{\gamma_{1}u+\gamma_{2}v+\gamma_{3}w+\gamma_{4}}{\delta_{1}u+\delta_{2}v+\delta_{3}w+\delta_{4}} \end{align} \left|\begin{align}k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right)+\delta_{i}^{2} & =k^{2}\\ (i=1,2,3)\\ k^{2}\left(\alpha_{4}^{2}+\beta_{4}^{2}+\gamma_{4}^{2}\right)+\delta_{4}^{2} & =1\\ \alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}+\delta_{i}\delta_{h} & =0\\ (i,h=1,2,3,4;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1b}}) ''(n=3)'' with ''k''²=-1.</p> Alternatively, Woods (1903, published 1905) – citing Killing (1885) – used the invariant quadratic form in terms of Weierstrass coordinates and its transformation (with <math>k=\sqrt{-1}k</math> for hyperbolic space):<ref group=M>Woods (1903/05), pp. 45–46; p. 48)</ref> :<math>\begin{matrix}x_{0}^{2}+k^{2}\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\right)=1\\ ds^{2}=\frac{1}{k^{2}}dx_{0}^{2}+dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}\\ \hline \begin{align}x_{1}^{\prime} & =\alpha_{1}x_{1}+\alpha_{2}x_{2}+\alpha_{3}x_{3}+\alpha_{0}x_{0}\\ x_{2}^{\prime} & =\beta_{1}x_{1}+\beta_{2}x_{2}+\beta_{3}x_{3}+\beta_{0}x_{0}\\ x_{3}^{\prime} & =\gamma_{1}x_{1}+\gamma_{2}x_{2}+\gamma_{3}x_{3}+\gamma_{0}x_{0}\\ x_{0}^{\prime} & =\delta_{1}x_{1}+\delta_{2}x_{2}+\delta_{3}x_{3}+\delta_{0}x_{0} \end{align} \left|\begin{align}\delta_{0}^{2}+k^{2}\left(\alpha_{0}^{2}+\beta_{0}^{2}+\gamma_{0}^{2}\right) & =1\\ \delta_{i}^{2}+k^{2}\left(\alpha_{i}^{2}+\beta_{i}^{2}+\gamma_{i}^{2}\right) & =k^{2}\\ (i=1,2,3)\\ \delta_{i}\delta_{h}+k^{2}\left(\alpha_{i}\alpha_{h}+\beta_{i}\beta_{h}+\gamma_{i}\gamma_{h}\right) & =0\\ (i,h=0,1,2,3;\ i\ne h) \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=3)'' with ''k''²=-1.</p> ==={{anchor|Liebmann}} Liebmann (1904–05) – Weierstrass coordinates=== [[w:Heinrich Liebmann]] (1904/05) – citing Killing (1885), Gérard (1892), Hausdorff (1899) – used the invariant quadratic form and its Lorentz transformation equivalent to ({{equationNote|1a}}) ''(n=2)''<ref group=M>Liebmann (1904/05), p. 168; pp. 175–176</ref> :<math>\begin{matrix}p^{\prime2}-x^{\prime2}-y^{\prime2}=1\\ \hline \begin{align}x_{1} & =\alpha_{11}x+\alpha_{12}y+\alpha_{13}p\\ y_{1} & =\alpha_{21}x+\alpha_{22}y+\alpha_{23}p\\ x_{1} & =\alpha_{31}x+\alpha_{32}y+\alpha_{33}p\\ \\ x & =\alpha_{11}x_{1}+\alpha_{21}y_{1}-\alpha_{31}p_{1}\\ y & =\alpha_{12}x_{1}+\alpha_{22}y_{1}-\alpha_{32}p_{1}\\ p & =-\alpha_{13}x_{1}-\alpha_{23}y_{1}+\alpha_{33}p_{1} \end{align} \left|\begin{align}\alpha_{33}^{2}-\alpha_{13}^{2}-\alpha_{23}^{2} & =1\\ -\alpha_{31}^{2}+\alpha_{11}^{2}+\alpha_{21}^{2} & =1\\ -\alpha_{32}^{2}+\alpha_{12}^{2}+\alpha_{22}^{2} & =1\\ \alpha_{31}\alpha_{32}-\alpha_{11}\alpha_{12}-\alpha_{21}\alpha_{22} & =0\\ \alpha_{32}\alpha_{33}-\alpha_{12}\alpha_{13}-\alpha_{22}\alpha_{23} & =0\\ \alpha_{33}\alpha_{31}-\alpha_{23}\alpha_{11}-\alpha_{23}\alpha_{21} & =0 \end{align} \right. \end{matrix}</math> <p style="background-color:Beige;border:1px solid black">This is equivalent to Lorentz transformation ({{equationNote|1a}}) ''(n=2)''.</p> ==References== ===Historical mathematical sources=== {{reflist|3|group=M}} *{{#section:History of Topics in Special Relativity/mathsource|apo}} *{{#section:History of Topics in Special Relativity/mathsource|bour56att}} *{{#section:History of Topics in Special Relativity/mathsource|chal82sec}} *{{#section:History of Topics in Special Relativity/mathsource|chas29}} *{{#section:History of Topics in Special Relativity/mathsource|cox81hom}} *{{#section:History of Topics in Special Relativity/mathsource|cox82hom}} *{{#section:History of Topics in Special Relativity/mathsource|fris76}} *{{#section:History of Topics in Special Relativity/mathsource|gau98}} *{{#section:History of Topics in Special Relativity/mathsource|gau18}} *{{#section:History of Topics in Special Relativity/mathsource|ger92}} *{{#section:History of Topics in Special Relativity/mathsource|gud30}} *{{#section:History of Topics in Special Relativity/mathsource|haus99}} *{{#section:History of Topics in Special Relativity/mathsource|hill82}} *{{#section:History of Topics in Special Relativity/mathsource|jac27}} *{{#section:History of Topics in Special Relativity/mathsource|jac32a}} *{{#section:History of Topics in Special Relativity/mathsource|jac32b}} *{{#section:History of Topics in Special Relativity/mathsource|jac33}} *{{#section:History of Topics in Special Relativity/mathsource|kil77}} *{{#section:History of Topics in Special Relativity/mathsource|kil79}} *{{#section:History of Topics in Special Relativity/mathsource|kil84}} *{{#section:History of Topics in Special Relativity/mathsource|kil85}} *{{#section:History of Topics in Special Relativity/mathsource|kil93}} *{{#section:History of Topics in Special Relativity/mathsource|kil97}} *{{#section:History of Topics in Special Relativity/mathsource|klei71}} *{{#section:History of Topics in Special Relativity/mathsource|klei73}} *{{#section:History of Topics in Special Relativity/mathsource|lag73}} *{{#section:History of Topics in Special Relativity/mathsource|leb37}} *{{#section:History of Topics in Special Relativity/mathsource|lie85}} *{{#section:History of Topics in Special Relativity/mathsource|lie90}} *{{#section:History of Topics in Special Relativity/mathsource|lie93}} *{{#section:History of Topics in Special Relativity/mathsource|lieb04}} *{{#section:History of Topics in Special Relativity/mathsource|lop}} *{{#section:History of Topics in Special Relativity/mathsource|pic82}} *{{#section:History of Topics in Special Relativity/mathsource|pic84a}} *{{#section:History of Topics in Special Relativity/mathsource|pic84b}} *{{#section:History of Topics in Special Relativity/mathsource|pic84c}} *{{#section:History of Topics in Special Relativity/mathsource|poin81a}} *{{#section:History of Topics in Special Relativity/mathsource|poin81b}} *{{#section:History of Topics in Special Relativity/mathsource|poin87}} *{{#section:History of Topics in Special Relativity/mathsource|sal62}} *{{#section:History of Topics in Special Relativity/mathsource|som63}} *{{#section:History of Topics in Special Relativity/mathsource|wedd47}} *{{#section:History of Topics in Special Relativity/mathsource|wern89}} *{{#section:History of Topics in Special Relativity/mathsource|woo01}} *{{#section:History of Topics in Special Relativity/mathsource|woo03}} ===Secondary sources=== {{reflist|3}} {{#section:History of Topics in Special Relativity/secsource|L1}} [[Category:Special Relativity]] [[Category:History of Physics]] o8sli64ia7a8aa5r3fkxzhgrjn5fh6d 0 267587 2416150 2408841 2022-08-18T13:16:51Z D.H 52339 /* Historical mathematical sources */ wikitext text/x-wiki ==Historical mathematical sources== <onlyinclude> *<section begin=apo />{{Citation|author=Apollonius of Perga|editor=Thomas Little Heath|date=1896|orig-year=~200 BC|title=Treatise on Conic Sections|publisher=Cambridge University Press|url=https://books.google.com/books?id=OcxZUnwuTWIC}}<section end=apo /> *<section begin=bach69 />{{Cite journal |author=Bachmann, 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Soc.|volume=4|pages=[https://archive.org/details/proceedingsofcam4188083camb/page/194 194]–196|url=https://archive.org/details/proceedingsofcam4188083camb}}<section end=cox82quab /> *<section begin=dar73 />{{Cite book|author=Darboux, G.|year=1873|title=Sur une classe remarquable de courbes et de surfaces algébriques et sur la théorie des imaginaires|url=https://archive.org/details/suruneclasserema00darb|publisher=Gauthier-Villars}}<section end=dar73 /> *<section begin=dar81cou />{{Cite journal|author=Darboux, G.|year=1881|title=[[s:fr:Sur les modes de transformation qui conservent les lignes de courbure|Sur les modes de transformation qui conservent les lignes de courbure]]|journal=Compt. Rend.|volume=92|pages=286-289}}<section end=dar81cou /> *<section begin=dar83cou />{{Cite journal|author=Darboux, G.|year=1883|title=Sur les surfaces dont la courbure totale est constante|journal=Compt. Rend.|volume=12|pages=848–850|url=https://gallica.bnf.fr/ark:/12148/bpt6k3053w/f848.image}}<section end=dar83cou /> *<section begin=dar87cou />{{Cite book|author=Darboux, G.|year=1887|title=Leçons sur la théorie générale des surfaces. Première partie|pages=[https://archive.org/details/leconsurlagendes01darbrich/page/n265 254]–256|publisher=Gauthier-Villars|location=Paris|url=https://archive.org/details/leconsurlagendes01darbrich}}<section end=dar87cou /> *<section begin=dar94cou />{{Cite book|author=Darboux, G.|year=1894|origyear=1891|title=Leçons sur la théorie générale des surfaces. Troisième partie|publisher=Gauthier-Villars|location=Paris|url=https://archive.org/details/leonssurlathorie03darb}} This third part of his lectures was initially published in three steps: [https://archive.org/details/in.ernet.dli.2015.377305 première fascicule] (1890), [https://gallica.bnf.fr/ark:/12148/bpt6k213929b/f585.image deuxième fascicule] (1891), and [https://gallica.bnf.fr/ark:/12148/bpt6k170701/f80.image troisième fascicule] (1895). The discussion of the Lie transform appears in the deuxième fascicule published in 1891.<section end=dar94cou /> *<section begin=eis05 />{{Cite journal|author=Eisenhart, L. P.|year=1905|journal=American Journal of Mathematics|volume=27|issue=2|title=Surfaces with the same Spherical Representation of their Lines of Curvature as Pseudospherical Surfaces|pages=113–172|doi=10.2307/2369977}}<section end=eis05 /> *<section begin=eli03 />{{Cite journal|author=Elliott, E.B.|year=1903|journal=Messenger of mathematics|volume=33|issue=1|title=On ternariants for the special cyclical subgroup of linear transformations|pages=108–112|url=https://books.google.com/books?id=cxUMAAAAYAAJ&pg=RA1-PA108}}<section end=eli03 /> *<section begin=esch74 />{{Cite journal|author=Escherich, G. von|year=1874|title=Die Geometrie auf den Flächen constanter negativer Krümmung|journal=Wiener Sitzungsberichte IIa|volume=69|pages=497–526|url=https://books.google.com/books?id=6e0DAAAAIAAJ&pg=PA497}}<section end=esch74 /> *<section begin=eul35 />{{Cite journal|author=Euler, L.|year=1740|origyear=1735|title=De motu planetarum et orbitarum determinatione|journal=Novi Commentarii Academiae Scientiarum Petropolitanae|volume=7|pages=67–85|url=http://eulerarchive.maa.org/pages/E037.html}}<section end=eul35 /> *<section begin=eul48a />{{Cite journal|author=Euler, L.|year=1748a|title=Memoire sur la plus grande equation des planetes|journal=Memoires de l'academie des sciences de Berlin|volume=2|pages=225–248|url=http://eulerarchive.maa.org/pages/E105.html}}<section end=eul48a /> *<section begin=eul48b />{{Cite book|author=Euler, L.|year=1748b|title=Introductio in analysin infinitorum, volume 1|location=Lausanne|url=http://eulerarchive.maa.org/pages/E101.html}}<section end=eul48b /> *<section begin=eul71 />{{Cite journal|author=Euler, L.|year=1771|title=Problema algebraicum ob affectiones prorsus singulares memorabile|journal=Novi Commentarii Academiae Scientiarum Petropolitanae|volume=15|pages=75–106|url=http://eulerarchive.maa.org/pages/E407.html}}<section end=eul71 /> *<section begin=eul77 />{{Cite journal|author=Euler, L.|year=1777|title=De proiectione geographica superficiei sphaericae|journal=Acta Academiae Scientiarum Imperialis Petropolitanae|pages=133–142|url=http://eulerarchive.maa.org/pages/E491.html}}<section end=eul77 /> *<section begin=fri91 />{{Cite journal |author=Fricke, R.|year=1891|title=Ueber eine besondere Classe discontinuirlicher Gruppen reeller linearer Substitutionen|journal=Mathematische Annalen|volume=38|pages=50–81|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002252910|doi=10.1007/bf01212693}}<section end=fri91 /> *<section begin=fri93 />{{Cite journal |author=Fricke, R.|year=1893|title=Ueber indefinite quadratische Formen mit drei und vier Veränderlichen|journal=Gött. Nachr.|pages=705–721|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002527030}}<section end=fri93 /> *<section begin=fri97 />{{Cite book|author=Fricke R. |author2=Klein, F.|year=1897|title=Vorlesungen über die Theorie der automorphen Functionen – Erster Band: Die gruppentheoretischen Grundlagen|location=Leipzig|publisher=Teubner|url=https://archive.org/details/vorlesungenber01fricuoft}}<section end=fri97 /> *<section begin=fris76 />{{Cite book|author=Frischauf, J.|year=1876|title=Elemente der absoluten Geometrie|location=Leipzig|publisher=Teubner|url=http://www.gutenberg.org/2/9/8/0/29806/}}<section end=fris76 /> *<section begin=fro77 />{{Cite journal |author=Frobenius, G.|year=1877 |title= Ueber lineare Substitutionen und bilineare Formen|journal=Journal für die Reine und Angewandte Mathematik |volume=84|pages=1–63|url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002156709}}<section end=fro77 /> *<section begin=gau98 />{{Cite book|author=Gauss, C.F |year=1801|origyear=1798|title=Disquisitiones arithmeticae|publisher=Fleischer |location=Leipzig|url=https://archive.org/details/disquisitionesa00gaus}}<section end=gau98 /> *<section begin=gau00 />{{Cite book|author=Gauss, C.F. |year=1863|origyear=1800|title=Werke – Zweiter Band|location=Göttingen|url=https://books.google.com/books?id=kVpEAQAAMAAJ&pg=PA311}}<section end=gau00 /> *<section begin=gau18 />{{Cite book|author=Gauss, C.F. |year=1818|title=Determinatio attractionis, quam in punctum quodvis positionis datae exerceret planet|location=Göttingen|doi=10.3931/e-rara-8856|publisher=Dietrich}}<section end=gau18 /> *<section begin=ger92 />{{Cite book|author=Gérard, L. |year=1892|title=Sur la géométrie non-Euclidienne|location=Paris|publisher=Gauthier-Villars|url=https://archive.org/details/surlagomtrienon01grgoog}}<section end=ger92 /> *<section begin=glai78 />{{Cite journal|author=Glaisher, JWL|year=1878|title=On a special form of determinant, and on ceetain functions of n variables analogous to the sine and cosine|journal=The Quarterly Journal of Pure and Applied Mathematics|volume=16|pages=15–33|url=https://books.google.com/books?id=eXo_AQAAIAAJ&pg=PA15}}<section end=glai78 /> *<section begin=gord68lin />{{Cite journal|author=Gordan, Paul|year=1868|title=Ueber eine das Hyperboloid betreffende Aufgabe|journal=Zeitschr. Math. Phys.|volume=13|pages=59-63|url=https://books.google.com/books?id=N3Fpk7rAFDYC&pg=PA59}}<section end=gord68lin /> *<section begin=gras44 />{{Cite book|author=Grassmann, H.|year=1844|title=Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik|location=Leipzig|publisher=Wiegand|url=https://books.google.com/books?id=bKgAAAAAMAAJ}}<section end=gras44 /> *<section begin=gud30 />{{Cite book|author=Gudermann, C.|year=1830|title=Grundriss der analytischen Sphärik|publisher=DüMont-Schauberg|location=Cologne|url=https://archive.org/details/grundrissderana01gudegoog}}<section end=gud30 /> *<section begin=guen80 />{{Cite book|author=Günther, S.|origyear=1880|year=1881|title=Die Lehre von den gewöhnlichen und verallgemeinerten Hyperbelfunktionen|location=Halle|url=https://archive.org/details/dielehrevondeng00fortgoog|publisher=L. Nebert}}<section end=guen80 /> *<section begin=ham44 />{{Cite journal |author=Hamilton, W.R.|year=1847|origyear=1844 |title=On Quaternions|journal=Proceedings of the Royal Irish Academy|volume=III|issue=48|pages= 1–16|url=https://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Quatern2/}} – Issue 48 for 1844-1845, full volume III published in 1847.<section end=ham44 /> *<section begin=haus99 />{{Cite journal|author=Hausdorff, F.|year=1899|title=Analytische Beiträge zur nichteuklidischen Geometrie|journal=Leipziger Math.-Phys. Berichte|volume=51|pages=161–214|url=https://hdl.handle.net/2027/hvd.32044092889328}}<section end=haus99 /> *<section begin=hask />{{Cite journal|author=Haskell, M.W.|year=1895|title=On the introduction of the notion of hyperbolic functions|journal=Bull. Amer. Math. Soc.|volume=1|issue=6|pages=155-159 |url=https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-1/issue-6/On-the-introduction-of-the-notion-of-hyperbolic-functions/bams/1183414352.full}}<section end=hask /> *<section begin=helm66 />{{Cite book|author=Helmholtz, H. von |year=1867|origyear=1866|title=Handbuch der physiologischen Optik|location=Leipzig|publisher=Voss|url=https://archive.org/details/handbuchderphysi00helm}}<section end=helm66 /> *<section begin=herm53 />{{Cite journal|author=Hermite, C.|origyear=1853|year=1854a|title=Sur la théorie des formes quadratiques ternaires indéfinies|journal=Journal für die Reine und Angewandte Mathematik|volume=47|pages=307–312|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002148455}}<section end=herm53 /> *<section begin=herm54 />{{Cite journal|author=Hermite, C.|year=1854b|title=Remarques sur un memoire de M. Cayley relatif aux determinants gauches|journal=The Cambridge and Dublin Mathematical Journal|volume=9|pages=63–67|url=https://books.google.com/books?id=7R5bAAAAYAAJ&pg=PA63}}<section end=herm54 /> *<section begin=hill82 />{{Cite journal |author=Hill, G. W.|year=1882|title=On Gauss's Method of Computing Secular Perturbations|journal=Astronomical Papers of the American Ephemeris|volume=I |pages=[https://archive.org/details/astronomicalpap00newcgoog/page/n347 315]–361|url=https://archive.org/details/astronomicalpap00newcgoog}}<section end=hill82 /> *<section begin=jac27 />{{Cite journal |author=Jacobi, C.G.J.|year=1827|title=De singulari quadam duplicis Integeralis transformatione|journal=Journal für die Reine und Angewandte Mathematik|volume=2|pages=234–242|url=http://www.digizeitschriften.de/dms/resolveppn/?PID=PPN243919689_0002%7Clog27}}<section end=jac27 /> *<section begin=jac32a />{{Cite journal |author=Jacobi, C.G.J.|year=1832a|title=De transformatione integralis duplicis indefinit...|journal=Journal für die Reine und Angewandte Mathematik|volume=8|pages=253–279|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002138433}}<section end=jac32a /> *<section begin=jac32b />{{Cite journal |author=Jacobi, C.G.J.|year=1832b|title=De transformatione integralis duplicis indefinit...|journal=Journal für die Reine und Angewandte Mathematik|volume=8|pages=321–357|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002138506}}<section end=jac32b /> *<section begin=jac33 />{{Cite journal |author=Jacobi, C.G.J.|year=1834|origyear=1833|title=De binis quibuslibet functionibus homogeneis secundi ordinis per substitutiones lineares...|journal=Journal für die Reine und Angewandte Mathematik|volume=12|pages=1–69|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002139871}}. All papers (1827, 1832a, 1832b, 1834) were reprinted with corrections in [https://archive.org/details/gesammeltewerke01weiegoog Jacobi's collected papers vol. 3].<section end=jac33 /> *<section begin=kep09 />{{Cite book|author=Kepler, J.|year=1609|title=Astronomia nova|location=Heidelberg|publisher=Voegelin|doi=10.3931/e-rara-558}}. Reprinted with corrections and comments in [http://publikationen.badw.de/002334739 Kepler's collected papers vol. 3].<section end=kep09 /> *<section begin=kil77 />{{Cite journal|author=Killing, W.|year=1878|origyear=1877|title=Ueber zwei Raumformen mit constanter positiver Krümmung|journal=Journal für die Reine und Angewandte Mathematik|volume=86|pages=72–83|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002157187}}<section end=kil77 /> *<section begin=kil79 />{{Cite journal|author=Killing, W.|year=1880|origyear=1879|title=Die Rechnung in den Nicht-Euklidischen Raumformen|journal=Journal für die Reine und Angewandte Mathematik|volume=89|pages=265–287|url=http://www.digizeitschriften.de/dms/resolveppn/?PID=PPN243919689_0089%7Clog27}}<section end=kil79 /> *<section begin=kil84 />{{Cite journal|author=Killing, W.|year=1885|origyear=1884|title=Die Mechanik in den Nicht-Euklidischen Raumformen|journal=Journal für die Reine und Angewandte Mathematik|volume=98|pages=1–48|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002159392}}<section end=kil84 /> *<section begin=kil85 />{{Cite book|author=Killing, W.|year=1885|title=Die nicht-euklidischen Raumformen|location=Leipzig|publisher=Teubner|url=https://archive.org/details/dienichteuklidis00killuoft}}<section end=kil85 /> *<section begin=kil93 />{{Cite book|author=Killing, W.|year=1893|title=Einführung in die Grundlagen der Geometrie I|location=Paderborn|publisher=Schöningh|url=https://archive.org/details/einfhrungindieg01killgoog}}<section end=kil93 /> *<section begin=kil97 />{{Cite book|author=Killing, W.|year=1898|origyear=1897|title=Einführung in die Grundlagen der Geometrie II|location=Paderborn|publisher=Schöningh|url=https://archive.org/details/einfhrungindieg02killgoog}}<section end=kil97 /> *<section begin=klei71 />{{Cite journal|author=Klein, F.|year=1871|title=Ueber die sogenannte Nicht-Euklidische Geometrie|journal=Mathematische Annalen|volume=4|pages=573–625|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002241692|doi=10.1007/bf02100583}}<section end=klei71 /> *<section begin=klei72a />{{Cite journal|author=Klein, F.|year=1872|origyear=1871|title=Ueber Liniengeometrie und metrische Geometrie|journal=Mathematische Annalen|volume=5|pages=257–277|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002241854|doi=10.1007/bf01444841}}<section end=klei72a /> *<section begin=klei72b />{{Cite book|author=Klein, F.|year=1872|title=Vergleichende Betrachtungen über neuere geometrische Forschungen|location=Erlangen|publisher=Deichert|url=https://quod.lib.umich.edu/u/umhistmath/abn7632.0001.001?view=toc}}<section end=klei72b /> *<section begin=klei73 />{{Cite journal|author=Klein, F.|year=1873|title=Ueber die sogenannte Nicht-Euklidische Geometrie|journal=Mathematische Annalen|volume=6|pages=112–145|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002242222|doi=10.1007/bf01443189}}<section end=klei73 /> *<section begin=klei75 />{{Cite journal|author=Klein, F.|year=1875|title=Ueber binäre Formen mit linearen Transformationen in sich selbst|journal=Mathematische Annalen|volume=9|pages=183–208|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002243237|doi=10.1007/bf01443373}}<section end=klei75 /> *<section begin=klei79 />{{Cite journal|author=Klein, F.|year=1879|origyear=1878|title=Ueber die Transformation der elliptischen Functionen und die Auflösung der Gleichungen fünften Grades|journal=Mathematische Annalen|volume=14|pages=111–172|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002244616|doi=10.1007/bf02297507}}<section end=klei79 /> *<section begin=klei82 />{{Cite journal|author=Klein, F.|year=1883|origyear=1882|title=Neue Beiträge zur Riemann'schen Functionentheorie|journal=Mathematische Annalen|volume=21|pages=141–218|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002247232|doi=10.1007/bf01442920}}<section end=klei82 /> *<section begin=klei84 />{{Cite book|author=Klein, F.|year=1884|title= Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade|location=Leipzig|publisher=Teubner|url=https://archive.org/details/cu31924059413439}}; English translation: [https://archive.org/details/cu31924059413439 Lectures on the ikosahedron and the solution of equations of the fifth degree] (1888)<section end=klei84 /> *<section begin=klei90a />{{Cite book|author=Klein, F. |author2=Fricke, R.|year=1890a|title=Vorlesungen über die Theorie der elliptische Modulfunctionen|location=Leipzig|publisher=Teubner|url=https://archive.org/details/vorlesungenuberd028637mbp}}<section end=klei90a /> *<section begin=klei90b />{{Cite journal|author=Klein, F.|year=1890b|title=Zur Nicht-Euklidischen Geometrie|journal=Mathematische Annalen|volume=37|pages=544–572|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002252856|doi=10.1007/bf01724772}}<section end=klei90b /> *<section begin=klei93a />{{Cite book|author=Klein, F.|editor=Schilling, Fr.|year=1893a|origyear=1890|title=Nicht-Euklidische Geometrie I, Vorlesung gehalten während des Wintersemesters 1889–90|location=Göttingen|url=https://archive.org/details/nichteuklidische01klei|publisher=Göttingen}}<section end=klei93a /> *<section begin=klei93b />{{Cite book|author=Klein, F.|editor=Schilling, Fr.|year=1893b|origyear=1890|title=Nicht-Euklidische Geometrie II, Vorlesung gehalten während des Sommersemesters 1890|location=Göttingen|url=https://archive.org/details/nichteuklidische02klei|publisher=Göttingen}}<section end=klei93b /> *<section begin=klei93c />{{Cite book|author=Klein, F.|editor=Schilling, Fr.|year=1893c|title=Einleitung in die höhere Geometrie I, Vorlesung gehalten während des Wintersemesters 1892–93|location=Göttingen|url=https://archive.org/details/eininhohere01kleirich}}<section end=klei93c /> *<section begin=klei93d />{{Cite book|author=Klein, F.|editor=Schilling, Fr.|year=1893d|title=Einleitung in die höhere Geometrie II, Vorlesung gehalten während des Sommersemesters 1893|location=Göttingen|url=https://archive.org/details/eininhohere02kleirich}}<section end=klei93d /> *<section begin=klei96 />{{Cite book|author=Klein, F.|year=1897|origyear=1896|title=The Mathematical Theory of the Top|publisher=Scribner|location=New York|url=https://archive.org/details/mathematicaltheo00kleiuoft}}<section end=klei96 /> *<section begin=lag70 />{{Cite journal|author=Lagrange, J. L.|year=1771|origyear=1770|title=Sur le Problème de Kepler|journal=Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin|pages=204–233|url=https://books.google.com/books?id=hmpUAAAAYAAJ&pg=PA204}}<section end=lag70 /> *<section begin=lag73 />{{Cite journal|author=Lagrange, J. L.|year=1775|origyear=1773|title=Recherches d'arithmetique|journal=Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Berlin|pages=265–314|url=https://books.google.com/books?id=bi5RAAAAYAAJ&pg=PA265}}<section end=lag73 /> *<section begin=lagu80 />{{Cite journal|author=Laguerre, Edmond|year=1880|title=Sur la géométrie de direction|journal=Bulletin de la Société Mathématique de France|volume=8|pages=196-208|doi=10.24033/bsmf.207}}<section end=lagu80 /> *<section begin=lagu82 />{{Cite journal|author=Laguerre, Edmond|year=1882|title=[[s:fr:Transformations par semi-droites réciproques|Transformations par semi-droites réciproques]]|journal=[[Nouvelles Annales de Mathématiques|Nouvelles annales de mathématiques]]|volume=1|pages=542–556}}<section end=lagu82 /> *<section begin=lais74a />{{Cite book|author=Laisant, C. A. |year=1874a|title=Essai sur les fonctions hyperboliques|url=https://books.google.com/books?id=xAzOAAAAMAAJ |publisher=Gauthier-Villars}}<section end=lais74a /> *<section begin=lais74b />{{Cite book|author=Laisant, C. A. |year=1874b|origyear=1854|chapter=Étude de l'hyperbole|title=French translation of Giusto Bellavitis (1854), Exposition de la méthode des equipollences|url=https://books.google.com/books?id=ptPUAAAAMAAJ |publisher=Gauthier-Villars|pages=133–136}}<section end=lais74b /> *<section begin=lam67 />{{Cite journal|author=Lambert, J. H. |year=1768|origyear=1767|journal=Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin|volume=17|title=Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques|pages=265–322|url=https://books.google.com/books?id=AmpUAAAAYAAJ&pg=PA265}}<section end=lam67 /> *<section begin=lam70 />{{Cite journal|author=Lambert, J. H. |year=1770|journal=Histoire de l'Académie Royale des Sciences et des Belles-Lettres de Berlin|volume=24|title=Observations trigonométriques|pages=327–354|url=https://books.google.com/books?id=LG1UAAAAYAAJ&pg=PA327}}<section end=lam70 /> *<section begin=leb37 />{{Cite journal |author=Lebesgue, V.A. |year=1837|title=Thèses de Mécanique et d'Astronomie|journal=Journal de Mathématiques Pures et Appliquées|series=1|volume=2|pages=1–69|url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1837_1_2_A30_0.pdf}}<section end=leb37 /> *<section begin=lie71a />{{Cite journal|author=Lie, S.|year=1871a|title=Ueber diejenige Theorie eines Raumes mit beliebig vielen Dimensionen, die der Krümmungs-Theorie des gewöhnlichen Raumes entspricht |journal=Göttinger Nachrichten|pages=191–209|url=https://books.google.com/books?id=mH41bQAAMAAJ&pg=PA191}}<section end=lie71a /> *<section begin=lie71b />{{Cite journal|author=Lie, S.|year=1872|origyear=1871|title=Ueber Complexe, insbesondere Linien- und Kugel-Complexe, mit Anwendung auf die Theorie partieller Differentialgleichungen |journal=Mathematische Annalen|volume=5|pages=145–256|url=https://books.google.com/books?id=TdZSAAAAcAAJ&pg=PA145|doi=10.1007/bf01446331}} English translation by David Delphenich: [http://neo-classical-physics.info/uploads/3/0/6/5/3065888/lie_-_line_and_sphere_complexes.pdf On complexes – in particular, line and sphere complexes – with applications to the theory of partial differential equations]<section end=lie71b /> *<section begin=lie79a />{{Cite journal|author=Lie, S.|year=1880|origyear=1879|journal=Archiv for Mathematik|volume=4|issue=4|title=Über Flächen, deren Krümmungsradien durch eine Relation verknüpft sind|pages=507–512}} Reprinted in [https://archive.org/details/gesammabhand03lierich Lie's collected papers, Vol. 3, pp. 387–391].<section end=lie79a /> *<section begin=lie79b />{{Cite journal|author=Lie, S.|year=1881|origyear=1879|journal=Fortschritte der Matheamtik|volume=11|title=Selbstanzeige: Über Flächen, deren Krümmungsradien durch eine Relation verknüpft sind|pages=529–531}} Reprinted in [https://archive.org/details/gesammabhand03lierich Lie's collected papers, Vol. 3, pp. 392–393].<section end=lie79b /> *<section begin=lie80 />{{Cite journal|author=Lie, S.|year=1881|origyear=1880|journal=Archiv for Mathematik|volume=6|issue=1|title=Diskussion der Differentialgleichung d2dz/dxdy|pages=112–124}} Reprinted in [https://archive.org/details/gesammabhand03lierich Lie's collected papers, Vol. 3, pp. 469–478].<section end=lie80 /> *<section begin=lie83 />{{Cite journal|author=Lie, S.|year=1884|origyear=1883|journal=Christ. Forh.|title=Untersuchungen über Differentialgleichungen IV}} Reprinted in [https://archive.org/details/gesammabhand03lierich Lie's collected papers, Vol. 3, pp. 556–560].<section end=lie83 /> *<section begin=lie84 />{{Cite journal|author=Lie, S.|year=1884|title=Ueber Differentialinvarianten|journal=Math. Ann.|volume=24|pages=537–578|url=http://resolver.sub.uni-goettingen.de/purl?PPN235181684_0024}}<section end=lie84 /> *<section begin=lie85 />{{Cite journal|author=Lie, Sophus|year=1886|origyear=1885|title=Untersuchungen über Transformationsgruppen II|journal=Archiv for Mathematik og Naturvidenskab|pages=[https://archive.org/details/archivformathema1018841886oslo/page/353 353]–413|volume=10|issue=4|url=https://archive.org/details/archivformathema1018841886oslo}}<section end=lie85 /> *<section begin=lie90 />{{Cite journal|author=Lie, S.|year=1890|title=Ueber die Grundlagen der Geometrie I|journal=Leipz. Ber.|pages=284–321|url=https://archive.org/details/bub_gb_3szAAAAMAAJ}}<section end=lie90 /> *<section begin=lie93 />{{Cite book|author=Lie, S. |year=1893|title=Theorie der Transformationsgruppen (Dritter und letzter Abschnitt, unter Mitwirkung von Prof. Dr. Friedrich Engel)|location=Leipzig|url=https://archive.org/details/theotransformation03liesrich|publisher=Leipzig, B.G. Teubner}}<section end=lie93 /> *<section begin=lieb04 />{{Cite book|author=Liebmann, H. |year=1905|origyear=1904|title=Nichteuklidische Geometrie|location=Leipzig|publisher=Göschen|url=https://archive.org/details/nichteuklidisch00liebgoog}}<section end=lieb04 /> *<section begin=lind90 />{{Cite book|author=Lindemann, F. |author2=Clebsch, A.|year=1891|origyear=1890|title=Vorlesungen über Geometrie von Clebsch II|location=Leipzig|publisher=Teubner|url=https://archive.org/details/vorlesungenber21cleb}}<section end=lind90 /> *<section begin=liou50 />{{Cite journal|author=Liouville, Joseph|year=1850|title=Théorème sur l'équation dx²+dy²+dz²=λ(dα²+dβ²+dγ²) |journal=Journal de Mathématiques Pures et Appliquées|volume=15|pages=103|url=https://books.google.com/books?id=wlZOAAAAMAAJ&pg=PA103}}<section end=liou50 /> *<section begin=lip86 />{{Cite book |author=Lipschitz, R.|year=1886|origyear=1885|title=Untersuchungen ueber die Summen von Quadraten|location=Bonn|publisher=Cohen|url=https://archive.org/details/untersuchungenu00lipsgoog}}<section end=lip86 /> *<section begin=lop />{{Cite book |author=l'Hôpital, G.|year=1707|title=Traité analytique des sections coniques|location=Paris|url=https://books.google.com/books?id=J-RJAAAAMAAJ}}<section end=lop /> *<section begin=mac92 />{{Cite journal |author=Macfarlane, A.|year=1892|title=The Imaginary of Algebra|journal=Proceedings of the American Association for the Advancement of Science|volume=41|pages=[https://archive.org/details/proceedingsamer26sciegoog/page/n150 33]–55|url=https://archive.org/details/proceedingsamer26sciegoog}}<section end=mac92 /> *<section begin=mac93 />{{Cite book |author=Macfarlane, A.|year=1893|title=The Fundamental Theorems of Analysis Generalized for Space|publisher=Cushing|location=Boston|url=https://archive.org/stream/cu31924001506769#page/n89/mode/2up}}<section end=mac93 /> *<section begin=mac94 />{{Cite book |author=Macfarlane, A.|year=1894|title=The principles of elliptic and hyperbolic analysis|location=Boston|url=https://archive.org/details/principlesofelli00macfrich|publisher=Boston, J.S. Cushing & co., printers}}<section end=mac94 /> *<section begin=mac00 />{{Cite journal |author=Macfarlane, A.|year=1902|origyear=1900|title=Hyperbolic quaternions|journal=Proceedings of the Royal Society of Edinburgh|volume=23|pages=[https://archive.org/details/proceedings2318991901roya/page/169 169]–180|url=https://archive.org/details/proceedings2318991901roya|doi=10.1017/s0370164600010385}}<section end=mac00 /> *<section begin=pic82 />{{Cite journal|author=Picard, E.|year=1882|title=Sur une classe de groupes discontinus de substitutions linéaires et sur les fonctions de deux variables indépendantes restant invariables par ces substitutions|journal=Acta Mathematica|volume=1|pages=297–320|url=https://projecteuclid.org/euclid.acta/1485801068|doi=10.1007/BF02592137|doi-access=free}}<section end=pic82 /> *<section begin=pic84a />{{Cite journal|author=Picard, E.|year=1884a|title=Mémoire sur les formes quadratiques binaires indéfinies à indéterminées conjuguées|journal=Annales Scientifiques de l'École Normale Supérieure|volume=1|pages=9–54|url=http://www.numdam.org/item/ASENS_1884_3_1__9_0/|doi=10.24033/asens.231|doi-access=free}}<section end=pic84a /> *<section begin=pic84b />{{Cite journal|author=Picard, E.|year=1884b|title=Sur certaines substitutions lineaires|journal=Comptes Rendus|volume=98|pages=416–417|url=https://gallica.bnf.fr/ark:/12148/bpt6k30546/f409.image}}<section end=pic84b /> *<section begin=pic84c />{{Cite journal|author=Picard, E.|year=1884c|title=Sur les formes quadratiques ternaires indéfinies á indéterminées conjuguées et sur les fonctions hyperfuchsiennes correspondantes|journal=Acta Mathematica|volume=5|pages=121–182|url=https://projecteuclid.org/euclid.acta/1485803353|doi=10.1007/BF02421555|doi-access=free}}<section end=pic84c /> *<section begin=plue65lin />{{Cite journal|author=Plücker, Julius|year=1865|title=On a new geometry of space|journal=Phil. Trans. R. Soc.|volume=155|pages=725–791|url=https://books.google.com/books?id=xVNFAAAAcAAJ&pg=PA725|doi=10.1098/rstl.1865.0017}}<section end=plue65lin /> *<section begin=poc91 />{{Cite book|author=Pockels, F. |year=1891|title=Über die partielle Differentialgleichung Δu+k²u=0 und deren Auftreten in der mathematischen Physik |location=Leipzig|publisher=Teubner|url=https://archive.org/details/berdiepartiell02pockuoft}}<section end=poc91 /> *<section begin=poin81a />{{Cite journal|author=Poincaré, H.|year=1881a|title=Sur les applications de la géométrie non-euclidienne à la théorie des formes quadratiques|journal=Association Française Pour l'Avancement des Sciences|volume=10|pages=132–138|url=http://aline.clsh.univ-nancy2.fr/chp/hp-pdf/hp1881af.pdf}}<section end=poin81a /> *<section begin=poin81b />{{Cite journal|author=Poincaré, H.|year=1881b|title=Sur les fonctions fuchsiennes|journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences|volume=92|pages=333–335|url=http://aline.clsh.univ-nancy2.fr/chp/hp-pdf/hp1881cr.pdf}}<section end=poin81b /> *<section begin=poin83 />{{Cite journal|author=Poincaré, H.|year=1883|title=Mémoire sur les groupes kleinéens|journal=Acta Mathematica|volume=3|pages=49–92|url=http://aline.clsh.univ-nancy2.fr/chp/hp-pdf/hp1883ama.pdf|doi=10.1007/bf02422441}}<section end=poin83 /> *<section begin=poin86 />{{Cite journal|author=Poincaré, H.|year=1886|title=Sur les fonctions fuchsiennes et les formes quadratiques ternaires indéfinies|journal=Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences|volume=102|pages=735–737|url=http://aline.clsh.univ-nancy2.fr/chp/hp-pdf/hp1886crb.pdf}}<section end=poin86 /> *<section begin=poin87 />{{Cite journal|author=Poincaré, H.|year=1887|title=Sur les hypothèses fondamentales de la géométrie|journal=Bulletin de la Société Mathématique de France|volume=15|pages=203–216|url=http://aline.clsh.univ-nancy2.fr/chp/hp-pdf/hp1887sm.pdf}}<section end=poin87 /> *<section begin=reyn />{{Cite book |author=Reynaud, A.|year=1819|title=Traité d'application de l'algèbre à la géométrie, et de trigonométrie|publisher=Courcier|location=Paris|url=https://books.google.com/books?id=Qt7qbSwu7iMC}}<section end=reyn /> *<section begin=rib70 />{{Cite journal|author=Ribaucour, Albert|year=1870|title=[[s:fr:Sur la déformation des surfaces|Sur la déformation des surfaces]]|journal=Comptes Rendus|volume=70|pages=330–333}}<section end=rib70 /> *<section begin=ric57 />{{Cite book|author=Riccati, V.|year=1757|title=Opusculorum ad res physicas, et mathematicas pertinentium|publisher=Aquinatis|url=http://resolver.sub.uni-goettingen.de/purl?PPN61578593X}}<section end=ric57 /> *<section begin=rod40 />{{Cite journal |author=Rodrigues, O.|year=1840 |journal=Journal de Mathématiques Pures et Appliquées |title= Des lois géométriques qui régissent les déplacements d'un système solide dans l'espace|volume=5|pages=380–440|url=http://portail.mathdoc.fr/JMPA/afficher_notice.php?id=JMPA_1840_1_5_A39_0}}<section end=rod40 /> *<section begin=sal62 />{{Cite book|author=Salmon, G. |year=1862|title=A Treatise on the Analytic Geometry of Three Dimensions|location=Dublin|url=https://books.google.com/books?id=oqALAAAAYAAJ}}<section end=sal62 /> *<section begin=schef99 />{{Cite journal|author=Scheffers, G. |year=1899|journal=Leipziger Math.-Phys. Berichte |title=Synthetische Bestimmung aller Berührungstransformationen der Kreise in der Ebene|volume=51|pages=145–160|url=https://hdl.handle.net/2027/hvd.32044092889328}}<section end=schef99 /> *<section begin=schu85 />{{Cite journal|author=Schur, F.|origyear=1885|year=1886|title=Ueber die Deformation der Räume constanten Riemann'schen Krümmungsmaasses|journal=Mathematische Annalen|volume=27|pages=163–176|url=http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002249243}}<section end=schu85 /> *<section begin=schu00 />{{Cite journal|author=Schur, F.|origyear=1900|year=1902|title=Ueber die Grundlagen der Geometrie|journal=Mathematische Annalen|volume=55|pages=265–292|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002258560|doi=10.1007/bf01444974}}<section end=schu00 /> *<section begin=schu09 />{{Cite book |author=Schur, F.|year=1909|title=Grundlagen der Geometrie|location=Leipzig|publisher=Teubner|url=https://archive.org/details/grundlagendergeo00schuuoft}}<section end=schu09 /> *<section begin=sel73 />{{Cite journal|author=Selling, Eduard|origyear=1873|year=1874|title=Ueber die binären und ternären quadratischen Formen|journal=Journal für die Reine und Angewandte Mathematik|volume=77|pages=143–229|url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002155559}}<section end=sel73 /> *<section begin=smi00 />{{Cite journal|author=Smith, Percey F.|year=1900|journal=Annals of Mathematics|title=On a Transformation of Laguerre|volume=1|issue=1/4|pages=153–172|url=https://archive.org/details/jstor-1967282|doi=10.2307/1967282}}<section end=smi00 /> *<section begin=som63 />{{Cite journal |author=Somov, I. |year=1863|title=Mémoire sur un cas particulier de l'homographie|journal=Mémoires de l'Acad. Imp. Des Sciences de St.-Petersbourg|series=7|volume=5|issue=9|url=https://books.google.com/books?id=PphFAAAAcAAJ&printsec=frontcover}}<section end=som63 /> *<section begin=spei />{{Citation|author=Speidell, E.|date=1688|title=Logarithmotechnia|location=London|url=https://books.google.com/books?id=kMdlAAAAcAAJ}}<section end=spei /> *<section begin=ste83 />{{Cite journal|author=Stephanos, C.|year=1883|title=Sur la théorie des quaternions|journal=Mathematische Annalen|volume=7|issue=4|pages=589–592|doi=10.1007/bf01443267 |url= http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002247798}}<section end=ste83 /> *<section begin=tau26 />{{Cite book|author=Taurinus, Franz Adolph|title=Geometriae prima elementa. Recensuit et novas observationes adjecit|year=1826|location=Köln|publisher=Bachem|url=http://nbn-resolving.de/urn:nbn:de:bvb:355-ubr16230-7}}; Partial German translation: {{Cite book|author=Taurinus, Franz Adolph|chapter=Geometriae. Prima elementa. Recensuit et novas observationes adjecit|editor=Engel, F |editor2=Stäckel, P.|title=Die Theorie der Parallellinien von Euklid bis auf Gauss|year=1895|origyear=1826|pages=267–286|location=Leipzig|publisher=Teubner|chapter-url=https://archive.org/details/dietheoriederpar00stuoft}}<section end=tau26 /> *<section begin=vah01 />{{Cite journal |author=Vahlen, K.Th.|year=1902 |origyear=1901|title=Ueber Bewegungen und complexe Zahlen|journal= Mathematische Annalen|volume=55|pages=585–593|url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002258714|doi=10.1007/bf01450354}}<section end=vah01 /> *<section begin=vah05 />{{Cite book |author=Vahlen, K.Th.|year=1905|title=Abstrakte Geometrie|location=Leipzig|publisher=Teubner|url=https://archive.org/details/abstraktegeometr00vahluoft}}<section end=vah05 /> *<section begin=wed75 />{{Cite book|author=Wedekind, L.|year=1875|title=Beiträge zur geometrischen Interpretation binärer Formen|location=Erlangen|url=https://books.google.com/books?id=6f6GDCKl3ZYC}}<section end=wed75 /> *<section begin=wedd47 />{{Cite journal|author=Weddle, T.|year=1847|title=On conjugate hyperboloids|journal=The Cambridge and Dublin Mathematical Journal|volume=2|pages=274–281|url=https://books.google.com/books?id=zC46AAAAMAAJ&pg=PA274}}<section end=wedd47 /> *<section begin=wern89 />{{cite book|author=Werner, H.|title=Bestimmung der grössten Untergruppen derjenigen projectiven Gruppe, welche eine Gleichung zweiten Grades in n Veränderlichen invariant lässt|year=1889|publisher=Teubner|location=Leipzig|url=https://archive.org/details/bestimmungdergr00werngoog}}<section end=wern89 /> *<section begin=wes99 />{{Cite journal|author=Wessel, C.|year=1799|origyear=1797|title=Om directionens analytiske betegning|journal=Royal Danish Academy of Sciences and Letters|volume=7|issue=4|pages=469–518|url=https://books.google.com/books?id=wmjNAAAAMAAJ&pg=PA469}}; French translation (1897): [http://gallica.bnf.fr/ark:/12148/bpt6k99681g/f1.image Essai sur la représentation analytique de la direction]<section end=wes99 /> *<section begin=whis />{{Citation|author=Whiston, W.|date=1716|orig-year=1710|title=Sir Isaac Newton's Mathematick Philosophy More Easily Demonstrated|location=London|url=https://books.google.com/books?id=t3hbAAAAQAAJ}}; See also the [https://archive.org/details/praelectionesph01whisgoog Latin original (1710)]<section end=whis /> *<section begin=whit98 />{{cite book|author=Whitehead, A.|title=A Treatise on Universal Algebra|year=1898|publisher=Cambridge University Press|url=https://archive.org/details/atreatiseonuniv00goog}}<section end=whit98 /> *<section begin=woo95 />{{Cite book|author=Woods, F. S.|year=1895|title=Ueber Pseudominimalflächen|publisher=Kästner|location=Göttingen|url=https://hdl.handle.net/2027/uc1.b2620512}}<section end=woo95 /> *<section begin=woo01 />{{Cite journal|author=Woods, F. S.|year=1901|title=Space of constant curvature|journal=The Annals of Mathematics|volume=3|issue=1/4|pages=71–112|url=https://www.jstor.org/stable/1967636}}<section end=woo01 /> *<section begin=woo03 />{{Cite journal|author=Woods, F. S.|year=1905|origyear=1903|title=Forms of non-Euclidean space|journal=The Boston Colloquium: Lectures on Mathematics for the Year 1903|pages=[https://archive.org/details/in.ernet.dli.2015.161268/page/n40 31]–74|url=https://archive.org/details/in.ernet.dli.2015.161268}}<section end=woo03 /></onlyinclude> ad3ch9a660z4wr9v9uhi9no8q0xn6wb 0 267737 2416306 2415574 2022-08-18T21:37:57Z Saltrabook 1417466 wikitext text/x-wiki ==Introduction == The Maritime Health Research and Education-NET MAHRE-Net is a non-profit network of researchers, seafarers and other workers, maritime students composed of four parts: # Research, based on standardized, health questionnaires, # Screening for T2 Diabetes mellitus and Hypertension in the fit-for-duty medical examinations # Health promotion program integrated with the T2DM & HTN screening program # Systematic Literature Reviews and Reviews of Systematic Reviews. The primary target study populations include maritime students, seafarers, fishermen, port workers, offshore workers, divers, and their social relations, and other industries. The aim is to provide a foundation for the evidence base for the identification of health risks to foster safe and healthy preventive strategies and policies within the UN Global Sustainable Goals.<ref>‘THE 17 GOALS | Sustainable Development. Accessed 1 May 2021. https://sdgs.un.org/goals</ref><ref>[[Maritime Health Research and Education-NET/Contribution to UNs 17 Sustainable Development Goals|Contribution to UNs 17 Sustainable Development Goals ]]</ref> The program research program includes the permanent monitoring of the main topics of the EU-Occupational Health strategy: # Mental health # Ergonomics # Screening for Hypertension and Diabetes Type 2, accurate diagnosis and prevention # Safety climate # COVID-19 and other infectious disease We will follow and support the young people from the maritime schools in their care in the cohort design strategies. The method is that we ask the classes of maritime (or other) students to fill out a standardized questionnaire in one of the four themes at the beginning of their studies on their mobile phones. The surveys in the maritime schools complete part of the diagnostics of a global mental health program at the schools and workplaces in the WHO health-promoting school-framework for improving the health and well being'''<ref> [https://learningportal.iiep.unesco.org/en/library/the-who-health-promoting-school-framework-for-improving-the-health-and-well-being-of?back_url=/en/library/search/occupational%20health%20research The WHO health-promoting school-framework for improving the health and well being]</ref>,<ref>https://pubmed.ncbi.nlm.nih.gov/24737131/</ref> <ref>https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-015-1360-y</ref>''' When they start their practice times at sea, they complete the same questions to identify the influence on their well-being on board. We measure how many of them leave the sea profession and we ask them how they think the profession can continue to be attractive to young people. It is intended to suggest and assist in the implementation of preventive measures based on the results. Later, we continue to ask them at some year intervals with the same questionnaires to assess whether the efforts have helped. We give the same questionnaires to the maritime students in other countries for comparison and learn from their proposals to get the best working conditions. Also, we ask what is needed of teaching to help the industry give them the best condition to stay safe in the job. Different cohort data sources like pre-employment medical health examination data can be tried out for feasibility and validity. A "Cohort" is defined in epidemiological science as a group of people who share the same characteristic, in this case, more or less the same birth years and we take several contacts to them over their life. Cohorts are also started with maritime workers through unions, other organizations, and shipping companies: ==[[/EU Consortium for Maritime Health Research and Education/]] == ==[[/Presentations pptx /|Presentations]] == ==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|International T2 Diabetes Mellitus and Hypertension Research Group]] == == [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] == ==[[/Students Research Projects|Education 2: Students Thesis Research]] == ==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The International Maritime Health Journal Club]]== ==[[/Standard Questionnaire Based studies/|Questionnaire Based studies]] == ==[[/Systematic Reviews/]] == ==[[/Systematic reviews of systematic reviews /]] == ==[[/Organisation / ]] == ==[[/Publications out and on the way/]] == ==[[/ BACKGROUND - open/| Literature background- open]] == == Objectives == # International prospective exposure and health risk cohort studies with maritime students and workers # All other industries are invited to promote the similar research and education goals # Using the standard protocols with different themes # Harmonise exposure and outcome information by using standard questionnaires # Objective and subjective assessments of workplace hazards exposures # Develop and validate Job-exposure Matrices # Systematic reviews and pooled studies from the cohort rounds # Cohort-Linkage to pre-entry and follow-up health exams and other health registers # Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension] # International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing VCT@WORK'''] # Make training materials based on the cohort study and the clinical study outcomes and other scientific sources # OHS training to maritime doctors, seafarers, fishermen, students, and others # Integrate research methodology in the supervision of student’s thesis work # Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data # Keep the Excel data file copies safely (producing country and supervisor) # Disseminate the knowledge in publications and organize seminars/webinars/symposia # There is a close relation to the [https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL Maritime Health Portal] ==Contributions to the Industry == # The maritime doctors and other maritime health professionals receive updated knowledge on the health risks at sea for the specific jobs and work areas # [https://stami.no/slik-skal-stami-sikre-at-kunnskapen-nar-virksomhetene/ Systematic dissemination of knowledge in work environment-e.g. STAMI Specific project] # Job-Exposure Matrices will support the needed evidence to the health examinations according to the requirements in the ILO/WHO Guidelines # The companies receive updated knowledge that enables for strategic and thus more cost-effective prevention efforts also in the Job-Exposure Matrices # The MAHRE-Net supports the international organizations with the updated scientific evidence for updating the international conventions and regulations # The MAHRE-Net supports the Flag states to comply with their obligations to monitor the working and living environments regularly according to ILO Conventions: MLC2006 for seafarers and C188 for fishermen. ==Contributions to the Health Risk Prevention == The cohort studies can be seen as the diagnostic part of the prevention related to each of the specific items. Guidelines for the prevention for each of the standard questionnaires will be included in the Cohort Protocols. ==Phases of the preventive program for single industry== # Epidemiological standardized questionnaire studies # Occupational therapists investigate troubled job positions # Occupational Medical Doctors supply with their patients from the workplace # Statistics of work accidents in the specific areas of the workplace years # Dialogue with companies to improve safety and ergonomics positions # Improve and continue if they have done well ==Ethical requirements == The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect "personal data" as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version''] Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials<ref>https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse</ref> ==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] == The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section. ==[[/Contribution to UNs 17 Sustainable Development Goals /]] == Goal 3: Good health and well-being for all workers <br> Goal 4: Quality Education<br> Goal 5: Gender Equity<br> Goal 8: Decent Work and Economic Growth<br> Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)<br> Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)<br> Goal 14: Life underwater observations on compliance with good waste management <br> Goal 17: Partnerships to achieve the Goals<br> ==Integrated Mental Health and Work Policy OECD's recommendation == To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement; Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities. Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace <ref>https://legalinstruments.oecd.org/public/doc/334/334.en.pdf</ref> ==Contribution to quality education == ===== For the maritime workers and the industry===== The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases. ===== For the maritime doctors===== Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. <br> https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases<br> ===== For the students ===== Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data. They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research. ==Strengths and Weaknesses == In many countries, the interest for a seafaring carrier among young people is rather low. It is a strength to start with the young seafarers at the maritime schools because they can bring fresh perspective and a different way of thinking to the maritime business and help to attract the young seafarers. According to Unicef, most of them are eager to learn, build their experience and apply their skills in the workforce.[https://www.unicef.org/thailand/stories/6-top-benefits-hiring-young-talent Unicef: 6 top benefits hiring young talents ] The method used is easy to implement in a low budget. It is a strength to use the method that immediately identifies risk elements in the work environment that is not seen by the shipping inspectors in the harbors to be amended for the benefit of the seafarers and the companies. In contrast to the register-based studies, these studies identify actual risk elements in the work environment that will never be learned from the register-based studies. Simple frequency analysis is very useful to start preventive work. In addition, the results in graphics are very useful for basic and advanced education. By using standardized short questionnaires, a good response rate is obtained. The General Data Protection Regulation (GDPR) enforced in EU since 2018 can be very complicated to manage in the research. The general survey data we use focuses on the general exposure data in the work environment and not personally identifiable information. Still, an assessment on GDPR data is always needed and in most cases, our survey research do not apply to the same ethical rules as for clinical database research. The students learn how to apply the research methods in their later professional tasks and search the scientific-based knowledge for solving practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for their professional positions as leaders on board. Among the weaknesses is that the response rate might be too low from the start, that they change their mail address so we have no contact and they are not willing to participate or have no time in the later rounds. Another weakness is that the cross-sectional design cannot identify the causal relations in the single studies. However, by comparison of different questionnaire rounds, the health risk hazards might be present in some cohort parts and not in others and thereby contribute to identifying the causal relations. ==The preventive maritime occupational health strategy == The maritime occupational and environmental health strategy in development will be based on the need for new knowledge of the main OH indicators. A continued accumulation of new knowledge from MAHRE-Net constitutes the base for continued development of a preventive maritime OH strategy for "constant care" of the workers at the sea.<ref>https://www.maersk.com/about/core-values    </ref> Based on the national and international OH strategies, e.g. the EU-OSHA [https://ec.europa.eu/info/law/better-regulation/have-your-say/initiatives/12673-EU-Strategic-Framework-on-Health-and-Safety-at-Work-2021-2027- EU Strategy] <ref>https://www.dropbox.com/s/kralxs9yvl569x0/The%20implementation%20of%20Occupational%20Safety%20and%20Health%20in%20the%20EUST_14630_2019_INIT_en.pdf?dl=0</ref>, The US-CDC and the WHO, ILO OH strategies we selected the four most important OH areas for constant systematic monitoring in the MAHRE-Net: 1. safety-climate, 2. mental health, 3. musculoskeletal risk and 4. Chemical risks. Besides these four main occupational health indicators the following public health indicators can be/are included in each of the surveys: height/weight for Body Mass Index, Smoking, General Health, Alcohol and questions on prevention The validated and international standard questionnaires are available with no extra costs for the student's thesis writing that seems to be an ideal way to make good progress despite the scarce financials. We profit from the results of the outstanding scientists from the Nordic countries and other countries who developed and validated the standardized questionnaires over the latest thirty years. We offer supervision and methodological support for the Bachelor and MScPubHealth graduation thesis and the thesis students in the maritime universities and other health educations like nurses, medical doctors, pharmaceuticals, etc. Data from the Radio Medical services and the seafarer's health examinations are also included. The intention is to obtain a win-win situation with the students getting inspired to continue to do more advanced studies. However, other types of research design apart from the monitor program are encouraged to be made by enthusiastic researchers based on paid clinic time, private time, or funds. Projects with pooling of data from many countries and trends analysis and combining of different questionnaires will require experienced researchers and fundings. One more very important research issue was added in May 2021, to establish early diagnosis of diseases, especially pre-hypertension and pre-diabetes, with evidence that in that time window of disease development, good effect of prevention. Altogether the research activities in MAHRE-Net are intended to be very wide from the most basic levels to the highest levels of competencies with constant education and learning in a preventive perspective. ==[[/ MARITIME HEALTH PORTAL/| Maritime Health Portal]] == ==Sharing research data== As a researcher, you are increasingly encouraged, to make your research data available and usable <ref>https://www.elsevier.com/authors/tools-and-resources/research-data</ref>. However, interviews with researchers revealed a reluctance to share data included a lack of confidence in the utility of the data <ref>https://f1000research.com/articles/7-1641</ref>Data-sharing is the desired default in the field of public health and a source of much ethical deliberation. Sharing data potentially contributes to the most efficient source of scientific data, but is fraught with contextual challenges which make stakeholders, particularly those in under-resourced contexts hesitant or slow to share <ref> Carrillo-Larco, Rodrigo M., J. Jaime Miranda, and Andre P. Kengne. ‘Data Pooling Efforts in Africa and Latin America’. The Lancet Global Health 5, no. 1 (1 January 2017): e37. https://doi.org/10.1016/S2214-109X(16)30297-2</ref> For example, the Global Body-Mass Index (BMI) Mortality Collaboration1 published their work on BMI as a predictor of all-cause mortality. The investigators pooled individual participant data from 239 prospective studies, with none originating from Latin America or Africa <ref>Body-mass index and all-cause mortality: individual-participant-data meta-analysis of 239 prospective studies in four continents. Lancet. 2016; 388: 776-786</ref>, <ref>Anane-Sarpong, Evelyn, Tenzin Wangmo, Claire Leonie Ward, Osman Sankoh, Marcel Tanner, and Bernice Simone Elger. ‘“You Cannot Collect Data Using Your Own Resources and Put It on Open Access”: Perspectives from Africa about Public Health Data-Sharing’. Developing World Bioethics 18, no. 4 (December 2018): 394–405. https://doi.org/10.1111/dewb.12159</ref> Also collaboration through OMEGA-NET will enhance the scientific output from individual studies and facilitate pooled studies, data sharing, and transfer of tools and skills to make greater and more efficient use of existing cohorts. Researchers from countries outside Europe can participate in COST Actions based on ascertained mutual benefit. Mehlum, Ingrid. <ref>1673f Network on the Coordination and Harmonisation of European Occupational Cohorts (Omega-Net). Occup Environ Med. Vol. 75, 2018. https://doi.org/10.1136/oemed-2018-ICOHabstracts.356.</ref> ==Links to relevant organizations, documents, and funds == [http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health] [http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health] [[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]] [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST] [http://dimopex.eu/about/ DiMoPEx (CA 15129)] [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications] [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda] [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values] [https://www.europeansurveyresearch.org The European Survey Research Association] [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)] [https://ec.europa.eu/esf/main.jsp?catId=67&langId=en&newsId=9691 The European Social Fund] == https://www.fi-compass.eu/esif/emff<br> [https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] <br> [https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]<br> [https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] <br> [https://www.seafarerstrust.org/about/ ITF Seafarers Trust]<br> [https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]<br> [https://novonordiskfonden.dk/en/ Novo Nordisk Found] ==References== 14l7dsd48t649dnk34gt44apxnh1czc 2416307 2416306 2022-08-18T21:39:16Z Saltrabook 1417466 wikitext text/x-wiki ==Introduction == The Maritime Health Research and Education-NET MAHRE-Net is a non-profit network of researchers, seafarers and other workers, maritime students composed of four parts: # Research, based on standardized, health questionnaires, # Screening for T2 Diabetes mellitus and Hypertension in the fit-for-duty medical examinations # Health promotion program integrated with the T2DM & HTN screening program # Systematic Literature Reviews and Reviews of Systematic Reviews. The primary target study populations include maritime students, seafarers, fishermen, port workers, offshore workers, divers, and their social relations, and other industries. The aim is to provide a foundation for the evidence base for the identification of health risks to foster safe and healthy preventive strategies and policies within the UN Global Sustainable Goals.<ref>‘THE 17 GOALS | Sustainable Development. Accessed 1 May 2021. https://sdgs.un.org/goals</ref><ref>[[Maritime Health Research and Education-NET/Contribution to UNs 17 Sustainable Development Goals|Contribution to UNs 17 Sustainable Development Goals ]]</ref> The program research program includes the permanent monitoring of the main topics of the EU-Occupational Health strategy: # Mental health # Ergonomics # Screening for Hypertension and Diabetes Type 2, accurate diagnosis and prevention # Safety climate # COVID-19 and other infectious disease We will follow and support the young people from the maritime schools in their care in the cohort design strategies. The method is that we ask the classes of maritime (or other) students to fill out a standardized questionnaire in one of the four themes at the beginning of their studies on their mobile phones. The surveys in the maritime schools complete part of the diagnostics of a global mental health program at the schools and workplaces in the WHO health-promoting school-framework for improving the health and well being'''<ref> [https://learningportal.iiep.unesco.org/en/library/the-who-health-promoting-school-framework-for-improving-the-health-and-well-being-of?back_url=/en/library/search/occupational%20health%20research The WHO health-promoting school-framework for improving the health and well being]</ref>,<ref>https://pubmed.ncbi.nlm.nih.gov/24737131/</ref> <ref>https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-015-1360-y</ref>''' When they start their practice times at sea, they complete the same questions to identify the influence on their well-being on board. We measure how many of them leave the sea profession and we ask them how they think the profession can continue to be attractive to young people. It is intended to suggest and assist in the implementation of preventive measures based on the results. Later, we continue to ask them at some year intervals with the same questionnaires to assess whether the efforts have helped. We give the same questionnaires to the maritime students in other countries for comparison and learn from their proposals to get the best working conditions. Also, we ask what is needed of teaching to help the industry give them the best condition to stay safe in the job. Different cohort data sources like pre-employment medical health examination data can be tried out for feasibility and validity. A "Cohort" is defined in epidemiological science as a group of people who share the same characteristic, in this case, more or less the same birth years and we take several contacts to them over their life. Cohorts are also started with maritime workers through unions, other organizations, and shipping companies: ==[[/EU Consortium for Maritime Health Research and Education/]] == ==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|International T2 Diabetes Mellitus and Hypertension Research Group]] == == [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] == ==[[/Students Research Projects|Education 2: Students Thesis Research]] == ==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The International Maritime Health Journal Club]]== ==[[/Standard Questionnaire Based studies/|Questionnaire Based studies]] == ==[[/Systematic Reviews/]] == ==[[/Systematic reviews of systematic reviews /]] == ==[[/Organisation / ]] == ==[[/Presentations pptx /|Presentations]] == ==[[/Publications out and on the way/]] == ==[[/ BACKGROUND - open/| Literature background- open]] == == Objectives == # International prospective exposure and health risk cohort studies with maritime students and workers # All other industries are invited to promote the similar research and education goals # Using the standard protocols with different themes # Harmonise exposure and outcome information by using standard questionnaires # Objective and subjective assessments of workplace hazards exposures # Develop and validate Job-exposure Matrices # Systematic reviews and pooled studies from the cohort rounds # Cohort-Linkage to pre-entry and follow-up health exams and other health registers # Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension] # International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing VCT@WORK'''] # Make training materials based on the cohort study and the clinical study outcomes and other scientific sources # OHS training to maritime doctors, seafarers, fishermen, students, and others # Integrate research methodology in the supervision of student’s thesis work # Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data # Keep the Excel data file copies safely (producing country and supervisor) # Disseminate the knowledge in publications and organize seminars/webinars/symposia # There is a close relation to the [https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL Maritime Health Portal] ==Contributions to the Industry == # The maritime doctors and other maritime health professionals receive updated knowledge on the health risks at sea for the specific jobs and work areas # [https://stami.no/slik-skal-stami-sikre-at-kunnskapen-nar-virksomhetene/ Systematic dissemination of knowledge in work environment-e.g. STAMI Specific project] # Job-Exposure Matrices will support the needed evidence to the health examinations according to the requirements in the ILO/WHO Guidelines # The companies receive updated knowledge that enables for strategic and thus more cost-effective prevention efforts also in the Job-Exposure Matrices # The MAHRE-Net supports the international organizations with the updated scientific evidence for updating the international conventions and regulations # The MAHRE-Net supports the Flag states to comply with their obligations to monitor the working and living environments regularly according to ILO Conventions: MLC2006 for seafarers and C188 for fishermen. ==Contributions to the Health Risk Prevention == The cohort studies can be seen as the diagnostic part of the prevention related to each of the specific items. Guidelines for the prevention for each of the standard questionnaires will be included in the Cohort Protocols. ==Phases of the preventive program for single industry== # Epidemiological standardized questionnaire studies # Occupational therapists investigate troubled job positions # Occupational Medical Doctors supply with their patients from the workplace # Statistics of work accidents in the specific areas of the workplace years # Dialogue with companies to improve safety and ergonomics positions # Improve and continue if they have done well ==Ethical requirements == The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect "personal data" as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version''] Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials<ref>https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse</ref> ==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] == The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section. ==[[/Contribution to UNs 17 Sustainable Development Goals /]] == Goal 3: Good health and well-being for all workers <br> Goal 4: Quality Education<br> Goal 5: Gender Equity<br> Goal 8: Decent Work and Economic Growth<br> Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)<br> Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)<br> Goal 14: Life underwater observations on compliance with good waste management <br> Goal 17: Partnerships to achieve the Goals<br> ==Integrated Mental Health and Work Policy OECD's recommendation == To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement; Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities. Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace <ref>https://legalinstruments.oecd.org/public/doc/334/334.en.pdf</ref> ==Contribution to quality education == ===== For the maritime workers and the industry===== The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases. ===== For the maritime doctors===== Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. <br> https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases<br> ===== For the students ===== Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data. They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research. ==Strengths and Weaknesses == In many countries, the interest for a seafaring carrier among young people is rather low. It is a strength to start with the young seafarers at the maritime schools because they can bring fresh perspective and a different way of thinking to the maritime business and help to attract the young seafarers. According to Unicef, most of them are eager to learn, build their experience and apply their skills in the workforce.[https://www.unicef.org/thailand/stories/6-top-benefits-hiring-young-talent Unicef: 6 top benefits hiring young talents ] The method used is easy to implement in a low budget. It is a strength to use the method that immediately identifies risk elements in the work environment that is not seen by the shipping inspectors in the harbors to be amended for the benefit of the seafarers and the companies. In contrast to the register-based studies, these studies identify actual risk elements in the work environment that will never be learned from the register-based studies. Simple frequency analysis is very useful to start preventive work. In addition, the results in graphics are very useful for basic and advanced education. By using standardized short questionnaires, a good response rate is obtained. The General Data Protection Regulation (GDPR) enforced in EU since 2018 can be very complicated to manage in the research. The general survey data we use focuses on the general exposure data in the work environment and not personally identifiable information. Still, an assessment on GDPR data is always needed and in most cases, our survey research do not apply to the same ethical rules as for clinical database research. The students learn how to apply the research methods in their later professional tasks and search the scientific-based knowledge for solving practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for their professional positions as leaders on board. Among the weaknesses is that the response rate might be too low from the start, that they change their mail address so we have no contact and they are not willing to participate or have no time in the later rounds. Another weakness is that the cross-sectional design cannot identify the causal relations in the single studies. However, by comparison of different questionnaire rounds, the health risk hazards might be present in some cohort parts and not in others and thereby contribute to identifying the causal relations. ==The preventive maritime occupational health strategy == The maritime occupational and environmental health strategy in development will be based on the need for new knowledge of the main OH indicators. A continued accumulation of new knowledge from MAHRE-Net constitutes the base for continued development of a preventive maritime OH strategy for "constant care" of the workers at the sea.<ref>https://www.maersk.com/about/core-values    </ref> Based on the national and international OH strategies, e.g. the EU-OSHA [https://ec.europa.eu/info/law/better-regulation/have-your-say/initiatives/12673-EU-Strategic-Framework-on-Health-and-Safety-at-Work-2021-2027- EU Strategy] <ref>https://www.dropbox.com/s/kralxs9yvl569x0/The%20implementation%20of%20Occupational%20Safety%20and%20Health%20in%20the%20EUST_14630_2019_INIT_en.pdf?dl=0</ref>, The US-CDC and the WHO, ILO OH strategies we selected the four most important OH areas for constant systematic monitoring in the MAHRE-Net: 1. safety-climate, 2. mental health, 3. musculoskeletal risk and 4. Chemical risks. Besides these four main occupational health indicators the following public health indicators can be/are included in each of the surveys: height/weight for Body Mass Index, Smoking, General Health, Alcohol and questions on prevention The validated and international standard questionnaires are available with no extra costs for the student's thesis writing that seems to be an ideal way to make good progress despite the scarce financials. We profit from the results of the outstanding scientists from the Nordic countries and other countries who developed and validated the standardized questionnaires over the latest thirty years. We offer supervision and methodological support for the Bachelor and MScPubHealth graduation thesis and the thesis students in the maritime universities and other health educations like nurses, medical doctors, pharmaceuticals, etc. Data from the Radio Medical services and the seafarer's health examinations are also included. The intention is to obtain a win-win situation with the students getting inspired to continue to do more advanced studies. However, other types of research design apart from the monitor program are encouraged to be made by enthusiastic researchers based on paid clinic time, private time, or funds. Projects with pooling of data from many countries and trends analysis and combining of different questionnaires will require experienced researchers and fundings. One more very important research issue was added in May 2021, to establish early diagnosis of diseases, especially pre-hypertension and pre-diabetes, with evidence that in that time window of disease development, good effect of prevention. Altogether the research activities in MAHRE-Net are intended to be very wide from the most basic levels to the highest levels of competencies with constant education and learning in a preventive perspective. ==[[/ MARITIME HEALTH PORTAL/| Maritime Health Portal]] == ==Sharing research data== As a researcher, you are increasingly encouraged, to make your research data available and usable <ref>https://www.elsevier.com/authors/tools-and-resources/research-data</ref>. However, interviews with researchers revealed a reluctance to share data included a lack of confidence in the utility of the data <ref>https://f1000research.com/articles/7-1641</ref>Data-sharing is the desired default in the field of public health and a source of much ethical deliberation. Sharing data potentially contributes to the most efficient source of scientific data, but is fraught with contextual challenges which make stakeholders, particularly those in under-resourced contexts hesitant or slow to share <ref> Carrillo-Larco, Rodrigo M., J. Jaime Miranda, and Andre P. Kengne. ‘Data Pooling Efforts in Africa and Latin America’. The Lancet Global Health 5, no. 1 (1 January 2017): e37. https://doi.org/10.1016/S2214-109X(16)30297-2</ref> For example, the Global Body-Mass Index (BMI) Mortality Collaboration1 published their work on BMI as a predictor of all-cause mortality. The investigators pooled individual participant data from 239 prospective studies, with none originating from Latin America or Africa <ref>Body-mass index and all-cause mortality: individual-participant-data meta-analysis of 239 prospective studies in four continents. Lancet. 2016; 388: 776-786</ref>, <ref>Anane-Sarpong, Evelyn, Tenzin Wangmo, Claire Leonie Ward, Osman Sankoh, Marcel Tanner, and Bernice Simone Elger. ‘“You Cannot Collect Data Using Your Own Resources and Put It on Open Access”: Perspectives from Africa about Public Health Data-Sharing’. Developing World Bioethics 18, no. 4 (December 2018): 394–405. https://doi.org/10.1111/dewb.12159</ref> Also collaboration through OMEGA-NET will enhance the scientific output from individual studies and facilitate pooled studies, data sharing, and transfer of tools and skills to make greater and more efficient use of existing cohorts. Researchers from countries outside Europe can participate in COST Actions based on ascertained mutual benefit. Mehlum, Ingrid. <ref>1673f Network on the Coordination and Harmonisation of European Occupational Cohorts (Omega-Net). Occup Environ Med. Vol. 75, 2018. https://doi.org/10.1136/oemed-2018-ICOHabstracts.356.</ref> ==Links to relevant organizations, documents, and funds == [http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health] [http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health] [[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]] [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST] [http://dimopex.eu/about/ DiMoPEx (CA 15129)] [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications] [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda] [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values] [https://www.europeansurveyresearch.org The European Survey Research Association] [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)] [https://ec.europa.eu/esf/main.jsp?catId=67&langId=en&newsId=9691 The European Social Fund] == https://www.fi-compass.eu/esif/emff<br> [https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] <br> [https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]<br> [https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] <br> [https://www.seafarerstrust.org/about/ ITF Seafarers Trust]<br> [https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]<br> [https://novonordiskfonden.dk/en/ Novo Nordisk Found] ==References== cj7cx6jj60tsikqydqugj64gc69ytlg 2416308 2416307 2022-08-18T21:39:55Z Saltrabook 1417466 /* Introduction */ wikitext text/x-wiki ==Introduction == The Maritime Health Research and Education-NET MAHRE-Net is a non-profit network of researchers, seafarers and other workers, maritime students composed of four parts: # Research, based on standardized, health questionnaires, # Screening for T2 Diabetes mellitus and Hypertension in the fit-for-duty medical examinations # Health promotion program integrated with the T2DM & HTN screening program # Systematic Literature Reviews and Reviews of Systematic Reviews. The primary target study populations include maritime students, seafarers, fishermen, port workers, offshore workers, divers, and their social relations, and other industries. The aim is to provide a foundation for the evidence base for the identification of health risks to foster safe and healthy preventive strategies and policies within the UN Global Sustainable Goals.<ref>‘THE 17 GOALS | Sustainable Development. Accessed 1 May 2021. https://sdgs.un.org/goals</ref><ref>[[Maritime Health Research and Education-NET/Contribution to UNs 17 Sustainable Development Goals|Contribution to UNs 17 Sustainable Development Goals ]]</ref> The program research program includes the permanent monitoring of the main topics of the EU-Occupational Health strategy: # Mental health # Ergonomics # Screening for Hypertension and Diabetes Type 2, accurate diagnosis and prevention at the medical examinations # Safety climate # COVID-19 and other infectious disease We will follow and support the young people from the maritime schools in their care in the cohort design strategies. The method is that we ask the classes of maritime (or other) students to fill out a standardized questionnaire in one of the four themes at the beginning of their studies on their mobile phones. The surveys in the maritime schools complete part of the diagnostics of a global mental health program at the schools and workplaces in the WHO health-promoting school-framework for improving the health and well being'''<ref> [https://learningportal.iiep.unesco.org/en/library/the-who-health-promoting-school-framework-for-improving-the-health-and-well-being-of?back_url=/en/library/search/occupational%20health%20research The WHO health-promoting school-framework for improving the health and well being]</ref>,<ref>https://pubmed.ncbi.nlm.nih.gov/24737131/</ref> <ref>https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-015-1360-y</ref>''' When they start their practice times at sea, they complete the same questions to identify the influence on their well-being on board. We measure how many of them leave the sea profession and we ask them how they think the profession can continue to be attractive to young people. It is intended to suggest and assist in the implementation of preventive measures based on the results. Later, we continue to ask them at some year intervals with the same questionnaires to assess whether the efforts have helped. We give the same questionnaires to the maritime students in other countries for comparison and learn from their proposals to get the best working conditions. Also, we ask what is needed of teaching to help the industry give them the best condition to stay safe in the job. Different cohort data sources like pre-employment medical health examination data can be tried out for feasibility and validity. A "Cohort" is defined in epidemiological science as a group of people who share the same characteristic, in this case, more or less the same birth years and we take several contacts to them over their life. Cohorts are also started with maritime workers through unions, other organizations, and shipping companies: ==[[/EU Consortium for Maritime Health Research and Education/]] == ==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|International T2 Diabetes Mellitus and Hypertension Research Group]] == == [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] == ==[[/Students Research Projects|Education 2: Students Thesis Research]] == ==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The International Maritime Health Journal Club]]== ==[[/Standard Questionnaire Based studies/|Questionnaire Based studies]] == ==[[/Systematic Reviews/]] == ==[[/Systematic reviews of systematic reviews /]] == ==[[/Organisation / ]] == ==[[/Presentations pptx /|Presentations]] == ==[[/Publications out and on the way/]] == ==[[/ BACKGROUND - open/| Literature background- open]] == == Objectives == # International prospective exposure and health risk cohort studies with maritime students and workers # All other industries are invited to promote the similar research and education goals # Using the standard protocols with different themes # Harmonise exposure and outcome information by using standard questionnaires # Objective and subjective assessments of workplace hazards exposures # Develop and validate Job-exposure Matrices # Systematic reviews and pooled studies from the cohort rounds # Cohort-Linkage to pre-entry and follow-up health exams and other health registers # Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension] # International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing VCT@WORK'''] # Make training materials based on the cohort study and the clinical study outcomes and other scientific sources # OHS training to maritime doctors, seafarers, fishermen, students, and others # Integrate research methodology in the supervision of student’s thesis work # Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data # Keep the Excel data file copies safely (producing country and supervisor) # Disseminate the knowledge in publications and organize seminars/webinars/symposia # There is a close relation to the [https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL Maritime Health Portal] ==Contributions to the Industry == # The maritime doctors and other maritime health professionals receive updated knowledge on the health risks at sea for the specific jobs and work areas # [https://stami.no/slik-skal-stami-sikre-at-kunnskapen-nar-virksomhetene/ Systematic dissemination of knowledge in work environment-e.g. STAMI Specific project] # Job-Exposure Matrices will support the needed evidence to the health examinations according to the requirements in the ILO/WHO Guidelines # The companies receive updated knowledge that enables for strategic and thus more cost-effective prevention efforts also in the Job-Exposure Matrices # The MAHRE-Net supports the international organizations with the updated scientific evidence for updating the international conventions and regulations # The MAHRE-Net supports the Flag states to comply with their obligations to monitor the working and living environments regularly according to ILO Conventions: MLC2006 for seafarers and C188 for fishermen. ==Contributions to the Health Risk Prevention == The cohort studies can be seen as the diagnostic part of the prevention related to each of the specific items. Guidelines for the prevention for each of the standard questionnaires will be included in the Cohort Protocols. ==Phases of the preventive program for single industry== # Epidemiological standardized questionnaire studies # Occupational therapists investigate troubled job positions # Occupational Medical Doctors supply with their patients from the workplace # Statistics of work accidents in the specific areas of the workplace years # Dialogue with companies to improve safety and ergonomics positions # Improve and continue if they have done well ==Ethical requirements == The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect "personal data" as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version''] Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials<ref>https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse</ref> ==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] == The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section. ==[[/Contribution to UNs 17 Sustainable Development Goals /]] == Goal 3: Good health and well-being for all workers <br> Goal 4: Quality Education<br> Goal 5: Gender Equity<br> Goal 8: Decent Work and Economic Growth<br> Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)<br> Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)<br> Goal 14: Life underwater observations on compliance with good waste management <br> Goal 17: Partnerships to achieve the Goals<br> ==Integrated Mental Health and Work Policy OECD's recommendation == To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement; Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities. Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace <ref>https://legalinstruments.oecd.org/public/doc/334/334.en.pdf</ref> ==Contribution to quality education == ===== For the maritime workers and the industry===== The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases. ===== For the maritime doctors===== Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. <br> https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases<br> ===== For the students ===== Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data. They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research. ==Strengths and Weaknesses == In many countries, the interest for a seafaring carrier among young people is rather low. It is a strength to start with the young seafarers at the maritime schools because they can bring fresh perspective and a different way of thinking to the maritime business and help to attract the young seafarers. According to Unicef, most of them are eager to learn, build their experience and apply their skills in the workforce.[https://www.unicef.org/thailand/stories/6-top-benefits-hiring-young-talent Unicef: 6 top benefits hiring young talents ] The method used is easy to implement in a low budget. It is a strength to use the method that immediately identifies risk elements in the work environment that is not seen by the shipping inspectors in the harbors to be amended for the benefit of the seafarers and the companies. In contrast to the register-based studies, these studies identify actual risk elements in the work environment that will never be learned from the register-based studies. Simple frequency analysis is very useful to start preventive work. In addition, the results in graphics are very useful for basic and advanced education. By using standardized short questionnaires, a good response rate is obtained. The General Data Protection Regulation (GDPR) enforced in EU since 2018 can be very complicated to manage in the research. The general survey data we use focuses on the general exposure data in the work environment and not personally identifiable information. Still, an assessment on GDPR data is always needed and in most cases, our survey research do not apply to the same ethical rules as for clinical database research. The students learn how to apply the research methods in their later professional tasks and search the scientific-based knowledge for solving practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for their professional positions as leaders on board. Among the weaknesses is that the response rate might be too low from the start, that they change their mail address so we have no contact and they are not willing to participate or have no time in the later rounds. Another weakness is that the cross-sectional design cannot identify the causal relations in the single studies. However, by comparison of different questionnaire rounds, the health risk hazards might be present in some cohort parts and not in others and thereby contribute to identifying the causal relations. ==The preventive maritime occupational health strategy == The maritime occupational and environmental health strategy in development will be based on the need for new knowledge of the main OH indicators. A continued accumulation of new knowledge from MAHRE-Net constitutes the base for continued development of a preventive maritime OH strategy for "constant care" of the workers at the sea.<ref>https://www.maersk.com/about/core-values    </ref> Based on the national and international OH strategies, e.g. the EU-OSHA [https://ec.europa.eu/info/law/better-regulation/have-your-say/initiatives/12673-EU-Strategic-Framework-on-Health-and-Safety-at-Work-2021-2027- EU Strategy] <ref>https://www.dropbox.com/s/kralxs9yvl569x0/The%20implementation%20of%20Occupational%20Safety%20and%20Health%20in%20the%20EUST_14630_2019_INIT_en.pdf?dl=0</ref>, The US-CDC and the WHO, ILO OH strategies we selected the four most important OH areas for constant systematic monitoring in the MAHRE-Net: 1. safety-climate, 2. mental health, 3. musculoskeletal risk and 4. Chemical risks. Besides these four main occupational health indicators the following public health indicators can be/are included in each of the surveys: height/weight for Body Mass Index, Smoking, General Health, Alcohol and questions on prevention The validated and international standard questionnaires are available with no extra costs for the student's thesis writing that seems to be an ideal way to make good progress despite the scarce financials. We profit from the results of the outstanding scientists from the Nordic countries and other countries who developed and validated the standardized questionnaires over the latest thirty years. We offer supervision and methodological support for the Bachelor and MScPubHealth graduation thesis and the thesis students in the maritime universities and other health educations like nurses, medical doctors, pharmaceuticals, etc. Data from the Radio Medical services and the seafarer's health examinations are also included. The intention is to obtain a win-win situation with the students getting inspired to continue to do more advanced studies. However, other types of research design apart from the monitor program are encouraged to be made by enthusiastic researchers based on paid clinic time, private time, or funds. Projects with pooling of data from many countries and trends analysis and combining of different questionnaires will require experienced researchers and fundings. One more very important research issue was added in May 2021, to establish early diagnosis of diseases, especially pre-hypertension and pre-diabetes, with evidence that in that time window of disease development, good effect of prevention. Altogether the research activities in MAHRE-Net are intended to be very wide from the most basic levels to the highest levels of competencies with constant education and learning in a preventive perspective. ==[[/ MARITIME HEALTH PORTAL/| Maritime Health Portal]] == ==Sharing research data== As a researcher, you are increasingly encouraged, to make your research data available and usable <ref>https://www.elsevier.com/authors/tools-and-resources/research-data</ref>. However, interviews with researchers revealed a reluctance to share data included a lack of confidence in the utility of the data <ref>https://f1000research.com/articles/7-1641</ref>Data-sharing is the desired default in the field of public health and a source of much ethical deliberation. Sharing data potentially contributes to the most efficient source of scientific data, but is fraught with contextual challenges which make stakeholders, particularly those in under-resourced contexts hesitant or slow to share <ref> Carrillo-Larco, Rodrigo M., J. Jaime Miranda, and Andre P. Kengne. ‘Data Pooling Efforts in Africa and Latin America’. The Lancet Global Health 5, no. 1 (1 January 2017): e37. https://doi.org/10.1016/S2214-109X(16)30297-2</ref> For example, the Global Body-Mass Index (BMI) Mortality Collaboration1 published their work on BMI as a predictor of all-cause mortality. The investigators pooled individual participant data from 239 prospective studies, with none originating from Latin America or Africa <ref>Body-mass index and all-cause mortality: individual-participant-data meta-analysis of 239 prospective studies in four continents. Lancet. 2016; 388: 776-786</ref>, <ref>Anane-Sarpong, Evelyn, Tenzin Wangmo, Claire Leonie Ward, Osman Sankoh, Marcel Tanner, and Bernice Simone Elger. ‘“You Cannot Collect Data Using Your Own Resources and Put It on Open Access”: Perspectives from Africa about Public Health Data-Sharing’. Developing World Bioethics 18, no. 4 (December 2018): 394–405. https://doi.org/10.1111/dewb.12159</ref> Also collaboration through OMEGA-NET will enhance the scientific output from individual studies and facilitate pooled studies, data sharing, and transfer of tools and skills to make greater and more efficient use of existing cohorts. Researchers from countries outside Europe can participate in COST Actions based on ascertained mutual benefit. Mehlum, Ingrid. <ref>1673f Network on the Coordination and Harmonisation of European Occupational Cohorts (Omega-Net). Occup Environ Med. Vol. 75, 2018. https://doi.org/10.1136/oemed-2018-ICOHabstracts.356.</ref> ==Links to relevant organizations, documents, and funds == [http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health] [http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health] [[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]] [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST] [http://dimopex.eu/about/ DiMoPEx (CA 15129)] [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications] [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda] [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values] [https://www.europeansurveyresearch.org The European Survey Research Association] [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)] [https://ec.europa.eu/esf/main.jsp?catId=67&langId=en&newsId=9691 The European Social Fund] == https://www.fi-compass.eu/esif/emff<br> [https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] <br> [https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]<br> [https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] <br> [https://www.seafarerstrust.org/about/ ITF Seafarers Trust]<br> [https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]<br> [https://novonordiskfonden.dk/en/ Novo Nordisk Found] ==References== 07gb3wshcfi27tyi6arn4yx3bt3mtdj 2416309 2416308 2022-08-18T21:40:21Z Saltrabook 1417466 /* Introduction */ wikitext text/x-wiki ==Introduction == The Maritime Health Research and Education-NET MAHRE-Net is a non-profit network of researchers, seafarers and other workers, maritime students composed of four parts: # Research, based on standardized, health questionnaires, # Screening for T2 Diabetes mellitus and Hypertension in the fit-for-duty medical examinations # Health promotion program integrated with the T2DM & HTN screening program # Systematic Literature Reviews and Reviews of Systematic Reviews. The primary target study populations include maritime students, seafarers, fishermen, port workers, offshore workers, divers, and their social relations, and other industries. The aim is to provide a foundation for the evidence base for the identification of health risks to foster safe and healthy preventive strategies and policies within the UN Global Sustainable Goals.<ref>‘THE 17 GOALS | Sustainable Development. Accessed 1 May 2021. https://sdgs.un.org/goals</ref><ref>[[Maritime Health Research and Education-NET/Contribution to UNs 17 Sustainable Development Goals|Contribution to UNs 17 Sustainable Development Goals ]]</ref> The program research program includes the permanent monitoring of the main topics of the EU-Occupational Health strategy: # Mental health # Ergonomics # Screening for Hypertension and Diabetes Type 2, accurate diagnosis and prevention at the medical examinations # COVID-19 and other infectious disease # Safety climate We will follow and support the young people from the maritime schools in their care in the cohort design strategies. The method is that we ask the classes of maritime (or other) students to fill out a standardized questionnaire in one of the four themes at the beginning of their studies on their mobile phones. The surveys in the maritime schools complete part of the diagnostics of a global mental health program at the schools and workplaces in the WHO health-promoting school-framework for improving the health and well being'''<ref> [https://learningportal.iiep.unesco.org/en/library/the-who-health-promoting-school-framework-for-improving-the-health-and-well-being-of?back_url=/en/library/search/occupational%20health%20research The WHO health-promoting school-framework for improving the health and well being]</ref>,<ref>https://pubmed.ncbi.nlm.nih.gov/24737131/</ref> <ref>https://bmcpublichealth.biomedcentral.com/articles/10.1186/s12889-015-1360-y</ref>''' When they start their practice times at sea, they complete the same questions to identify the influence on their well-being on board. We measure how many of them leave the sea profession and we ask them how they think the profession can continue to be attractive to young people. It is intended to suggest and assist in the implementation of preventive measures based on the results. Later, we continue to ask them at some year intervals with the same questionnaires to assess whether the efforts have helped. We give the same questionnaires to the maritime students in other countries for comparison and learn from their proposals to get the best working conditions. Also, we ask what is needed of teaching to help the industry give them the best condition to stay safe in the job. Different cohort data sources like pre-employment medical health examination data can be tried out for feasibility and validity. A "Cohort" is defined in epidemiological science as a group of people who share the same characteristic, in this case, more or less the same birth years and we take several contacts to them over their life. Cohorts are also started with maritime workers through unions, other organizations, and shipping companies: ==[[/EU Consortium for Maritime Health Research and Education/]] == ==[[/The International Type 2 Diabetes Mellitus and Hypertension Research Group/|International T2 Diabetes Mellitus and Hypertension Research Group]] == == [[Maritime Health Research and Education-NET/EDUCATION/Education module links|Education 1: Research Methodology]] == ==[[/Students Research Projects|Education 2: Students Thesis Research]] == ==[[/Education 3: The SDG17 International Maritime Health Journal Club/|Education 3: The International Maritime Health Journal Club]]== ==[[/Standard Questionnaire Based studies/|Questionnaire Based studies]] == ==[[/Systematic Reviews/]] == ==[[/Systematic reviews of systematic reviews /]] == ==[[/Organisation / ]] == ==[[/Presentations pptx /|Presentations]] == ==[[/Publications out and on the way/]] == ==[[/ BACKGROUND - open/| Literature background- open]] == == Objectives == # International prospective exposure and health risk cohort studies with maritime students and workers # All other industries are invited to promote the similar research and education goals # Using the standard protocols with different themes # Harmonise exposure and outcome information by using standard questionnaires # Objective and subjective assessments of workplace hazards exposures # Develop and validate Job-exposure Matrices # Systematic reviews and pooled studies from the cohort rounds # Cohort-Linkage to pre-entry and follow-up health exams and other health registers # Use routine health exams for early diagnosis and primary/secondary prevention of [https://en.wikipedia.org/wiki/Prediabetes Pre-diabetes] and [https://en.wikipedia.org/wiki/Prehypertension Pre-hypertension] # International workplace research- intervention plan based on the [https://www.ilo.org/wcmsp5/groups/public/---dgreports/---gender/documents/publication/wcms_762676.pdf '''ILO Guidelines integrated health testing VCT@WORK'''] # Make training materials based on the cohort study and the clinical study outcomes and other scientific sources # OHS training to maritime doctors, seafarers, fishermen, students, and others # Integrate research methodology in the supervision of student’s thesis work # Adapt to the OMEGA-NET on data sharing and reporting cohort meta-data # Keep the Excel data file copies safely (producing country and supervisor) # Disseminate the knowledge in publications and organize seminars/webinars/symposia # There is a close relation to the [https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL Maritime Health Portal] ==Contributions to the Industry == # The maritime doctors and other maritime health professionals receive updated knowledge on the health risks at sea for the specific jobs and work areas # [https://stami.no/slik-skal-stami-sikre-at-kunnskapen-nar-virksomhetene/ Systematic dissemination of knowledge in work environment-e.g. STAMI Specific project] # Job-Exposure Matrices will support the needed evidence to the health examinations according to the requirements in the ILO/WHO Guidelines # The companies receive updated knowledge that enables for strategic and thus more cost-effective prevention efforts also in the Job-Exposure Matrices # The MAHRE-Net supports the international organizations with the updated scientific evidence for updating the international conventions and regulations # The MAHRE-Net supports the Flag states to comply with their obligations to monitor the working and living environments regularly according to ILO Conventions: MLC2006 for seafarers and C188 for fishermen. ==Contributions to the Health Risk Prevention == The cohort studies can be seen as the diagnostic part of the prevention related to each of the specific items. Guidelines for the prevention for each of the standard questionnaires will be included in the Cohort Protocols. ==Phases of the preventive program for single industry== # Epidemiological standardized questionnaire studies # Occupational therapists investigate troubled job positions # Occupational Medical Doctors supply with their patients from the workplace # Statistics of work accidents in the specific areas of the workplace years # Dialogue with companies to improve safety and ergonomics positions # Improve and continue if they have done well ==Ethical requirements == The ethical rules for database research in the respective Universities and the ICOH Code of ethics are complied with. Confidentiality in handling personal information is done according to the rules set out by the national Data Protection Agencies. The European General Data Protection Regulation [https://gdpr-info.eu/ GDPR ]is complied with. All supervisors and the students are obliged to be familiar with the GDPR through a course. None of the research projects collect "personal data" as defined in the GDPR regulation and no personally sensitive information is included. All questionnaires ask for informed consent as the first question. The supervisors take care to secure that the data is processed under the Act on medical confidentiality as guidelines for good epidemiological practice. The participants' anonymity will be protected in every way and this will be indicated in the project description. It will be ensured that the electronic table is locked so that the information cannot be seen by anyone other than the researchers. The researchers respect individual ownership of the data and share publications and the data where this is convenient and keep always good partnerships as described in [https://allea.org/code-of-conduct/#toggle-id-18 ''The European Code of Conduct for Research Integrity for self-regulation in all research in 18 translations''] [https://www.allea.org/wp-content/uploads/2017/05/ALLEA-European-Code-of-Conduct-for-Research-Integrity-2017.pdf ''The English version''] Types of experiments not to be notified: Questionnaire and interview surveys; Registry research surveys; Quality assurance projects; Non-interventional drug trials<ref>https://komite.regionsyddanmark.dk/i-tvivl-om-anmeldelse</ref> ==[https://www.elsevier.com/about/policies/publishing-ethics Publishing Ethics] == The Elsevier publishers Guidelines include duties for the Publishers, Editors, Reviewers and the Authors corresponding to the international well-agreed different types of duties. Authorship should be limited to those who have made a significant contribution to the conception, design, execution, or interpretation of the reported study. Only those who made substantial contributions should be listed as co-authors. Others who have participated in certain substantive aspects of the paper e.g. language editing or medical writing should be recognised in the acknowledgements section. ==[[/Contribution to UNs 17 Sustainable Development Goals /]] == Goal 3: Good health and well-being for all workers <br> Goal 4: Quality Education<br> Goal 5: Gender Equity<br> Goal 8: Decent Work and Economic Growth<br> Goal 10: Reduced Inequity (Compliance with MLC2006 and the C188)<br> Goal 12: Responsible Consumption and Production (Ships’ SOx and NOx emissions)<br> Goal 14: Life underwater observations on compliance with good waste management <br> Goal 17: Partnerships to achieve the Goals<br> ==Integrated Mental Health and Work Policy OECD's recommendation == To monitor and improve the overall school and preschool climate to promote social-emotional learning, mental health, and wellbeing of all children and students through whole-of-school-based interventions and the prevention of mental stress, bullying, and aggression at school, using effective indicators of comprehensive school health and student achievement; Promote and enforce psychosocial risk assessment and risk prevention in the workplace to ensure that all companies have complied with their legal responsibilities. Develop a strategy for addressing the stigma, discrimination, and misconceptions faced by many workers living with mental health conditions at their workplace <ref>https://legalinstruments.oecd.org/public/doc/334/334.en.pdf</ref> ==Contribution to quality education == ===== For the maritime workers and the industry===== The updated scientific evidence on the prevalent health risk exposures and health conditions on board will qualify the prioritization of the preventive actions in the Safety Committees on board, in the companies, and the worker's organizations. The workers will benefit from the updated maritime doctors to better understand their possible claims and symptoms that call for adequate clinical and laboratory diagnostics and possible notification as occupational diseases. ===== For the maritime doctors===== Guidelines for early diagnosis of hypertension and diabetes type 2, the use of the Excel reporting scheme and follow-up of the new diagnosed seafarers. The outcomes of the cohort studies will be an important part of the continuing training of the Maritime Medical doctors and the training for fishermen and seafarers. <br> https://en.wikiversity.org/wiki/Maritime_Health_Research_and_Education-NET/MARITIME_HEALTH_PORTAL Without this knowledge, the medical doctors cannot perform their obligations adequately and give adequate advice for the seafarers and fishermen according to the ILO/IMO Guidelines on the Medical Examinations of Seafarers and act adequately with possible notification of occupational diseases<br> ===== For the students ===== Preferably we use our maritime health and safety research outcomes as the basis for our teaching for the MSc.Pub Health and the Maritime students. They learn the research methods in occupational maritime health with an assessment of reliability, generalisability and different types of bias in the scientific context including clearance of the ownership of the data. They learn how to apply the research methods in their coming professional tasks and search the scientific-based knowledge to solve practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for use in their professional positions as leaders on board. Personal data as defined in the EU GDPR regulation is not used in this research. ==Strengths and Weaknesses == In many countries, the interest for a seafaring carrier among young people is rather low. It is a strength to start with the young seafarers at the maritime schools because they can bring fresh perspective and a different way of thinking to the maritime business and help to attract the young seafarers. According to Unicef, most of them are eager to learn, build their experience and apply their skills in the workforce.[https://www.unicef.org/thailand/stories/6-top-benefits-hiring-young-talent Unicef: 6 top benefits hiring young talents ] The method used is easy to implement in a low budget. It is a strength to use the method that immediately identifies risk elements in the work environment that is not seen by the shipping inspectors in the harbors to be amended for the benefit of the seafarers and the companies. In contrast to the register-based studies, these studies identify actual risk elements in the work environment that will never be learned from the register-based studies. Simple frequency analysis is very useful to start preventive work. In addition, the results in graphics are very useful for basic and advanced education. By using standardized short questionnaires, a good response rate is obtained. The General Data Protection Regulation (GDPR) enforced in EU since 2018 can be very complicated to manage in the research. The general survey data we use focuses on the general exposure data in the work environment and not personally identifiable information. Still, an assessment on GDPR data is always needed and in most cases, our survey research do not apply to the same ethical rules as for clinical database research. The students learn how to apply the research methods in their later professional tasks and search the scientific-based knowledge for solving practical problems in their professional life. The maritime students get interested in searching and using the scientific-based maritime knowledge for their professional positions as leaders on board. Among the weaknesses is that the response rate might be too low from the start, that they change their mail address so we have no contact and they are not willing to participate or have no time in the later rounds. Another weakness is that the cross-sectional design cannot identify the causal relations in the single studies. However, by comparison of different questionnaire rounds, the health risk hazards might be present in some cohort parts and not in others and thereby contribute to identifying the causal relations. ==The preventive maritime occupational health strategy == The maritime occupational and environmental health strategy in development will be based on the need for new knowledge of the main OH indicators. A continued accumulation of new knowledge from MAHRE-Net constitutes the base for continued development of a preventive maritime OH strategy for "constant care" of the workers at the sea.<ref>https://www.maersk.com/about/core-values    </ref> Based on the national and international OH strategies, e.g. the EU-OSHA [https://ec.europa.eu/info/law/better-regulation/have-your-say/initiatives/12673-EU-Strategic-Framework-on-Health-and-Safety-at-Work-2021-2027- EU Strategy] <ref>https://www.dropbox.com/s/kralxs9yvl569x0/The%20implementation%20of%20Occupational%20Safety%20and%20Health%20in%20the%20EUST_14630_2019_INIT_en.pdf?dl=0</ref>, The US-CDC and the WHO, ILO OH strategies we selected the four most important OH areas for constant systematic monitoring in the MAHRE-Net: 1. safety-climate, 2. mental health, 3. musculoskeletal risk and 4. Chemical risks. Besides these four main occupational health indicators the following public health indicators can be/are included in each of the surveys: height/weight for Body Mass Index, Smoking, General Health, Alcohol and questions on prevention The validated and international standard questionnaires are available with no extra costs for the student's thesis writing that seems to be an ideal way to make good progress despite the scarce financials. We profit from the results of the outstanding scientists from the Nordic countries and other countries who developed and validated the standardized questionnaires over the latest thirty years. We offer supervision and methodological support for the Bachelor and MScPubHealth graduation thesis and the thesis students in the maritime universities and other health educations like nurses, medical doctors, pharmaceuticals, etc. Data from the Radio Medical services and the seafarer's health examinations are also included. The intention is to obtain a win-win situation with the students getting inspired to continue to do more advanced studies. However, other types of research design apart from the monitor program are encouraged to be made by enthusiastic researchers based on paid clinic time, private time, or funds. Projects with pooling of data from many countries and trends analysis and combining of different questionnaires will require experienced researchers and fundings. One more very important research issue was added in May 2021, to establish early diagnosis of diseases, especially pre-hypertension and pre-diabetes, with evidence that in that time window of disease development, good effect of prevention. Altogether the research activities in MAHRE-Net are intended to be very wide from the most basic levels to the highest levels of competencies with constant education and learning in a preventive perspective. ==[[/ MARITIME HEALTH PORTAL/| Maritime Health Portal]] == ==Sharing research data== As a researcher, you are increasingly encouraged, to make your research data available and usable <ref>https://www.elsevier.com/authors/tools-and-resources/research-data</ref>. However, interviews with researchers revealed a reluctance to share data included a lack of confidence in the utility of the data <ref>https://f1000research.com/articles/7-1641</ref>Data-sharing is the desired default in the field of public health and a source of much ethical deliberation. Sharing data potentially contributes to the most efficient source of scientific data, but is fraught with contextual challenges which make stakeholders, particularly those in under-resourced contexts hesitant or slow to share <ref> Carrillo-Larco, Rodrigo M., J. Jaime Miranda, and Andre P. Kengne. ‘Data Pooling Efforts in Africa and Latin America’. The Lancet Global Health 5, no. 1 (1 January 2017): e37. https://doi.org/10.1016/S2214-109X(16)30297-2</ref> For example, the Global Body-Mass Index (BMI) Mortality Collaboration1 published their work on BMI as a predictor of all-cause mortality. The investigators pooled individual participant data from 239 prospective studies, with none originating from Latin America or Africa <ref>Body-mass index and all-cause mortality: individual-participant-data meta-analysis of 239 prospective studies in four continents. Lancet. 2016; 388: 776-786</ref>, <ref>Anane-Sarpong, Evelyn, Tenzin Wangmo, Claire Leonie Ward, Osman Sankoh, Marcel Tanner, and Bernice Simone Elger. ‘“You Cannot Collect Data Using Your Own Resources and Put It on Open Access”: Perspectives from Africa about Public Health Data-Sharing’. Developing World Bioethics 18, no. 4 (December 2018): 394–405. https://doi.org/10.1111/dewb.12159</ref> Also collaboration through OMEGA-NET will enhance the scientific output from individual studies and facilitate pooled studies, data sharing, and transfer of tools and skills to make greater and more efficient use of existing cohorts. Researchers from countries outside Europe can participate in COST Actions based on ascertained mutual benefit. Mehlum, Ingrid. <ref>1673f Network on the Coordination and Harmonisation of European Occupational Cohorts (Omega-Net). Occup Environ Med. Vol. 75, 2018. https://doi.org/10.1136/oemed-2018-ICOHabstracts.356.</ref> ==Links to relevant organizations, documents, and funds == [http://www.icohweb.org/site/homepage.asp ICOH International Commission on Occupational Health] [http://www.icoh-epicoh.org/ EPICOH Scientific Committee on Epidemiology in Occupational Health] [[Wikipedia: European_Cooperation_in_Science_and_Technology|COST explained in Wikipedia]] [https://omeganetcohorts.eu/ The OMEGA-NET Cohorts and COST] [http://dimopex.eu/about/ DiMoPEx (CA 15129)] [https://omeganetcohorts.eu/resources/scientific-publications/ Links to OMEGA-NET Scientific Publications] [https://www.heraresearcheu.eu/ The HERA network for an environmental, climate and health research agenda] [https://www.cost.eu/who-we-are/mission-vision-and-values/ The COST mission vision and values] [https://www.europeansurveyresearch.org The European Survey Research Association] [https://www.eurofound.europa.eu/surveys/european-working-conditions-surveys-ewcs European Working Conditions Surveys (EWCS)] [https://ec.europa.eu/esf/main.jsp?catId=67&langId=en&newsId=9691 The European Social Fund] == https://www.fi-compass.eu/esif/emff<br> [https://www.norden.org/en/funding-opportunities/nordic-council-ministers-funding-programme-ngo-co-operation-baltic-sea-region Nordic Council Ministers Funding NGO Co-Operation Baltic Sea Region] <br> [https://www.norden.org/en/information/about-funding-nordic-council-ministers Funding Nordic Council Ministers]<br> [https://www.norden.org/en/funding-opportunities/nordic-council-ministers-open-call-funding-opportunity-nordic-russian-co Nordic Council Ministers Funding-opportunity Nordic-Russian Co-Operation] <br> [https://www.seafarerstrust.org/about/ ITF Seafarers Trust]<br> [https://terravivagrants.org/grant-makers/cross-cutting/nippon-foundation/ Nippon Foundation]<br> [https://novonordiskfonden.dk/en/ Novo Nordisk Found] ==References== iusz3v0qtii7druygtah8dzcyq2upj6 0 269289 2416317 2413033 2022-08-18T22:04:44Z Maxim Masiutin 2902665 /* Future Directions */ wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref><ref name="pmid23386646" /> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405" /> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939"/><ref name="Nishiyama2011"/><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706" /> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity.<ref name="pmid18308097" /> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} te8qt9ve2q4u9ivxe7q0yrtnse79guo 2416320 2416317 2022-08-18T22:14:13Z Maxim Masiutin 2902665 /* Future Directions */ added DOIs to Urology published in 2022 wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref><ref name="pmid23386646" /> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405" /> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939"/><ref name="Nishiyama2011"/><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706" /> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, the article had a discrepancy in steroid levels reported for the control subjects. <ref>{{Cite journal|last=Masiutin|first=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|volume=0|issue=0|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last=Dimitrakoff|first=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|volume=0|issue=0|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We therefore hypothesize that that 5α-dihydrotestosterone produced by the “backdoor” pathway and 11-oxygenated androgens may be relevant in CP/CPPS. ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} lwvforbkdfojzp2zjhjzbf52vbdlwxk 2416321 2416320 2022-08-18T22:15:25Z Maxim Masiutin 2902665 /* Future Directions */ wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref><ref name="pmid23386646" /> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405" /> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939"/><ref name="Nishiyama2011"/><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706" /> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, the article had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last=Masiutin|first=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|volume=0|issue=0|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last=Dimitrakoff|first=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|volume=0|issue=0|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} l4ix85y6kaumgqj4on7oh75sp9e8uug 2416322 2416321 2022-08-18T22:16:33Z Maxim Masiutin 2902665 /* Future Directions */ wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref><ref name="pmid23386646" /> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405" /> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939"/><ref name="Nishiyama2011"/><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706" /> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last=Masiutin|first=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|volume=0|issue=0|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last=Dimitrakoff|first=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|volume=0|issue=0|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} 62wv0qwgp2vm7wpkzptc4v3ajhdnb4h 2416362 2416322 2022-08-18T23:54:39Z Maxim Masiutin 2902665 fixed Cite error: Invalid <ref> tag; no text was provided for refs named pmid32203405 wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref><ref name="pmid23386646" /> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939"/><ref name="Nishiyama2011"/><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706" /> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last=Masiutin|first=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|volume=0|issue=0|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last=Dimitrakoff|first=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|volume=0|issue=0|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} 6grtp0isqsntgmhe6zr6gttu77m6ufq 2416363 2416362 2022-08-18T23:55:43Z Maxim Masiutin 2902665 wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref><ref name="pmid23386646" /> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939">{{cite journal|last1=Dozmorov|first1=Mikhail G.|last2=Yang|first2=Qing|last3=Matwalli|first3=Adam|last4=Hurst|first4=Robert E.|last5=Culkin|first5=Daniel J.|last6=Kropp|first6=Bradley P.|last7=Lin|first7=Hsueh-Kung|year=2007|title=5alpha-androstane-3alpha,17beta-diol selectively activates the canonical PI3K/AKT pathway: a bioinformatics-based evidence for androgen-activated cytoplasmic signaling|journal=Genomic Med|volume=1|issue=3–4|pages=139–46|doi=10.1007/s11568-008-9018-9|pmc=2269037|pmid=18923939}}</ref><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706" /> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last=Masiutin|first=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|volume=0|issue=0|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last=Dimitrakoff|first=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|volume=0|issue=0|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} 21lgquxis3cyvl7vqyuto3s47rq0pxi 2416364 2416363 2022-08-18T23:59:47Z Maxim Masiutin 2902665 wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref><ref name="pmid23386646" /> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939">{{cite journal|last1=Dozmorov|first1=Mikhail G.|last2=Yang|first2=Qing|last3=Matwalli|first3=Adam|last4=Hurst|first4=Robert E.|last5=Culkin|first5=Daniel J.|last6=Kropp|first6=Bradley P.|last7=Lin|first7=Hsueh-Kung|year=2007|title=5alpha-androstane-3alpha,17beta-diol selectively activates the canonical PI3K/AKT pathway: a bioinformatics-based evidence for androgen-activated cytoplasmic signaling|journal=Genomic Med|volume=1|issue=3–4|pages=139–46|doi=10.1007/s11568-008-9018-9|pmc=2269037|pmid=18923939}}</ref><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706" /> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} kfa7nceehv4v2ebc4c2nuy7g0kwgca5 2416365 2416364 2022-08-19T00:05:04Z Maxim Masiutin 2902665 removed irrelevant reference wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939">{{cite journal|last1=Dozmorov|first1=Mikhail G.|last2=Yang|first2=Qing|last3=Matwalli|first3=Adam|last4=Hurst|first4=Robert E.|last5=Culkin|first5=Daniel J.|last6=Kropp|first6=Bradley P.|last7=Lin|first7=Hsueh-Kung|year=2007|title=5alpha-androstane-3alpha,17beta-diol selectively activates the canonical PI3K/AKT pathway: a bioinformatics-based evidence for androgen-activated cytoplasmic signaling|journal=Genomic Med|volume=1|issue=3–4|pages=139–46|doi=10.1007/s11568-008-9018-9|pmc=2269037|pmid=18923939}}</ref><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706" /> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} hsw8akywzp1znne7cs97s5xl9t0rgx2 2416366 2416365 2022-08-19T00:09:27Z Maxim Masiutin 2902665 removed irrelevant reference wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |pages=431 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|pages=868|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|pages=66|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=441 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-), therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939">{{cite journal|last1=Dozmorov|first1=Mikhail G.|last2=Yang|first2=Qing|last3=Matwalli|first3=Adam|last4=Hurst|first4=Robert E.|last5=Culkin|first5=Daniel J.|last6=Kropp|first6=Bradley P.|last7=Lin|first7=Hsueh-Kung|year=2007|title=5alpha-androstane-3alpha,17beta-diol selectively activates the canonical PI3K/AKT pathway: a bioinformatics-based evidence for androgen-activated cytoplasmic signaling|journal=Genomic Med|volume=1|issue=3–4|pages=139–46|doi=10.1007/s11568-008-9018-9|pmc=2269037|pmid=18923939}}</ref><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} 8uh1p3mdywvxssegveurd4k64is6613 2416371 2416366 2022-08-19T00:21:02Z Maxim Masiutin 2902665 added proper reference for "oxy" wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRDA51 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid18923939">{{cite journal|last1=Dozmorov|first1=Mikhail G.|last2=Yang|first2=Qing|last3=Matwalli|first3=Adam|last4=Hurst|first4=Robert E.|last5=Culkin|first5=Daniel J.|last6=Kropp|first6=Bradley P.|last7=Lin|first7=Hsueh-Kung|year=2007|title=5alpha-androstane-3alpha,17beta-diol selectively activates the canonical PI3K/AKT pathway: a bioinformatics-based evidence for androgen-activated cytoplasmic signaling|journal=Genomic Med|volume=1|issue=3–4|pages=139–46|doi=10.1007/s11568-008-9018-9|pmc=2269037|pmid=18923939}}</ref><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} dkgooi2amduaqsbw9hzto6tn6iqkl2x 2416373 2416371 2022-08-19T00:31:34Z Maxim Masiutin 2902665 removed irrelevant references and put back the relevant ones wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref><ref name="pmid9183566"/> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT<ref name="pmid21795608"/>}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} rp7h65n5gzjfrh4nbbojh1yb5c2lwud 2416374 2416373 2022-08-19T00:33:05Z Maxim Masiutin 2902665 /* 5α-Dione Pathway */ wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens<ref name="pmid27442248" /><ref name="pmid32203405" /><ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref><ref name="pmid25869556" /> (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4, 17OHP, A4 and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Wether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While KDHT is equipotent to DHT, circulating levels of KDHT are lower than DHT and SRD5A1 mediated transformation of KT to KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure see in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} cv5bh901m9au3d7ciih8e8mmtk1rglg 2416377 2416374 2022-08-19T00:52:32Z Maxim Masiutin 2902665 removed review, left just primary research articles, to put them to relevant places wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4,<ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref> 17OHP, A4<ref name="pmid25869556" /><ref name="pmid27442248" /> and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Whether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does (though some evidence suggests that they do<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref>). While 11KDHT is equipotent to DHT, circulating levels of 11KDHT are lower than DHT and SRD5A1-mediated transformation of 11KT to 11KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure seen in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17-OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then 11KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} 8xgf8h2fik7iqz5zv9zcxni0fk9uhaf 2416379 2416377 2022-08-19T00:53:58Z Maxim Masiutin 2902665 /* 11-Oxygenated Androgen Pathways */ wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4,<ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref> 17OHP, A4<ref name="pmid25869556" /><ref name="pmid27442248" /> and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Whether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does, though some evidence suggests that they do.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> While 11KDHT is equipotent to DHT, circulating levels of 11KDHT are lower than DHT and SRD5A1-mediated transformation of 11KT to 11KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure seen in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17-OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then 11KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} qo4z0h5376sl1mx7nv68fw97lo7syx9 2416380 2416379 2022-08-19T00:55:24Z Maxim Masiutin 2902665 /* 11-Oxygenated Androgen Pathways */ wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4,<ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref> 17OHP, A4<ref name="pmid25869556" /><ref name="pmid27442248" /> and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Whether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does, though some evidence suggests that 11KT does decline.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> While 11KDHT is equipotent to DHT, circulating levels of 11KDHT are lower than DHT and SRD5A1-mediated transformation of 11KT to 11KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure seen in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17-OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then 11KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> . ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} 9n0ocqltfiz8dqdcg5g61miyf7f0i0w 2416381 2416380 2022-08-19T00:56:21Z Maxim Masiutin 2902665 /* 11-Oxygenated Androgen Pathways */ wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4,<ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref> 17OHP, A4<ref name="pmid25869556" /><ref name="pmid27442248" /> and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Whether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does, though some evidence suggests that 11KT does decline.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> While 11KDHT is equipotent to DHT, circulating levels of 11KDHT are lower than DHT and SRD5A1-mediated transformation of 11KT to 11KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure seen in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2.<ref name="pmid23685396" /><ref name="Haru1980">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17-OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4, an important circulating androgen precursor, which is further transformed to 11KA4 and then 11KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT.<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} s4our44ycyczsxo97ji76w0fo92amdw 2416382 2416381 2022-08-19T01:09:34Z Maxim Masiutin 2902665 all these studies are about the fact that A4 can be converted by CYP11B1/B2 wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4,<ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref> 17OHP, A4<ref name="pmid25869556" /><ref name="pmid27442248" /> and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Whether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does, though some evidence suggests that 11KT does decline.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> While 11KDHT is equipotent to DHT, circulating levels of 11KDHT are lower than DHT and SRD5A1-mediated transformation of 11KT to 11KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure seen in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2. 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17-OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17-OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4,<ref name="pmid6970302">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref> <ref name="pmid23685396" /> an important circulating androgen precursor, which is further transformed to 11KA4 and then 11KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT.<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} sev81twhj7rw3r76ejjkz6g8jvurzmm 2416383 2416382 2022-08-19T01:10:07Z Maxim Masiutin 2902665 removed CRLF wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4,<ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref> 17OHP, A4<ref name="pmid25869556" /><ref name="pmid27442248" /> and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Whether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does, though some evidence suggests that 11KT does decline.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> While 11KDHT is equipotent to DHT, circulating levels of 11KDHT are lower than DHT and SRD5A1-mediated transformation of 11KT to 11KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure seen in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2. 2. 5α-reduction by SRD5A1/2 3. Reversible 3α-reduction/oxidation of the ketone/alcohol These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17-OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17-OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4,<ref name="pmid6970302">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref><ref name="pmid23685396" /> an important circulating androgen precursor, which is further transformed to 11KA4 and then 11KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT.<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> ==== C₂₁ Steroid Entry Points ==== Currently there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} 0y1jkaq5asdb8d8aqsh921p6kmftq7o 2416386 2416383 2022-08-19T01:22:04Z Maxim Masiutin 2902665 Jargon wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4,<ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref> 17OHP, A4<ref name="pmid25869556" /><ref name="pmid27442248" /> and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Whether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does, though some evidence suggests that 11KT does decline.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> While 11KDHT is equipotent to DHT, circulating levels of 11KDHT are lower than DHT and SRD5A1-mediated transformation of 11KT to 11KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure seen in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2, 2. 5α-reduction by SRD5A1/2, 3. reversible 3α-reduction/oxidation of the ketone/alcohol.{{Clarify}} These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol{{Clarify}} (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol{{Clarify}} to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17-OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17-OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4,<ref name="pmid6970302">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref><ref name="pmid23685396" /> an important circulating androgen precursor, which is further transformed to 11KA4 and then 11KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT.<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> ==== C₂₁ Steroid Entry Points ==== Currently, there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body''.''<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> <ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} 1wykebw8b8s2izgr58gpeprm0ep0to2 2416387 2416386 2022-08-19T01:26:38Z Maxim Masiutin 2902665 removed irrelevant study wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725" /> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4,<ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref> 17OHP, A4<ref name="pmid25869556" /><ref name="pmid27442248" /> and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Whether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does, though some evidence suggests that 11KT does decline.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> While 11KDHT is equipotent to DHT, circulating levels of 11KDHT are lower than DHT and SRD5A1-mediated transformation of 11KT to 11KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure seen in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2, 2. 5α-reduction by SRD5A1/2, 3. reversible 3α-reduction/oxidation of the ketone/alcohol.{{Clarify}} These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol{{Clarify}} (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol{{Clarify}} to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17-OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17-OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4,<ref name="pmid6970302">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref><ref name="pmid23685396" /> an important circulating androgen precursor, which is further transformed to 11KA4 and then 11KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT.<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> ==== C₂₁ Steroid Entry Points ==== Currently, there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body.<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} j9a8qtpu3x67tofz37xbcgslyz6f6ze 2416388 2416387 2022-08-19T01:27:09Z Maxim Masiutin 2902665 wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4,<ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref> 17OHP, A4<ref name="pmid25869556" /><ref name="pmid27442248" /> and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Whether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does, though some evidence suggests that 11KT does decline.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> While 11KDHT is equipotent to DHT, circulating levels of 11KDHT are lower than DHT and SRD5A1-mediated transformation of 11KT to 11KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure seen in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2, 2. 5α-reduction by SRD5A1/2, 3. reversible 3α-reduction/oxidation of the ketone/alcohol.{{Clarify}} These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol{{Clarify}} (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol{{Clarify}} to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17-OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17-OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4,<ref name="pmid6970302">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref><ref name="pmid23685396" /> an important circulating androgen precursor, which is further transformed to 11KA4 and then 11KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT.<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> ==== C₂₁ Steroid Entry Points ==== Currently, there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body.<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> In a 2021 study, However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} gi34zugck9w9unrn2c85w04fustytcs 2416389 2416388 2022-08-19T01:28:22Z Maxim Masiutin 2902665 /* 11-Oxygenated Androgens */ wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4,<ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref> 17OHP, A4<ref name="pmid25869556" /><ref name="pmid27442248" /> and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Whether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does, though some evidence suggests that 11KT does decline.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> While 11KDHT is equipotent to DHT, circulating levels of 11KDHT are lower than DHT and SRD5A1-mediated transformation of 11KT to 11KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure seen in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2, 2. 5α-reduction by SRD5A1/2, 3. reversible 3α-reduction/oxidation of the ketone/alcohol.{{Clarify}} These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol{{Clarify}} (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol{{Clarify}} to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17-OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17-OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4,<ref name="pmid6970302">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref><ref name="pmid23685396" /> an important circulating androgen precursor, which is further transformed to 11KA4 and then 11KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT.<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> ==== C₂₁ Steroid Entry Points ==== Currently, there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body.<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} 1lamyqk1ljg4g61qn8zh7boy57r7ziz 2416390 2416389 2022-08-19T01:29:38Z Maxim Masiutin 2902665 /* 11-Oxygenated Androgens */ wikitext text/x-wiki {{Article info | first1 = Maxim G | last1 = Masiutin | orcid1 = 0000-0002-8129-4500 | correspondence1 = maxim@masiutin.com | first2 = Maneesh K | last2 = Yadav | orcid2 = 0000-0002-4584-7606 | submitted = 4/22/2022 | contributors = | et_al = <!-- * The Wikipedia source page was https://en.wikipedia.org/wiki/Androgen_backdoor_pathway * No other people except the authors of the present article have contributed to the source page until this article was forked from that page on October 22, 2020 * When I added the "w1" attribute to the "Article info" box, the "et al." appears. The "et_al = false" attribute does not seem to work. There should be no "et al.". I have not found any way to remove the "et al." rather than removing the "w1" attribute. * Only when I remove both the "w1" attribute here and the link to Wikipedia entry in the Wikidate item, the "et al." disappears. | et_al = false | w1 = Androgen backdoor pathway --> | correspondence = | journal = WikiJournal of Medicine | license = | abstract = The term "backdoor pathway" is sometimes used to specify different androgen steroidogenic pathways that avoid testosterone as an intermediate product. Although the term was initially defined as a metabolic route by which the 5α-reduction of 17α-hydroxyprogesterone ultimately leads to 5α-dihydrotestosterone, several other routes towards potent androgens have been discovered, which are also described as backdoor pathways. Some of the routes lead to 11-oxygenated androgens that are clinically relevant agonists of the androgen receptor. This review aims to provide a clear, comprehensive description that includes all currently known metabolic routes. Patient comprehension and the clinical diagnosis of relevant conditions such as hyperandrogenism can be impaired by the lack of clear and consistent knowledge of alternative androgen pathways; the authors hope this review will accurately disseminate such knowledge to facilitate the beneficial treatment of such patients. | keywords = testosterone, 11-oxygenated androgen, 11-oxyandrogen, 11-ketotestosterone, hyperandrogenism }} ==Introduction== The classical view of androgen steroidogenesis involves the combination of adrenal and gonadal pathways that convert cholesterol to the androgen testosterone (T), which in turn converts to the potent androgen 5α-dihydrotestosterone (DHT). Broadly, androgens are understood to exert their primary effects through binding to cytosolic Androgen Receptor (AR) which is translocated to the nucleus upon androgen binding and ultimately results in the transcriptional regulation of a number of genes via Androgen Responsive Elements.<ref name="pmid12089231">{{Cite journal|last=Gelmann|first=Edward P.|year=2022|title=Molecular Biology of the Androgen Receptor|url=https://ascopubs.org/doi/10.1200/JCO.2002.10.018|journal=Journal of Clinical Oncology|language=en|volume=20|issue=13|pages=3001–3015|doi=10.1200/JCO.2002.10.018|pmid=12089231 |issn=0732-183X}}</ref> In 2003, a metabolic route to DHT that did not proceed through T was discovered in the tammar wallaby.<ref name="pmid12538619">{{cite journal|last1=Wilson|first1=Jean D.|last2=Auchus|first2=Richard J.|last3=Leihy|first3=Michael W.|last4=Guryev|first4=Oleg L.|last5=Estabrook|first5=Ronald W.|last6=Osborn|first6=Susan M.|last7=Shaw|first7=Geoffrey|last8=Renfree|first8=Marilyn B.|title=5alpha-androstane-3alpha,17beta-diol is formed in tammar wallaby pouch young testes by a pathway involving 5alpha-pregnane-3alpha,17alpha-diol-20-one as a key intermediate|journal=Endocrinology|year=2003 |volume=144|issue=2|pages=575–80|doi=10.1210/en.2002-220721|pmid=12538619|s2cid=84765868}}</ref> Shortly after this study, it was hypothesized that human steroidogenic enzymes are capable of catalyzing this pathway and the potential clinical relevance in conditions involving androgen biosynthesis was proposed.<ref name="pmid15519890">{{cite journal|last1=Auchus|first1=Richard J.|year=2004|title=The backdoor pathway to dihydrotestosterone|journal=Trends in Endocrinology and Metabolism: TEM|volume=15|issue=9|pages=432–8|doi=10.1016/j.tem.2004.09.004|pmid=15519890|s2cid=10631647}}</ref> Since then, steroidogenic androgen pathways to potent 11-oxygenated androgens have also been discovered and proposed as clinically relevant.<ref name="pmid27519632">{{cite journal |title=A new dawn for androgens: Novel lessons from 11-oxygenated C19 steroids |journal=Mol Cell Endocrinol |volume=441 |pages=76–85 |year=2017 |pmid=27519632 |doi=10.1016/j.mce.2016.08.014|last1=Pretorius |first1=Elzette |last2=Arlt |first2=Wiebke |last3=Storbeck |first3=Karl-Heinz |s2cid=4079662 |url=http://pure-oai.bham.ac.uk/ws/files/30346231/Pretorius_et_al_manuscript.pdf }}</ref> The discovery of these "alternative androgen pathways" can confound the search for clinical information when androgen steroidogenesis is relevant. Studies across different androgen pathways have also, confusingly, used different names for the same metabolic intermediates. In addition, pathways in studies sometimes differ in the precise initial/terminal molecules and the inclusion/exclusion of such points can hinder queries in electronic pathway databases. Alternative androgen pathways are now known to be responsible for the production of biologically active androgens in humans, and there is growing evidence that they play a role in clinical conditions associated with hyperandrogenism. While naming inconsistencies are notoriously common when it comes to biomolecules,<ref name="pmid30736318">{{cite journal|last1=Pham|first1=Nhung|last2=van Heck|first2=Ruben G. A.|last3=van Dam|first3=Jesse C. J.|last4=Schaap|first4=Peter J.|last5=Saccenti|first5=Edoardo|last6=Suarez-Diez|first6=Maria|year=2019|title=Consistency, Inconsistency, and Ambiguity of Metabolite Names in Biochemical Databases Used for Genome-Scale Metabolic Modelling|journal=Metabolites|volume=9|issue=2|page=28|doi=10.3390/metabo9020028|issn=2218-1989|pmc=6409771|pmid=30736318|doi-access=free}}</ref> understanding androgen steroidogenesis at the level of detail presented in this paper and establishing consensus names and pathway specifications would facilitate access to information towards diagnosis and patient comprehension. ==History== === Backdoor Pathways to 5α-Dihydrotestosterone === In 1987, Eckstein et al. incubated rat testicular microsomes in presence of radiolabeled steroids and demonstrated that 5α-androstane-3α,17β-diol can be produced in immature rat testes from progesterone (P4), 17α-hydroxyprogesterone (17-OHP) and androstenedione (A4) but preferentially from 17-OHP.<ref name="pmid3828389">{{cite journal|last1=Eckstein|first1=B.|last2=Borut|first2=A.|last3=Cohen|first3=S.|title=Metabolic pathways for androstanediol formation in immature rat testis microsomes|journal=Biochimica et Biophysica Acta (BBA) - General Subjects |year=1987 |url=https://pubmed.ncbi.nlm.nih.gov/3828389|volume=924|issue=1|pages=1–6|doi=10.1016/0304-4165(87)90063-8|issn=0006-3002|pmid=3828389}}</ref> While "androstanediol" was used to denote both 5α-androstane-3α,17β-diol and 5α-androstane-3β,17β-diol, we use "3α-diol" to abbreviate 5α-androstane-3α,17β-diol in this paper as it is a common convention and emphasizes it as the 3α-reduced derivative of DHT. Tammar wallaby pouch young do not show sexually dimorphic circulating levels of T and DHT during prostate development, which led Shaw et al. to hypothesize in 2000 that another pathway was responsible for AR activation in this species.<ref name="pmid11035809" /> While 3α-diol has a reduced AR binding affinity relative to DHT by 5 orders of magnitude and is generally described as AR inactive, it was known 3α-diol can be oxidized back to DHT via the action of a number of dehydrogenases.<ref name="pmid9183566">{{Cite journal|last=Penning|first=Trevor M.|year=1997|title=Molecular Endocrinology of Hydroxysteroid Dehydrogenases| url=https://academic.oup.com/edrv/article/18/3/281/2530742|journal=Endocrine Reviews|language=en|volume=18|issue=3|pages=281–305|doi=10.1210/edrv.18.3.0302|pmid=9183566 |s2cid=29607473 |issn=0163-769X}}</ref> Shaw et al. showed that prostate formation in these wallaby is caused by circulating 3α-diol (generated in the testes) and led to their prediction that 3α-diol acts in target tissues via conversion to DHT.<ref name="pmid11035809">{{cite journal|last1=Shaw|first1=G.|last2=Renfree|first2=M. B.|last3=Leihy|first3=M. W.|last4=Shackleton|first4=C. H.|last5=Roitman|first5=E.|last6=Wilson|first6=J. D.|year=2000|title=Prostate formation in a marsupial is mediated by the testicular androgen 5 alpha-androstane-3 alpha,17 beta-diol|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=97|issue=22|pages=12256–12259|bibcode=2000PNAS...9712256S|doi=10.1073/pnas.220412297|issn=0027-8424|pmc=17328|pmid=11035809|doi-access=free}}</ref> In 2003, Wilson et al. incubated the testes of tammar wallaby pouch young with radiolabeled progesterone to show that 5α reductase expression in this tissue enabled a novel pathway from 17-OHP to 3α-diol without T as an intermediate:<ref name="pmid12538619" />{{unbulleted list|<small>17α-hydroxyprogesterone (17OHP) → 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The authors hypothesized that a high level of 5α-reductase in the virilizing wallaby testes causes most C₁₉ steroids to be 5α-reduced to become ready DHT precursors. In 2004, Mahendroo et al. demonstrated that an overlapping novel pathway is operating in mouse testes, generalizing what had been demonstrated in tammar wallaby:<ref name="pmid15249131">{{cite journal|last1=Mahendroo|first1=Mala|last2=Wilson|first2=Jean D.|last3=Richardson|first3=James A.|last4=Auchus|first4=Richard J.|year=2004|title=Steroid 5alpha-reductase 1 promotes 5alpha-androstane-3alpha,17beta-diol synthesis in immature mouse testes by two pathways|url=https://pubmed.ncbi.nlm.nih.gov/15249131|journal=Molecular and Cellular Endocrinology|volume=222|issue=1–2|pages=113–120|doi=10.1016/j.mce.2004.04.009|issn=0303-7207|pmid=15249131|s2cid=54297812}}</ref>{{unbulleted list|<small>progesterone (P4) → 5α-dihydroprogesterone (5α-DHP) → 5α-pregnan-3α-ol-20-one (AlloP5)→ 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol)</small>}} The term "backdoor pathway" was coined by Auchus in 2004<ref name="pmid15519890" /> where it was defined as a route to DHT that: (1) bypasses conventional intermediates A4 and T; (2) involves 5α-reduction of the 21-carbon precursors (pregnanes) to 19-carbon products (androstanes) and (3) involves the 3α-oxidation of 3α-diol to DHT. This alternative pathway seems to explain how potent androgens are produced under certain normal and pathological conditions in humans when the canonical androgen biosynthetic pathway cannot fully explain the observed consequences. The pathway was described as:{{unbulleted list|<small>17α-hydroxyprogesterone (17-OHP) → 17-OH-DHP (5α-pregnan-17α-ol-3,20-dione) → 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) → 5α-androstan-3α-ol-17-one (AST) → 5α-androstane-3α,17β-diol (3α-diol) → 5α-dihydrotestosterone (DHT)</small>}} The clinical relevance of these results was demonstrated in 2012 for the first time when Kamrath et al. attributed the urinary metabolites to the androgen backdoor pathway from 17-OHP to DHT in patients with steroid 21-hydroxylase (CYP21A2) deficiency.<ref name="pmid22170725">{{cite journal|last1=Kamrath|first1=Clemens|last2=Hochberg|first2=Ze'ev|last3=Hartmann|first3=Michaela F.|last4=Remer|first4=Thomas|last5=Wudy|first5=Stefan A.|title=Increased activation of the alternative "backdoor" pathway in patients with 21-hydroxylase deficiency: evidence from urinary steroid hormone analysis|url=https://pubmed.ncbi.nlm.nih.gov/22170725|journal=The Journal of Clinical Endocrinology and Metabolism|year=2012 |volume=97|issue=3|pages=E367–375|doi=10.1210/jc.2011-1997|issn=1945-7197|pmid=22170725|s2cid=3162065 }}</ref> === 5α-Dione Pathway === In 2011, Chang et al. demonstrated that an alternative pathway to DHT was dominant and possibly essential in castration-resistant prostate cancer (CRPC) by presenting evidence from cell culture and xenograft models:<ref name="pmid21795608">{{cite journal|last1=Chang|first1=K.-H.|last2=Li|first2=R.|last3=Papari-Zareei|first3=M.|last4=Watumull|first4=L.|last5=Zhao|first5=Y. D.|last6=Auchus|first6=R. J.|last7=Sharifi|first7=N.|year=2011|title=Dihydrotestosterone synthesis bypasses testosterone to drive castration-resistant prostate cancer|journal=Proceedings of the National Academy of Sciences of the United States of America|publisher=Proceedings of the National Academy of Sciences|volume=108|issue=33|pages=13728–13733|bibcode=2011PNAS..10813728C|doi=10.1073/pnas.1107898108|issn=0027-8424|pmc=3158152|pmid=21795608|doi-access=free}}</ref>{{unbulleted list|<small>androstenedione (A4) → androstanedione (5α-dione) → 5α-dihydrotestosterone (DHT)</small>}} While this pathway was described as the "5α-dione pathway" in a 2012 review,<ref name="pmid22064602">{{cite journal |title=The 5α-androstanedione pathway to dihydrotestosterone in castration-resistant prostate cancer |journal=J Investig Med |volume=60 |issue=2 |pages=504–7 |year=2012 |pmid=22064602 |pmc=3262939 |doi=10.2310/JIM.0b013e31823874a4 |last1=Sharifi |first1=Nima }}</ref> the existence of such a pathway in the prostate was hypothesized in a 2008 review by Luu-The et al.<ref name="pmid18471780">{{cite journal|last1=Luu-The|first1=Van|last2=Bélanger|first2=Alain|last3=Labrie|first3=Fernand|year=2008|title=Androgen biosynthetic pathways in the human prostate|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=22|issue=2|pages=207–221|doi=10.1016/j.beem.2008.01.008|issn=1521-690X|pmid=18471780}}</ref> A modern outlook of the synthesis of the backdoor pathways to DHT and the 5α-dione pathway is shown in Figure 2. === 11-Oxygenated Androgen Pathways === 11-Oxygenated androgens are the products of another alternative androgen pathway found in humans. While the 11-oxygenated C₁₉ steroids 11OHA4 and 11KA4 were known since the 1950s to be products of the human adrenal, with negligible androgenic activity, but their role as substrates to potent androgens had been overlooked in humans though they were known to be the main androgens in teleost fishes.<ref name="pmid27519632" /> Rege et al. in 2013 measured 11-oxygenated androgens in healthy women and showed the 11-ketodihydrotestosterone (11KT) and 11β-hydroxytestosterone (11OHT) activation of human AR.<ref name="pmid23386646">{{cite journal|last1=Rege|first1=Juilee|last2=Nakamura|first2=Yasuhiro|last3=Satoh|first3=Fumitoshi|last4=Morimoto|first4=Ryo|last5=Kennedy|first5=Michael R.|last6=Layman|first6=Lawrence C.|last7=Honma|first7=Seijiro|last8=Sasano|first8=Hironobu|last9=Rainey|first9=William E.|year=2013|title=Liquid chromatography-tandem mass spectrometry analysis of human adrenal vein 19-carbon steroids before and after ACTH stimulation|journal=J Clin Endocrinol Metab|volume=98|issue=3|pages=1182–8|doi=10.1210/jc.2012-2912|pmc=3590473|pmid=23386646}}</ref> In 2013, Storbeck et al. demonstrated the existence of 11-oxygenated androgen pathways in androgen-dependent prostate cancer cell culture.<ref name="pmid23856005">{{cite journal|title=11β-Hydroxydihydrotestosterone and 11-ketodihydrotestosterone, novel C19 steroids with androgenic activity: a putative role in castration resistant prostate cancer? |journal=Mol Cell Endocrinol |volume=377 |issue=1–2 |pages=135–46 |pmid=23856005 |doi=10.1016/j.mce.2013.07.006 |s2cid=11740484 |last1=Storbeck |first1=Karl-Heinz |last2=Bloem |first2=Liezl M. |last3=Africander |first3=Donita |last4=Schloms |first4=Lindie |last5=Swart |first5=Pieter |last6=Swart |first6=Amanda C. |year=2013 }}</ref> The authors indicated that A4 is converted 1β-hydroxyandrostenedione (11OHA4) which can ultimately be converted into 11KT and 11KDHT as shown in Figure 4. The authors found that 11KT activity is comparable to that of T, and 11-ketodihydrotestosterone (11KDHT) activity is comparable to that of DHT, while the activities of 11OHT and 5α-dihydro-11β-hydroxytestosterone (11OHDHT) were observed to be about half of T and DHT, respectively. However, androgen activity in that study was only assessed at a single concentration of 1 nM.<ref name="pmid23856005" /> To confirm androgen activity of 11KT and 11KDHT, a study by Pretorius et al. performing full dose responses showed in 2016 that 11KT and 11KDHT both bind and activate the human AR with affinities, potencies, and efficacies that are similar to that of T and DHT, respectively.<ref name="pmid27442248">{{cite journal|last1=Pretorius|first1=Elzette|last2=Africander|first2=Donita J.|last3=Vlok|first3=Maré|last4=Perkins|first4=Meghan S.|last5=Quanson|first5=Jonathan|last6=Storbeck|first6=Karl-Heinz|year=2016|title=11-Ketotestosterone and 11-Ketodihydrotestosterone in Castration Resistant Prostate Cancer: Potent Androgens Which Can No Longer Be Ignored|journal=PLOS ONE|volume=11|issue=7|pages=e0159867|doi=10.1371/journal.pone.0159867|pmc=4956299|pmid=27442248|doi-access=free}}</ref> These findings were later confirmed in 2021<ref name="pmid34990809">{{cite journal|last1=Handelsman|first1=David J.|last2=Cooper|first2=Elliot R.|last3=Heather|first3=Alison K.|year=2022|title=Bioactivity of 11 keto and hydroxy androgens in yeast and mammalian host cells|journal=J Steroid Biochem Mol Biol|volume=218|issue=|pages=106049|doi=10.1016/j.jsbmb.2021.106049|pmid=34990809|s2cid=245635429}}</ref> and 2022.<ref name="pmid35046557">{{cite journal|last1=Snaterse|first1=Gido|last2=Mies|first2=Rosinda|last3=Van Weerden|first3=Wytske M.|last4=French|first4=Pim J.|last5=Jonker|first5=Johan W.|last6=Houtsmuller|first6=Adriaan B.|last7=Van Royen|first7=Martin E.|last8=Visser|first8=Jenny A.|last9=Hofland|first9=Johannes|year=2022|title=Androgen receptor mutations modulate activation by 11-oxygenated androgens and glucocorticoids|url=https://pure.eur.nl/ws/files/48975803/s41391_022_00491_z.pdf|journal=Prostate Cancer Prostatic Dis|doi=10.1038/s41391-022-00491-z|pmid=35046557|s2cid=246040148}}</ref> Bloem et al. in 2015<ref name="pmid25869556">{{cite journal|last1=Bloem|first1=Liezl M.|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=du Toit|first4=Therina|last5=Schloms|first5=Lindie|last6=Swart|first6=Amanda C.|year=2015|title=Advances in the analytical methodologies: Profiling steroids in familiar pathways-challenging dogmas|url=https://pubmed.ncbi.nlm.nih.gov/25869556|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=153|pages=80–92|doi=10.1016/j.jsbmb.2015.04.009|issn=1879-1220|pmid=25869556|s2cid=31332668}}</ref> demonstrated that androgen pathways towards those 11-keto and 11β-hydroxy androgens can bypass A4 and T to produce 11KDHT in pathways similar to a backdoor pathway to DHT. This similarity led to the description of pathways from P4 and 17OHP to 11-oxyandrogens as "backdoor" pathways,<ref name="pmid25869556" /> which was further characterized in subsequent studies as contributing to active and biologically relevant androgens.<ref name="pmid28774496">{{cite journal|last1=Barnard|first1=Lise|last2=Gent|first2=Rachelle|last3=Van Rooyen|first3=Desmaré|last4=Swart|first4=Amanda C.|year=2017|title=Adrenal C11-oxy C21 steroids contribute to the C11-oxy C19 steroid pool via the backdoor pathway in the biosynthesis and metabolism of 21-deoxycortisol and 21-deoxycortisone|url=https://www.sciencedirect.com/science/article/abs/pii/S0960076017302091|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=174|pages=86–95|doi=10.1016/j.jsbmb.2017.07.034|pmid=28774496|s2cid=24071400}}</ref><ref name="pmid29277707">{{cite journal|last1=van Rooyen|first1=Desmaré|last2=Gent|first2=Rachelle|last3=Barnard|first3=Lise|last4=Swart|first4=Amanda C.|year=2018|title=The in vitro metabolism of 11β-hydroxyprogesterone and 11-ketoprogesterone to 11-ketodihydrotestosterone in the backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=178|pages=203–212|doi=10.1016/j.jsbmb.2017.12.014|pmid=29277707|s2cid=3700135}}</ref><ref name="pmid32007561">{{cite journal|last1=Van Rooyen|first1=Desmaré|last2=Yadav|first2=Rahul|last3=Scott|first3=Emily E.|last4=Swart|first4=Amanda C.|year=2020|title=CYP17A1 exhibits 17αhydroxylase/17,20-lyase activity towards 11β-hydroxyprogesterone and 11-ketoprogesterone metabolites in the C11-oxy backdoor pathway|journal=The Journal of Steroid Biochemistry and Molecular Biology|volume=199|pages=105614|doi=10.1016/j.jsbmb.2020.105614|pmid=32007561|s2cid=210955834}}</ref> A diagram of 11-oxygenated androgen steroidogenesis is shown in Figure 4. ==Definition== We suggest the term "alternative androgen pathway" to refer to any pathway that produces potent androgens without a T intermediate. This subsumes all three groups of androgen pathways described in the previous section. A new term that describes the three groups pathways (as well as future discoveries) will allow a single entry point into scientific information when alternatives to canonical<ref name="pmid30763313">{{cite journal|last1=O'Shaughnessy|first1=Peter J.|last2=Antignac|first2=Jean Philippe|last3=Le Bizec|first3=Bruno|last4=Morvan|first4=Marie-Line|last5=Svechnikov|first5=Konstantin|last6=Söder|first6=Olle|last7=Savchuk|first7=Iuliia|last8=Monteiro|first8=Ana|last9=Soffientini|first9=Ugo|year=2019|title=Alternative (backdoor) androgen production and masculinization in the human fetus|journal=PLOS Biology|volume=17|issue=2|pages=e3000002|doi=10.1371/journal.pbio.3000002|pmc=6375548|pmid=30763313|last10=Johnston|first10=Zoe C.|last11=Bellingham|first11=Michelle|last12=Hough|first12=Denise|last13=Walker|first13=Natasha|last14=Filis|first14=Panagiotis|last15=Fowler|first15=Paul A.|editor-last1=Rawlins|editor-first1=Emma}}</ref><ref name="pmid31900912" /> androgen pathway must be considered. ==Nomenclature and Background== Complex naming rules for organic chemistry lead to the use of incorrect steroid names in studies. The presence of incorrect names impairs the ability to query information about androgen pathways. Since we were able to find many examples of incorrect names for molecules referred to in this paper in Google Scholar searches<ref name="google-pregnan17diol" /><ref name="google-pregnane17ol" />, we have added this expository section on steroid nomenclature to facilitate the use of correct names. Almost all biologically relevant steroids can be presented as a derivative of a parent hydrocarbon structure. These parent structures have specific names, such as pregnane, androstane, etc. The derivatives carry various functional groups called suffixes or prefixes after the respective numbers indicating their position in the steroid nucleus.<ref name="pmid2606099-parent-elisions" /> The widely-used steroid names such as progesterone, testosterone or cortisol can also be used as base names to derive new names, however, by adding prefixes only rather than suffixes, e.g., the steroid 17α-hydroxyprogesterone has a hydroxy group (-OH) at position 17 of the steroid nucleus comparing to progesterone. The letters α and β<ref name="pmid2606099-rs">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |year=1989 |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=431|chapter=3S-1.4|quote=3S-1.4. Orientation of projection formulae When the rings of a steroid are denoted as projections onto the plane of the paper, the formula is normally to be oriented as in 2a. An atom or group attached to a ring depicted as in the orientation 2a is termed α (alpha) if it lies below the plane of the paper or β (beta) if it lies above the plane of the paper. }}</ref> denote absolute stereochemistry at chiral centers (a specific nomenclature distinct from the R/S convention<ref name="norc-rs">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-91|quote-page=868|quote=P-91.2.1.1 Cahn-Ingold-Prelog (CIP) stereodescriptors Some stereodescriptors described in the Cahn-Ingold-Prelog (CIP) priority system, called ‘CIP stereodescriptors’, are recommended to specify the configuration of organic compounds, as described and exemplified in this Chapter and applied in Chapters P-1 through P-8, and in the nomenclature of natural products in Chapter P-10. The following stereodescriptors are used as preferred stereodescriptors (see P-92.1.2): (a) ‘R’ and ‘S’, to designate the absolute configuration of tetracoordinate (quadriligant) chirality centers;}}</ref> of organic chemistry). In steroids drawn from the standard perspective used in this paper, α-bonds are depicted on figures as dashed wedges and β-bonds as solid wedges. The molecule "11-deoxycortisol" is an example of a derived name that uses cortisol as a parent structure without an oxygen atom (hence "deoxy") attached to position 11 (as a part of a hydroxy group).<ref name="norc-deoxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-13.8.1.1|quote-page=66|quote=P-13.8.1.1 The prefix ‘de’ (not ‘des’), followed by the name of a group or atom (other than hydrogen), denotes removal (or loss) of that group and addition of the necessary hydrogen atoms, i.e., exchange of that group with hydrogen atoms. As an exception, ‘deoxy’, when applied to hydroxy compounds, denotes the removal of an oxygen atom from an –OH group with the reconnection of the hydrogen atom. ‘Deoxy’ is extensively used as a subtractive prefix in carbohydrate nomenclature (see P-102.5.3).}}</ref> The numbering of positions of carbon atoms in the steroid nucleus is set in a template found in the Nomenclature of Steroids<ref name="pmid2606099-numbering">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=430|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=3S-1.l. Numbering and ring letters Steroids are numbered and rings are lettered as in formula 1|quote-page=430}}</ref> that is used regardless of whether an atom is present in the steroid in question. Although the nomenclature defines more than 30 positions, we need just positions up to 21 for the steroids described here (see Figure 1). [[File:steroid-numbering-to-21-opt.svg|thumb|Numbering of carbon atoms up to position 21 (positions 18 and 19 are omitted) in a hypothetical steroid nucleus, as defined by the Nomenclature of Steroids]] Unsaturation (presence of double bonds between carbon atoms in the steroid nucleus) is indicated by changing -ane to -ene.<ref name="pmid2606099-unsaturation">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |pages=436–437 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099 |quote-page=436-437|quote=3S-2.5 Unsaturation Unsaturation is indicated by changing -ane to -ene, -adiene, -yne etc., or -an- to -en-, -adien-, -yn- etc. Examples: Androst-5-ene, not 5-androstene 5α-Cholest-6-ene 5β-Cholesta-7,9(11)-diene 5α-Cholest-6-en-3β-ol Notes 1) It is now recommended that the locant of a double bond is always adjacent to the syllable designating the unsaturation. [...] 3) The use of Δ (Greek capital delta) character is not recommended to designate unsaturation in individual names. It may be used, however, in generic terms, like ‘Δ⁵-steroids’}}</ref> This change was traditionally done in the parent name, adding a prefix to denote the position, with or without Δ (Greek capital delta), for example, 4-pregnene-11β,17α-diol-3,20-dione (also Δ⁴-pregnene-11β,17α-diol-3,20-dione) or 4-androstene-3,11,17-trione (also Δ⁴-androstene-3,11,17-trione). However, the Nomenclature of Steroids recommends the locant of a double bond to be always adjacent to the syllable designating the unsaturation, therefore, having it as a suffix rather than a prefix, and without the use of the Δ character, i.e. pregn-4-ene-11β,17α-diol-3,20-dione or androst-4-ene-3,11,17-trione. The double bond is designated by the lower-numbered carbon atom, i.e. "Δ⁴-" or "4-ene" means the double bond between positions 4 and 5. Saturation of double bonds (replacing a double bond between two carbon atoms with a single bond so that each of these atoms can attach one additional hydrogen atom) of a parent steroid can be done by adding "dihydro-" prefix,<ref name="norc">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=P-3|quote=P-31.2.2 General methodology ‘Hydro’ and ‘dehydro’ prefixes are associated with hydrogenation and dehydrogenation, respectively, of a double bond; thus, multiplying prefixes of even values, as ‘di’, ‘tetra’, etc. are used to indicate the saturation of double bond(s), for example ‘dihydro’, ‘tetrahydro’; or creation of double (or triple) bonds, as ‘didehydro’, etc. In names, they are placed immediately at the front of the name of the parent hydride and in front of any nondetachable prefixes. Indicated hydrogen atoms have priority over ‘hydro‘ prefixes for low locants. If indicated hydrogen atoms are present in a name, the ‘hydro‘ prefixes precede them.}}</ref> i.e. saturation of a double bond between positions 4 and 5 of testosterone with two hydrogen atoms may yield 4,5α-dihydrotestosterone or 4,5β-dihydrotestosterone. Generally, when there is no ambiguity, one number of a hydrogen position from a steroid with a saturated bond may be omitted, leaving only the position of the second hydrogen atom, e.g., 5α-dihydrotestosterone or 5β-dihydrotestosterone. Some steroids are traditionally grouped as Δ⁵-steroids (with a double bond between carbons 5 and 6 junctions (Figure 1)) and some as Δ⁴ steroids (with a double bond between carbons 4 and 5), respectively.<ref name="pmid21051590">{{cite journal |title=The molecular biology, biochemistry, and physiology of human steroidogenesis and its disorders |journal=Endocr Rev |volume=32 |issue=1 |pages=81–151 |pmid=21051590 |pmc=3365799 |doi=10.1210/er.2010-0013|last1=Miller |first1=Walter L. |last2=Auchus |first2=Richard J.|year=2011 }}</ref><ref name="pmid2606099-unsaturation"/> Canonical androgen synthesis is generally described as having a Δ⁵ pathway (from cholesterol to pregnenolone (P5) to 17α-hydroxypregnenolone (17OHP5) to DHEA to androstenediol (A5)) and of the Δ⁴ pathway (from P4 to 17-OHP to A4 to T). The abbreviations like "P4" and "A4" are used for convenience to designate them as Δ⁴-steroids, while "P5" and "A5" - as Δ⁵-steroids, respectively. The suffix -ol denotes a hydroxy group, while the suffix -one denotes an oxo group. When two or three identical groups are attached to the base structure at different positions, the suffix is ​​indicated as -diol or -triol for hydroxy, and -dione or -trione for oxo groups, respectively. For example, 5α-pregnane-3α,17α-diol-20-one has a hydrogen atom at the 5α position (hence the "5α-" prefix), two hydroxy groups (-OH) at the 3α and 17α positions (hence "3α,17α-diol" suffix) and an oxo group (=O) at the position 20 (hence the "20-one" suffix). However, erroneous use of suffixes can be found, e.g., "5α-pregnan-17α-diol-3,11,20-trione"<ref name="google-pregnan17diol">{{cite web | url=https://scholar.google.com/scholar?&q=%225%CE%B1-pregnan-17%CE%B1-diol-3%2C11%2C20-trione%22| title=Google Scholar search results for "5α-pregnan-17α-diol-3,11,20-trione" that is an incorrect name| year=2022}}</ref> [''sic''] — since it has just one hydroxy group (at 17α) rather than two, then the suffix should be -ol, rather than -diol, so that the correct name to be "5α-pregnan-17α-ol-3,11,20-trione". According to the rule set in the Nomenclature of Steroids, the terminal "e" in the parent structure name should be elided before the vowel (the presence or absence of a number does not affect such elision).<ref name="pmid2606099-parent-elisions">{{cite journal |title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989 |journal=Eur J Biochem |volume=186 |issue=3 |doi=10.1111/j.1432-1033.1989.tb15228.x |pmid=2606099|quote-page=441|quote=3S-4. FUNCTIONAL GROUPS 3S-4.0. General Nearly all biologically important steroids are derivatives of the parent hydrocarbons (cf. Table 1) carrying various functional groups. [...] Suffixes are added to the name of the saturated or unsaturated parent system (see 33-2.5), the terminal e of -ane, -ene, -yne, -adiene etc. being elided before a vowel (presence or absence of numerals has no effect on such elisions).}}</ref> This means, for instance, that if the suffix immediately appended to the parent structure name begins with a vowel, the trailing "e" is removed from that name. An example of such removal is "5α-pregnan-17α-ol-3,20-dione", where the last "e" of "pregnane" is dropped due to the vowel ("o") at the beginning of the suffix -ol. Some authors incorrectly use this rule, eliding the terminal "e" where it should be kept, or vice versa.<ref name="google-pregnane17ol">{{cite web | url=https://scholar.google.com/scholar?q=%225%CE%B1-pregnane-17%CE%B1-ol-3%2C20-dione%22| title=Google Scholar search results for "5α-pregnane-17α-ol-3,20-dione" that is an incorrect name| year=2022}}</ref> In the term "11-oxygenated" applied to a steroid, "oxygenated" refers to the presence of the oxygen atom in a group; this term is consistently used within the chemistry of the steroids<ref name="chemster">{{cite journal|last1=Makin|first1=H.L.J.|last2=Trafford|first2=D.J.H.|year=1972|title=The chemistry of the steroids|journal=Clinics in Endocrinology and Metabolism|volume=1|issue=2|pages=333–360|doi=10.1016/S0300-595X(72)80024-0}}</ref> since as early as 1950s.<ref name="pmid13167092">{{cite journal|last1=Bongiovanni|first1=A. M.|last2=Clayton|first2=G. W.|year=1954|title=Simplified method for estimation of 11-oxygenated neutral 17-ketosteroids in urine of individuals with adrenocortical hyperplasia|url=|journal=Proc Soc Exp Biol Med|volume=85|issue=3|pages=428–9|doi=10.3181/00379727-85-20905|pmid=13167092|s2cid=8408420}}</ref> Some studies use the term "11-oxyandrogens"<ref name="11oxyhs">{{cite journal|last1=Slaunwhite|first1=W.Roy|last2=Neely|first2=Lavalle|last3=Sandberg|first3=Avery A.|year=1964|title=The metabolism of 11-Oxyandrogens in human subjects|journal=Steroids|volume=3|issue=4|pages=391–416|doi=10.1016/0039-128X(64)90003-0}}</ref><ref name="pmid35611324" /> potentially as an abbreviation for 11-oxygenated androgens, to emphasize that they all have an oxygen atom attached to carbon at position 11.<ref name="pmid32203405">{{cite journal |title=11-Oxygenated androgens in health and disease |journal=Nat Rev Endocrinol |volume=16 |issue=5 |pages=284–296 |year=2020 |pmid=32203405 |pmc=7881526 |doi=10.1038/s41574-020-0336-x|last1=Turcu |first1=Adina F. |last2=Rege |first2=Juilee |last3=Auchus |first3=Richard J. |last4=Rainey |first4=William E. }}</ref><ref name="pmid33539964">{{cite journal|last1=Barnard|first1=Lise|last2=du Toit|first2=Therina|last3=Swart|first3=Amanda C.|title=Back where it belongs: 11β-hydroxyandrostenedione compels the re-assessment of C11-oxy androgens in steroidogenesis|url=https://pubmed.ncbi.nlm.nih.gov/33539964|journal=Molecular and Cellular Endocrinology|year=2021 |volume=525|pages=111189|doi=10.1016/j.mce.2021.111189|issn=1872-8057|pmid=33539964|s2cid=231776716 }}</ref> However, in chemical nomenclature, the prefix "oxy" refers to an ether, i.e., a compound with an oxygen atom connected to two alkyl or aryl groups (-O-),<ref name="norc-oxy">{{cite book|first1=Henri|last1=Favre|first2=Warren|last2=Powell|title=Nomenclature of Organic Chemistry - IUPAC Recommendations and Preferred Names 2013|publisher=The Royal Society of Chemistry|year=2014|isbn=978-0-85404-182-4|doi=10.1039/9781849733069|chapter=Appendix 2|quote-page=1112|quote=oxy* –O– P-15.3.1.2.1.1; P-63.2.2.1.1}}</ref> therefore, using the part "oxy" for a steroid may be misleading. The oxo group (=O) bound to a carbon atom forms a larger, ketone group (R₂C=O), hence the prefix "11-keto" used in the medical literature to denote an oxo group bound to carbon at position 11. However, the 1989 recommendations of the Joint Commission on Biochemical Nomenclature discourage the application of the prefix "keto" for steroid names, and favor the prefix "oxo" (e.g., 11-oxo steroids rather than 11-keto steroids), because keto denotes "R₂C=O", while only "=O" is attached in steroids to the carbon at a particular position. Therefore, the same carbon atom should not be specified twice.<ref name="pmid2606099-keto">{{cite journal|year=1989|title=IUPAC-IUB Joint Commission on Biochemical Nomenclature (JCBN). The nomenclature of steroids. Recommendations 1989|journal=Eur J Biochem|volume=186|issue=3|pages=429–58|doi=10.1111/j.1432-1033.1989.tb15228.x|pmid=2606099|quote=The prefix oxo- should also be used in connection with generic terms, e.g., 17-oxo steroids. The term ‘17-keto steroids’, often used in the medical literature, is incorrect because C-17 is specified twice, as the term keto denotes C=O|quote-page=430}}</ref> == Biochemistry == A more detailed description of each alternative androgen pathway described in the History section is provided below. Protein names are abbreviated by the standard gene names that they are encoded by (e.g., 5α-reductases type 1 is abbreviated by SRD5A1). Full enzyme names can be found in the Abbreviations section. === Backdoor Pathways to 5α-Dihydrotestosterone === While 5α-reduction is the last transformation in canonical androgen steroidogenesis, it is the first step in the backdoor pathways to 5α-dihydrotestosterone that acts on either 17-OHP or P4 which are ultimately converted to DHT.[[File:Androgen backdoor pathway.svg|thumb|left|The androgen backdoor pathways from 17α-hydroxyprogesterone or progesterone towards 5α-dihydrotestosterone roundabout testosterone and androstenedione (red arrows), as well as the "5α-dione" pathway that starts with 5α-reduction of androstenedione, embedded within canonical steroidogenesis (black arrows). Genes corresponding to the enzymes for catalysis are shown in boxed text with the associated arrow. Some additional proteins that are required for specific transformations (such as Steroidogenic acute regulatory protein (STAR), Cytochromes b₅, Cytochrome P450 reductase (POR)) are not shown for clarity.]] ====17α-Hydroxyprogesterone Pathway ==== [[File:Androgen backdoor pathway from 17-OHP to DHT.svg|thumb|right|The steroids involved in the metabolic pathway from 17α-hydroxyprogesterone to 5α-dihydrotestosterone with roundabout of testosterone. The red circle indicates the change in molecular structure compared to the precursor.]] The first step of this pathway is the 5α-reduction of 17-OHP to 5α-pregnan-17α-ol-3,20-dione (17-OH-DHP, since it is also known as 17α-hydroxy-dihydroprogesterone). The reaction is catalyzed by SRD5A1,<ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> though some authors{{specify}} suggest SRD5A2 may also catalyze this reaction in some contexts. 17-OH-DHP is then converted to 5α-pregnane-3α,17α-diol-20-one (5α-Pdiol) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2 and AKR1C4)<ref name="pmid30763313" /><ref name="pmid21802064">{{cite journal|last1=Flück|first1=Christa E.|last2=Meyer-Böni|first2=Monika|last3=Pandey|first3=Amit V.|last4=Kempná|first4=Petra|last5=Miller|first5=Walter L.|last6=Schoenle|first6=Eugen J.|last7=Biason-Lauber|first7=Anna|year=2011|title=Why boys will be boys: two pathways of fetal testicular androgen biosynthesis are needed for male sexual differentiation|journal=American Journal of Human Genetics|volume=89|issue=2|pages=201–218|doi=10.1016/j.ajhg.2011.06.009|issn=1537-6605|pmc=3155178|pmid=21802064}}</ref> or 17β-hydroxysteroid dehydrogenase type 6 (HSD17B6), that also has 3α-reduction activity.<ref name="pmid9188497">{{cite journal |title=Expression cloning and characterization of oxidative 17beta- and 3alpha-hydroxysteroid dehydrogenases from rat and human prostate |journal=J Biol Chem |volume=272 |issue=25 |pages=15959–66 |pmid=9188497 |doi=10.1074/jbc.272.25.15959|doi-access=free |last1=Biswas |first1=Michael G. |last2=Russell |first2=David W. |year=1997 }}</ref><ref name="pmid22114194">{{cite journal|title=Estrogen receptor β and 17β-hydroxysteroid dehydrogenase type 6, a growth regulatory pathway that is lost in prostate cancer |journal=Proc Natl Acad Sci U S A |volume=108 |issue=50 |pages=20090–4 |pmid=22114194 |pmc=3250130 |doi=10.1073/pnas.1117772108|doi-access=free |last1=Muthusamy |first1=Selvaraj |last2=Andersson |first2=Stefan |last3=Kim |first3=Hyun-Jin |last4=Butler |first4=Ryan |last5=Waage |first5=Linda |last6=Bergerheim |first6=Ulf |last7=Gustafsson |first7=Jan-Åke |year=2011 |bibcode=2011PNAS..10820090M }}</ref> 5α-Pdiol is also known as 17α-hydroxyallopregnanolone or 17-OH-allopregnanolone. 5α-Pdiol is then converted to 5α-androstan-3α-ol-17-one (AST) by 17,20-lyase activity of CYP17A1 which cleaves a side-chain (C17-C20 bond) from the steroid nucleus, converting a C₂₁ steroid (a pregnane) to C₁₉ steroid (an androstane or androgen). AST, in its turn, is 17β-reduced to 5α-androstane-3α,17β-diol (3α-diol) by 17β-hydroxysteroid dehydrogenase type 3 or type 5 (HSD17B3 and AKR1C3).<ref name="pmid31900912" /> The final step is 3α-oxidation of 3α-diol in target tissues to DHT by an enzyme that has 3α-hydroxysteroid oxidase activity, such as AKR1C2,<ref name="pmid12604227">{{cite journal |vauthors=Rizner TL, Lin HK, Penning TM |title=Role of human type 3 3alpha-hydroxysteroid dehydrogenase (AKR1C2) in androgen metabolism of prostate cancer cells |journal=Chem Biol Interact |volume=143-144 |issue= |pages=401–9 |date=February 2003 |pmid=12604227 |doi=10.1016/s0009-2797(02)00179-5}}</ref> HSD17B6, HSD17B10, RDH16, RDH5, and DHRS9.<ref name="pmid31611378"/> This oxidation is not required in the canonical pathway. The pathway can be summarized as:{{unbulleted list|17-OHP → 17-OH-DHP → 5α-Pdiol → AST → 3α-diol → DHT}} ====Progesterone Pathway==== The pathway from P4 to DHT is similar to that described above from 17-OHP to DHT, but the initial substrate for 5α-reductase here is P4 rather than 17-OHP. In male fetuses, placental P4 acts as a substrate during the biosynthesis of backdoor androgens, which occur in multiple tissues. Enzymes related to this backdoor pathway in the human male fetus are mainly expressed in non-gonadal tissues, and the steroids involved in this pathway are also primarily present in non-gonadal tissues.<ref name="pmid30763313"/> The first step in this pathway is 5α-reduction of P4 towards 5α-dihydroprogesterone (5α-DHP) by SRD5A1. 5α-DHP is then converted to 5α-pregnan-3α-ol-20-one (AlloP5) via 3α-reduction by a 3α-hydroxysteroid dehydrogenase isozyme (AKR1C2/AKR1C4). AlloP5 is then converted to 5α-Pdiol by the 17α-hydroxylase activity of CYP17A1. This metabolic pathway proceeds analogously to DHT as the 17α-Hydroxyprogesterone Pathway. The pathway can be summarized as:{{unbulleted list|P4 → 5α-DHP → AlloP5 → 5α-Pdiol → AST → 3α-diol → DHT}} === 5α-Dione Pathway === 5α-reduction is also the initial transformation of the 5α-dione pathway where A4 is converted to androstanedione (5α-dione) by SRD5A1 and then directly to DHT by either HSD17B3 or AKR1C3. While this pathway is unlikely to be biological relevance in healthy humans, it has been found operating in castration-resistant prostate cancer.<ref name="pmid21795608"/> The 5α-dione can also transformed into AST, which can then either converted back to 5α-dione or be transformed into DHT along the common part of the backdoor pathways to DHT (i.e., via 3α-diol).<ref name="pmid21795608"/><ref name="Nishiyama2011">{{cite journal|last1=Nishiyama|first1=Tsutomu|last2=Ishizaki|first2=Fumio|last3=Takizawa|first3=Itsuhiro|last4=Yamana|first4=Kazutoshi|last5=Hara|first5=Noboru|last6=Takahashi|first6=Kota|year=2011|title=5α-Androstane-3α 17β-diol Will Be a Potential Precursor of the Most Active Androgen 5α-Dihydrotestosterone in Prostate Cancer|journal=Journal of Urology|volume=185|issue=4S|doi=10.1016/j.juro.2011.02.378}}</ref> This pathway can be summarized as:{{unbulleted list|A4 → 5α-dione → DHT}} === 11-Oxygenated Androgen Pathways === [[File:Routes to 11-oxyandrogens.svg|thumb|Routes to 11-oxygenated androgens in humans|thumb|left|Abbreviated routes to 11-oxygenated androgens with transformations annotated with gene names of corresponding enzymes. Certain CYP17A1 mediated reactions that transform 11-oxygenated androgens classes (grey box) are omitted for clarity. Δ⁵ compounds that are transformed to Δ⁴ compounds are also omitted for clarity.]] Routes to 11-oxygenated androgens (Figure 4) also fall under our definition of alternative androgen pathways. These routes begin with four Δ⁴ steroid entry points (P4,<ref name="pmid30825506">{{cite journal|last1=Gent|first1=R.|last2=Du Toit|first2=T.|last3=Bloem|first3=L. M.|last4=Swart|first4=A. C.|year=2019|title=The 11β-hydroxysteroid dehydrogenase isoforms: pivotal catalytic activities yield potent C11-oxy C19 steroids with HSD11B2 favouring 11-ketotestosterone, 11-ketoandrostenedione and 11-ketoprogesterone biosynthesis|journal=J Steroid Biochem Mol Biol|volume=189|issue=|pages=116–126|doi=10.1016/j.jsbmb.2019.02.013|pmid=30825506|s2cid=73490363}}</ref> 17OHP, A4<ref name="pmid25869556" /><ref name="pmid27442248" /> and T) and can then enter a lattice-like organization of possible transformations between 19-carbon steroid products. Whether or not a steroid is subject to a given transformation path depends on health status and the expression of a given enzyme in the tissue where that steroid is synthesized or transported to. All the steroid products in this lattice have a hydroxy group (-OH) or an oxo group (=O) covalently bound to the carbon atom at position 11 (see Figure 1). Only four 11-oxygenated steroids are known to be androgenic: 11OHT, 11OHDHT, 11KT and 11KDHT with activities that are correspondingly comparable to T and DHT. The relative importance of the androgens depends on activity, circulating levels and stability. It may be that 11KT is the main androgen in women since it circulates at similar level to T but 11KT levels may not decline with age as T does, though some evidence suggests that 11KT does decline.<ref name="pmid31390028">{{cite journal|last1=Skiba|first1=Marina A.|last2=Bell|first2=Robin J.|last3=Islam|first3=Rakibul M.|last4=Handelsman|first4=David J.|last5=Desai|first5=Reena|last6=Davis|first6=Susan R.|year=2019|title=Androgens During the Reproductive Years: What Is Normal for Women?|journal=J Clin Endocrinol Metab|volume=104|issue=11|pages=5382–5392|doi=10.1210/jc.2019-01357|pmid=31390028|s2cid=199467054}}</ref> While 11KDHT is equipotent to DHT, circulating levels of 11KDHT are lower than DHT and SRD5A1-mediated transformation of 11KT to 11KDHT does not seem be significant. The other steroid products 11OHA4 and 11KA4 have been established as not having any androgen activity, but remain important molecules in this context since they act as androgen precursors. The complex lattice structure seen in Figure 4 can be understood broadly as the four Δ⁴ steroid entry points that can undergo a common sequence of three transformations: 1. 11β-hydroxylation by CYP11B1/2, 2. 5α-reduction by SRD5A1/2, 3. reversible 3α-reduction/oxidation of the ketone/alcohol.{{Clarify}} These steroids correspond to the "11OH" column in Figure 4. This sequence is replicated in the parallel column of "11K" steroids, in which are a result of 11β-reduction/oxidation of the ketone/alcohol{{Clarify}} (HSD11B1 catalyzes both oxidation and reduction while HSD11B2 only catalyzes the oxidation).<ref name="pmid23856005" /> There are additional transformations in the lattice that cross the derivatives of the entry points. AKR1C3 catalyzes (reversibly in some cases) 17β-reduction of the ketone/alcohol{{Clarify}} to transform between steroids that can be derived from T and A4. Steroids that can be derived from P4 can also be transformed to those that can be derived from 17-OHP via CYP17A1 17α-hydroxylase activity. Some members of the 17-OHP derived steroids can be transformed to A4 derived members via CYP17A1 17,20 lyase activity. The next sections describe what are understood to be the primary routes to androgens amongst the many possible routes visible in Figure 4. ==== C₁₉ Steroid Entry Points ==== A4 that is synthesized in the adrenal where it can undergo 11β-hydroxylation to yield 11OHA4,<ref name="pmid6970302">{{cite journal|last1=Haru|first1=Shibusawa|last2=Yumiko|first2=Sano|last3=Shoichi|first3=Okinaga|last4=Kiyoshi|first4=Arai|year=1980|title=Studies on 11β-hydroxylase of the human fetal adrenal gland|journal=Journal of Steroid Biochemistry|publisher=Elsevier BV|volume=13|issue=8|pages=881–887|doi=10.1016/0022-4731(80)90161-2|issn=0022-4731|pmid=6970302}}</ref><ref name="pmid22101210">{{cite journal|last1=Schloms|first1=Lindie|last2=Storbeck|first2=Karl-Heinz|last3=Swart|first3=Pieter|last4=Gelderblom|first4=Wentzel C.A.|last5=Swart|first5=Amanda C.|year=2012|title=The influence of Aspalathus linearis (Rooibos) and dihydrochalcones on adrenal steroidogenesis: quantification of steroid intermediates and end products in H295R cells|journal=J Steroid Biochem Mol Biol|volume=128|issue=3–5|pages=128–38|doi=10.1016/j.jsbmb.2011.11.003|pmid=22101210|s2cid=26099234}}</ref><ref name="pmid23685396" /> an important circulating androgen precursor, which is further transformed to 11KA4 and then 11KT (primarily outside the adrenal in peripheral tissue): {{unbulleted list|A4 → 11OHA4 → 11KA4 → 11KT}} This route is regarded as the primary 11-oxygenated androgen pathway in healthy humans. It is thought that the T entry point also operates in normal human physiology, but much less that A4: {{unbulleted list|T → 11OHT → 11OHA4 → 11KA4 → 11KT}}{{unbulleted list|T → 11OHT → 11KT}} The diminished role of these pathways is supported by that fact that the adrenal significantly more produces 11OHA4 than OHT.<ref name="pmid23386646" /><ref name="pmid29936123">{{cite journal|last1=Barnard|first1=Monique|last2=Quanson|first2=Jonathan L.|last3=Mostaghel|first3=Elahe|last4=Pretorius|first4=Elzette|last5=Snoep|first5=Jacky L.|last6=Storbeck|first6=Karl-Heinz|year=2018|title=11-Oxygenated androgen precursors are the preferred substrates for aldo-keto reductase 1C3 (AKR1C3): Implications for castration resistant prostate cancer|journal=J Steroid Biochem Mol Biol|volume=183|issue=|pages=192–201|doi=10.1016/j.jsbmb.2018.06.013|pmc=6283102|pmid=29936123}}</ref> ==== C₂₁ Steroid Entry Points ==== Currently, there is no good evidence for 11-oxygenated androgens from the C₂₁ steroid entry points (P4, 17OHP) operating in healthy humans. These entry points are relevant in the clinical context as discussed in the next section. ==Clinical Significance == === 11-Oxygenated Androgens === Measurements of the levels of circulating 11KT, in a 2021 study, Schiffer et al. identified 11KT biosynthesis in human peripheral blood mononuclear cells (in blood samples), which produced eight times the amount of 11KT compared to T. The lag time before isolation of cellular components from whole blood increased serum 11KT concentrations in a time-dependent manner, with a significant increase observed from two hours after blood collection. These results emphasize that care should be taken when performing lab tests—to avoid falsely elevated 11KT levels.<ref name="pmid33444228">{{cite journal|last1=Schiffer|first1=Lina|last2=Bossey|first2=Alicia|last3=Kempegowda|first3=Punith|last4=Taylor|first4=Angela E.|last5=Akerman|first5=Ildem|last6=Scheel-Toellner|first6=Dagmar|last7=Storbeck|first7=Karl-Heinz|last8=Arlt|first8=Wiebke|year=2021|title=Peripheral blood mononuclear cells preferentially activate 11-oxygenated androgens|journal=Eur J Endocrinol|volume=184|issue=3|pages=353–363|doi=10.1530/EJE-20-1077|issn=1479-683X|pmc=7923147|pmid=33444228}}</ref> Unlike T and A4, 11-oxygenated androgens are not known to be aromatized to estrogens in the human body.<ref name="pmid32862221">{{cite journal |last1=Nagasaki |first1=Keisuke |last2=Takase |first2=Kaoru |last3=Numakura |first3=Chikahiko |last4=Homma |first4=Keiko |last5=Hasegawa |first5=Tomonobu |last6=Fukami |first6=Maki |title=Foetal virilisation caused by overproduction of non-aromatisable 11-oxy C19 steroids in maternal adrenal tumour |journal=Human Reproduction |year=2020 |volume=35 |issue=11 |pages=2609–2612 |doi=10.1093/humrep/deaa221 |pmid=32862221 }}</ref><ref name="pmid33340399">{{cite journal|title = 11-Oxygenated Estrogens Are a Novel Class of Human Estrogens but Do not Contribute to the Circulating Estrogen Pool | journal = Endocrinology | volume = 162 | issue = 3 | pmid = 33340399 | pmc = 7814299 | doi = 10.1210/endocr/bqaa231 | last1 = Barnard | first1 = Lise | last2 = Schiffer | first2 = Lina | last3 = Louw Du-Toit | first3 = Renate | last4 = Tamblyn | first4 = Jennifer A. | last5 = Chen | first5 = Shiuan | last6 = Africander | first6 = Donita | last7 = Arlt | first7 = Wiebke | last8 = Foster | first8 = Paul A. | last9 = Storbeck | first9 = Karl-Heinz |year = 2021 }}</ref> The inability of aromatase to convert the 11-oxygenated androgens to estrogens may contribute to the 11-oxygenated androgens circulating at higher levels than other androgens in women, except perhaps DHEA.<ref name="pmid15994348">{{cite journal | title = Direct agonist/antagonist functions of dehydroepiandrosterone | journal = Endocrinology | year = 2005 | volume = 146 | issue = 11 | pages = 4568–76 | pmid = 15994348 | doi = 10.1210/en.2005-0368 | doi-access = free | last1 = Chen | first1 = Fang | last2 = Knecht | first2 = Kristin | last3 = Birzin | first3 = Elizabeth | last4 = Fisher | first4 = John | last5 = Wilkinson | first5 = Hilary | last6 = Mojena | first6 = Marina | last7 = Moreno | first7 = Consuelo Tudela | last8 = Schmidt | first8 = Azriel | last9 = Harada | first9 = Shun-Ichi | last10 = Freedman | first10 = Leonard P. | last11 = Reszka | first11 = Alfred A. }}</ref><ref name="pmid16159155">{{cite journal |title = Chemistry and structural biology of androgen receptor | journal = Chemical Reviews | volume = 105 | issue = 9 | pages = 3352–70 | pmid = 16159155 | pmc = 2096617 | doi = 10.1021/cr020456u | last1 = Gao | first1 = Wenqing | last2 = Bohl | first2 = Casey E. | last3 = Dalton | first3 = James T. | year = 2005 }}</ref> However, it is possible that 11-oxygenated estrogens may be produced in some conditions such as feminizing adrenal carcinoma.<ref name="MAHESH196351">{{cite journal|title = Isolation of estrone and 11β-hydroxy estrone from a feminizing adrenal carcinoma | journal = Steroids | volume = 1 | number = 1 | pages = 51–61 |year = 1963 |issn = 0039-128X| doi = 10.1016/S0039-128X(63)80157-9 | url = https://www.sciencedirect.com/science/article/pii/S0039128X63801579 |first1=Virendra |last1=Mahesh |first2=Walter |last2=Herrmann}}</ref> DHT, an androgen that can also be produced in a backdoor pathway, is also a non-aromatizable androgen.<ref name="pmid2943941">{{cite journal |title=Stimulation of aromatase activity by dihydrotestosterone in human skin fibroblasts |journal=J Steroid Biochem |volume=25 |issue=1 |pages=165–9 |year=1986 |pmid=2943941 |doi=10.1016/0022-4731(86)90296-7 |last1=Chabab |first1=Aziz |last2=Sultan |first2=Charles |last3=Fenart |first3=Odile |last4=Descomps |first4=Bernard }}</ref><ref name="pmid10332569">{{cite journal |title=Dihydrotestosterone: a rationale for its use as a non-aromatizable androgen replacement therapeutic agent |journal=Baillieres Clin Endocrinol Metab |volume=12 |issue=3 |pages=501–6 |year=1998 |pmid=10332569 |doi=10.1016/s0950-351x(98)80267-x |last1=Swerdloff |first1=Ronald S. |last2=Wang |first2=Christina }}</ref> Most{{dubious}} conditions highlighted in the following sections have had demonstrations of potential roles for 11-oxygenated androgens. === Hyperandrogenism === Alternative androgen pathways are not always considered in the clinical evaluation of patients with hyperandrogenism, i.e., androgen excess.<ref name="pmid32610579">{{cite journal|last1=Sumińska|first1=Marta|last2=Bogusz-Górna|first2=Klaudia|last3=Wegner|first3=Dominika|last4=Fichna|first4=Marta|year=2020|title=Non-Classic Disorder of Adrenal Steroidogenesis and Clinical Dilemmas in 21-Hydroxylase Deficiency Combined with Backdoor Androgen Pathway. Mini-Review and Case Report|journal=Int J Mol Sci|volume=21|issue=13|page=4622|doi=10.3390/ijms21134622|pmc=7369945|pmid=32610579|doi-access=free}}</ref> Hyperandrogenism may lead to symptoms like acne, hirsutism, alopecia, premature adrenarche, oligomenorrhea or amenorrhea, polycystic ovaries and infertility.<ref name="pmid16772149">{{cite journal|last1=Yildiz|first1=Bulent O.|year=2006|title=Diagnosis of hyperandrogenism: clinical criteria|journal=Best Practice & Research. Clinical Endocrinology & Metabolism|publisher=Elsevier BV|volume=20|issue=2|pages=167–176|doi=10.1016/j.beem.2006.02.004|issn=1521-690X|pmid=16772149}}</ref><ref name="pmid24184282">{{cite journal|last1=Peigné|first1=Maëliss|last2=Villers-Capelle|first2=Anne|last3=Robin|first3=Geoffroy|last4=Dewailly|first4=Didier|year=2013|title=Hyperandrogénie féminine|journal=Presse Medicale (Paris, France)|language=fr|publisher=Elsevier BV|volume=42|issue=11|pages=1487–1499|doi=10.1016/j.lpm.2013.07.016|issn=0755-4982|pmid=24184282|s2cid=28921380}}</ref> Not considering alternative androgen pathways in clinical hyperandrogenism investigations may obfuscate the condition.<ref name="pmid32610579" /> Despite the prevailing dogma that T and DHT are the primary human androgens, this paradigm applies only to healthy men.<ref name="pmid28234803">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2017|title=Clinical significance of 11-oxygenated androgens|journal=Curr Opin Endocrinol Diabetes Obes|volume=24|issue=3|pages=252–259|doi=10.1097/MED.0000000000000334|pmc=5819755|pmid=28234803}}</ref> Although T has been traditionally used as a biomarker of androgen excess,<ref name="pmid32912651">{{cite journal|last1=Yang|first1=Yabo|last2=Ouyang|first2=Nengyong|last3=Ye|first3=Yang|last4=Hu|first4=Qin|last5=Du|first5=Tao|last6=Di|first6=Na|last7=Xu|first7=Wenming|last8=Azziz|first8=Ricardo|last9=Yang|first9=Dongzi|year=2020|title=The predictive value of total testosterone alone for clinical hyperandrogenism in polycystic ovary syndrome|journal=Reprod Biomed Online|volume=41|issue=4|pages=734–742|doi=10.1016/j.rbmo.2020.07.013|pmid=32912651|s2cid=221625488|last10=Zhao|first10=Xiaomiao}}</ref> it correlates poorly with clinical findings of androgen excess.<ref name="pmid28234803" /> If the levels of T appear to be normal, ignoring the alternative androgen pathways may lead to diagnostic errors since hyperandrogenism may be caused by potent androgens such as DHT produced by a backdoor pathway and 11-oxygenated androgens also produced from 21-carbon steroid (pregnane) precursors in a backdoor pathway.<ref name="pmid33415088">{{cite journal|last1=Balsamo|first1=Antonio|last2=Baronio|first2=Federico|last3=Ortolano|first3=Rita|last4=Menabo|first4=Soara|last5=Baldazzi|first5=Lilia|last6=Di Natale|first6=Valeria|last7=Vissani|first7=Sofia|last8=Cassio|first8=Alessandra|year=2020|title=Congenital Adrenal Hyperplasias Presenting in the Newborn and Young Infant|journal=Frontiers in Pediatrics|publisher=Frontiers Media SA|volume=8|page=593315|doi=10.3389/fped.2020.593315|issn=2296-2360|pmc=7783414|pmid=33415088|doi-access=free}}</ref><ref name="pmid29277706">{{cite journal|last1=Kamrath|first1=Clemens|last2=Wettstaedt|first2=Lisa|last3=Boettcher|first3=Claudia|last4=Hartmann|first4=Michaela F.|last5=Wudy|first5=Stefan A.|year=2018|title=Androgen excess is due to elevated 11-oxygenated androgens in treated children with congenital adrenal hyperplasia|journal=The Journal of Steroid Biochemistry and Molecular Biology|publisher=Elsevier BV|volume=178|pages=221–228|doi=10.1016/j.jsbmb.2017.12.016|issn=0960-0760|pmid=29277706|s2cid=3709499}}</ref> It had been suggested that 11β-hydroxyandrostenedione (11OHA4) and its urinary metabolites could have clinical applications used as a biomarkers of adrenal origin of androgen excess in women. Increased adrenal 11OHA4 production was characterized, using changes in A4:11OHA4 and 11β-hydroxyandrosterone:11β-hydroxyetiocholanolone ratios, in cushing syndrome, hirsutism, CAH and PCOS.<ref name="pmid1623996">{{cite journal|last1=Carmina|first1=E.|last2=Stanczyk|first2=F. Z.|last3=Chang|first3=L.|last4=Miles|first4=R. A.|last5=Lobo|first5=R. A.|year=1992|title=The ratio of androstenedione:11 beta-hydroxyandrostenedione is an important marker of adrenal androgen excess in women|journal=Fertil Steril|volume=58|issue=1|pages=148–52|doi=10.1016/s0015-0282(16)55152-8|pmid=1623996}}</ref><ref name="pmid14417423">{{cite journal|last1=Lipsett|first1=Mortimer B.|last2=Riter|first2=Barbara|year=1960|title=Urinary ketosteroids and pregnanetriol in hirsutism|journal=J Clin Endocrinol Metab|volume=20|issue=2|pages=180–6|doi=10.1210/jcem-20-2-180|pmid=14417423}}</ref><ref name="pmid33340399" /><ref name="pmid3129451">{{cite journal|last1=Polson|first1=D. W.|last2=Reed|first2=M. J.|last3=Franks|first3=S.|last4=Scanlon|first4=M. J.|last5=James|first5=V. H. T.|year=1988|title=Serum 11 beta-hydroxyandrostenedione as an indicator of the source of excess androgen production in women with polycystic ovaries|journal=J Clin Endocrinol Metab|volume=66|issue=5|pages=946–50|doi=10.1210/jcem-66-5-946|pmid=3129451}}</ref> These ratios have still not been established as a standard clinical as a diagnostic tool. Congenital adrenal hyperplasia (CAH) is a well known disease of hyperandrogenism, but the contributions of the backdoor pathway to DHT remain under appreciated. CAH refers to a group of autosomal recessive disorders characterized by impaired cortisol biosynthesis<ref name="pmid28576284">{{cite journal|date=November 2017|title=Congenital adrenal hyperplasia|url=|journal=Lancet|volume=390|issue=10108|pages=2194–2210|doi=10.1016/S0140-6736(17)31431-9|pmid=28576284|vauthors=El-Maouche D, Arlt W, Merke DP}}</ref> caused by a deficiency in any of the enzymes required to produce cortisol in the adrenal.<ref name="pmid12930931">{{cite journal|date=August 2003|title=Congenital adrenal hyperplasia|url=|journal=N Engl J Med|volume=349|issue=8|pages=776–88|doi=10.1056/NEJMra021561|pmid=12930931|vauthors=Speiser PW, White PC}}</ref><ref name="pmid30272171">{{cite journal|year=2018|title=Congenital Adrenal Hyperplasia Due to Steroid 21-Hydroxylase Deficiency: An Endocrine Society Clinical Practice Guideline|journal=The Journal of Clinical Endocrinology and Metabolism|volume=103|issue=11|pages=4043–4088|doi=10.1210/jc.2018-01865|pmc=6456929|pmid=30272171}}</ref> This deficiency leads to an excessive accumulation of steroid precursors that are converted to androgens. In a 2016 study, Turcu et al.<ref name="pmid26865584">{{cite journal|last1=Turcu|first1=Adina F.|last2=Nanba|first2=Aya T.|last3=Chomic|first3=Robert|last4=Upadhyay|first4=Sunil K.|last5=Giordano|first5=Thomas J.|last6=Shields|first6=James J.|last7=Merke|first7=Deborah P.|last8=Rainey|first8=William E.|last9=Auchus|first9=Richard J.|year=2016|title=Adrenal-derived 11-oxygenated 19-carbon steroids are the dominant androgens in classic 21-hydroxylase deficiency|journal=Eur J Endocrinol|volume=174|issue=5|pages=601–9|doi=10.1530/EJE-15-1181|pmc=4874183|pmid=26865584}}</ref> showed that in classic CAH due to CYP21A2 deficiency both conventional and 11-oxygenated androgens were elevated 3-4 fold in CAH patients receiving glucocorticoid therapy compared to healthy controls. The levels of 11-oxygenated androgens correlated positively with conventional androgens in women but negatively in men. The levels of 11KT were 4 times higher compared to that of T in women with the condition.<ref name="pmid26865584" /> In CAH due to 21-hydroxylase<ref name="pmid22170725" /> (CYP17A1) or cytochrome P450 oxidoreductase (POR) deficiency,<ref name="pmid31611378" /><ref name="pmid35793998" /> the associated elevated 17-OHP levels result in flux through the backdoor pathway to DHT that begins with 5α-reduction. This pathway may be activated regardless of age and sex.<ref name="pmid26038201">{{cite journal|last1=Turcu|first1=Adina F.|last2=Auchus|first2=Richard J.|year=2015|title=Adrenal Steroidogenesis and Congenital Adrenal Hyperplasia|journal=Endocrinology and Metabolism Clinics of North America|publisher=Elsevier BV|volume=44|issue=2|pages=275–296|doi=10.1016/j.ecl.2015.02.002|issn=0889-8529|pmc=4506691703046|pmid=26038201}}</ref> Fetal excess of 17-OHP in CAH may contribute to DHT synthesis and lead to external genital virilization in newborn girls with CAH.<ref name="pmid31611378" /> P4 levels may also be elevated in CAH,<ref name="pmid25850025">{{cite journal|last1=Turcu|first1=A. F.|last2=Rege|first2=J.|last3=Chomic|first3=R.|last4=Liu|first4=J.|last5=Nishimoto|first5=H. K.|last6=Else|first6=T.|last7=Moraitis|first7=A. G.|last8=Palapattu|first8=G. S.|last9=Rainey|first9=W. E.|year=2015|title=Profiles of 21-Carbon Steroids in 21-hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=100|issue=6|pages=2283–2290|doi=10.1210/jc.2015-1023|pmc=4454804|pmid=25850025|last10=Auchus|first10=R. J.}}</ref><ref name="pmid31505456">{{cite journal|date=November 2019|title=Influence of hormones on the immunotolerogenic molecule HLA-G: a cross-sectional study in patients with congenital adrenal hyperplasia|url=|journal=Eur J Endocrinol|volume=181|issue=5|pages=481–488|doi=10.1530/EJE-19-0379|pmid=31505456|vauthors=Nguyen LS, Rouas-Freiss N, Funck-Brentano C, Leban M, Carosella ED, Touraine P, Varnous S, Bachelot A, Salem JE}}</ref> leading to androgen excess via the backdoor pathway from P4 to DHT.<ref name="pmid28188961">{{cite journal|date=May 2017|title=High serum progesterone associated with infertility in a woman with nonclassic congenital adrenal hyperplasia|url=|journal=J Obstet Gynaecol Res|volume=43|issue=5|pages=946–950|doi=10.1111/jog.13288|pmid=28188961|vauthors=Kawarai Y, Ishikawa H, Segawa T, Teramoto S, Tanaka T, Shozu M}}</ref> 17-OHP and P4 may also serve as substrates to 11-oxygenated androgens in CAH.<ref name="pmid28472487">{{cite journal|last1=Turcu|first1=Adina F|last2=Mallappa|first2=Ashwini|last3=Elman|first3=Meredith S|last4=Avila|first4=Nilo A|last5=Marko|first5=Jamie|last6=Rao|first6=Hamsini|last7=Tsodikov|first7=Alexander|last8=Auchus|first8=Richard J|last9=Merke|first9=Deborah P|year=2017|title=11-Oxygenated Androgens Are Biomarkers of Adrenal Volume and Testicular Adrenal Rest Tumors in 21-Hydroxylase Deficiency|journal=The Journal of Clinical Endocrinology and Metabolism|volume=102|issue=8|pages=2701–2710|doi=10.1210/jc.2016-3989|pmc=5546849|pmid=28472487}}</ref><ref name="pmid26865584" /><ref name="pmid29718004">{{cite journal|last1=White|first1=Perrin C.|year=2018|title=Update on diagnosis and management of congenital adrenal hyperplasia due to 21-hydroxylase deficiency|journal=Current Opinion in Endocrinology, Diabetes, and Obesity|volume=25|issue=3|pages=178–184|doi=10.1097/MED.0000000000000402|pmid=29718004|s2cid=26072848}}</ref><ref name="pmid34867794">{{cite journal|last1=Turcu|first1=Adina F.|last2=Mallappa|first2=Ashwini|last3=Nella|first3=Aikaterini A.|last4=Chen|first4=Xuan|last5=Zhao|first5=Lili|last6=Nanba|first6=Aya T.|last7=Byrd|first7=James Brian|last8=Auchus|first8=Richard J.|last9=Merke|first9=Deborah P.|year=2021|title=24-Hour Profiles of 11-Oxygenated C19 Steroids and Δ5-Steroid Sulfates during Oral and Continuous Subcutaneous Glucocorticoids in 21-Hydroxylase Deficiency|journal=Front Endocrinol (Lausanne)|volume=12|issue=|pages=751191|doi=10.3389/fendo.2021.751191|pmc=8636728|pmid=34867794|doi-access=free}}</ref> In males with CAH, 11-oxygenated androgens may lead to development of testicular adrenal rest tumors.<ref name="pmid25850025" /><ref name="pmid28472487" /><ref name="pmid34390337">{{cite journal|last1=Schröder|first1=Mariska A M.|last2=Turcu|first2=Adina F.|last3=o'Day|first3=Patrick|last4=Van Herwaarden|first4=Antonius E.|last5=Span|first5=Paul N.|last6=Auchus|first6=Richard J.|last7=Sweep|first7=Fred C G J.|last8=Claahsen-Van Der Grinten|first8=Hedi L.|year=2022|title=Production of 11-Oxygenated Androgens by Testicular Adrenal Rest Tumors|journal=J Clin Endocrinol Metab|volume=107|issue=1|pages=e272–e280|doi=10.1210/clinem/dgab598|pmc=8684463|pmid=34390337}}</ref> === Disorders of Sex Development === Both canonical and the backdoor androgen pathway to DHT are required for normal human male genital development.<ref name="pmid30943210">{{cite journal|last1=Miller|first1=Walter L.|last2=Auchus|first2=Richard J.|year=2019|title=The "backdoor pathway" of androgen synthesis in human male sexual development|journal=PLOS Biology|volume=17|issue=4|pages=e3000198|doi=10.1371/journal.pbio.3000198|pmc=6464227|pmid=30943210}}</ref><ref name="pmid35793998">{{cite journal|last1=Lee|first1=Hyun Gyung|last2=Kim|first2=Chan Jong|year=2022|title=Classic and backdoor pathways of androgen biosynthesis in human sexual development|journal=Ann Pediatr Endocrinol Metab|volume=27|issue=2|pages=83–89|doi=10.6065/apem.2244124.062|pmid=35793998|s2cid=250155674}}</ref> Deficiencies in the backdoor pathway to DHT from 17-OHP or from P4<ref name="pmid21802064"/><ref name="pmid23073980">{{cite journal|last1=Fukami|first1=Maki|last2=Homma|first2=Keiko|last3=Hasegawa|first3=Tomonobu|last4=Ogata|first4=Tsutomu|year=2013|title=Backdoor pathway for dihydrotestosterone biosynthesis: implications for normal and abnormal human sex development|journal=Developmental Dynamics|volume=242|issue=4|pages=320–9|doi=10.1002/dvdy.23892|pmid=23073980|s2cid=44702659}}</ref> lead to underverilization of the male fetus,<ref name="pmid24793988">{{cite journal |title=Steroidogenesis of the testis -- new genes and pathways |journal=Ann Endocrinol (Paris) |volume=75 |issue=2 |pages=40–7 |year=2014 |pmid=24793988 |doi=10.1016/j.ando.2014.03.002 |last1=Flück |first1=Christa E. |last2=Pandey |first2=Amit V. }}</ref><ref name="pmid8636249">{{cite journal |title=Prismatic cases: 17,20-desmolase (17,20-lyase) deficiency |journal=J Clin Endocrinol Metab |volume=81 |issue=2 |pages=457–9 |year=1996 |pmid=8636249 |doi=10.1210/jcem.81.2.8636249 |url=|last1=Zachmann |first1=M. }}</ref> as placental P4 is a precursor to DHT in the backdoor pathway.<ref name="pmid30763313"/> In 2011 Flück et al. described a case of five 46,XY (male) patients from two families with DSD, caused by mutations in AKR1C2 and/or AKR1C4, an enzyme required for a backdoor pathway to DHT, but not the canonical pathway of androgen biosynthesis. In these patients, mutations in the AKR1C1 and AKR1C3 were excluded, and disorders in the canonical pathway of androgen biosynthesis have also been excluded, however, they had genital ambiguity. The 46,XX (female) relatives of affected patients, having the same mutations, were phenotypically normal and fertile. Although both AKR1C2 and AKR1C4 are needed for DHT synthesis in a backdoor pathway (Figure 2), the study found that mutations in AKR1C2 only were sufficient for disruption.<ref name="pmid21802064"/> However, these AKR1C2/AKR1C4 variants leading to DSD are rare and have been only so far reported in just those two families.<ref name="pmid34711511">{{cite journal |title=Rare forms of genetic steroidogenic defects affecting the gonads and adrenals |journal=Best Pract Res Clin Endocrinol Metab |volume=36 |issue=1 |pages=101593 |year=2022 |pmid=34711511 |doi=10.1016/j.beem.2021.101593}}</ref> Isolated 17,20-lyase deficiency syndrome due to variants in CYP17A1, cytochrome b₅, and POR may also disrupt a backdoor pathway to DHT, as the 17,20-lyase activity of CYP17A1 is required for both canonical and backdoor androgen pathways (Figure 2). This rare deficiency can lead to DSD in both sexes with affected girls are asymptomatic until puberty, when they show amenorrhea.<ref name="pmid34711511"/> 11-oxygenated androgens may play important roles in DSDs.<ref name="pmid34171490">{{cite journal |title=Turning the spotlight on the C11-oxy androgens in human fetal development |journal=J Steroid Biochem Mol Biol |volume=212 |issue= |pages=105946 |pmid=34171490 |doi=10.1016/j.jsbmb.2021.105946|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2021 |s2cid=235603586 }}</ref><ref name="pmid34987475">{{cite journal|title=Disorders of Sex Development of Adrenal Origin |journal=Front Endocrinol (Lausanne) |volume=12 |issue= |pages=770782 |pmid=34987475 |pmc=8720965 |doi=10.3389/fendo.2021.770782 |doi-access=free |last1=Finkielstain |first1=Gabriela P. |last2=Vieites |first2=Ana |last3=Bergadá |first3=Ignacio |last4=Rey |first4=Rodolfo A. |year=2021 }}</ref><ref name="pmid31611378">{{cite journal|last1=Reisch|first1=Nicole|last2=Taylor|first2=Angela E.|last3=Nogueira|first3=Edson F.|last4=Asby|first4=Daniel J.|last5=Dhir|first5=Vivek|last6=Berry|first6=Andrew|last7=Krone|first7=Nils|last8=Auchus|first8=Richard J.|last9=Shackleton|first9=Cedric H. L.|title=Alternative pathway androgen biosynthesis and human fetal female virilization|journal=Proceedings of the National Academy of Sciences of the United States of America|year=2019 |volume=116|issue=44|pages=22294–22299|doi=10.1073/pnas.1906623116|issn=1091-6490|pmc=6825302|pmid=31611378|doi-access=free }}</ref> 11-oxygenated androgen fetal biosynthesis may coincide with the key stages of production of cortisol — at weeks 8–9, 13–24, and from 31 and onwards. In these stages, impaired CYP17A1 and CYP21A2 activity lead to increased ACTH due to cortisol deficiency and the accumulation of precursors that serve as substrates for CYP11B1 in pathways to 11-oxygenated androgens, which cause abnormal female fetal development.<ref name="pmid34171490"/> === Polycystic Ovary Syndrome === In PCOS, DHT may be produced in the backdoor androgen pathway from upregulation of SRD5A1 activity.<ref name="pmid27471004">{{cite journal |title=Genes and proteins of the alternative steroid backdoor pathway for dihydrotestosterone synthesis are expressed in the human ovary and seem enhanced in the polycystic ovary syndrome |journal=Mol Cell Endocrinol |volume=441 |issue= |pages=116–123 |pmid=27471004 |doi=10.1016/j.mce.2016.07.029|last1=Marti |first1=Nesa |last2=Galván |first2=José A. |last3=Pandey |first3=Amit V. |last4=Trippel |first4=Mafalda |last5=Tapia |first5=Coya |last6=Müller |first6=Michel |last7=Perren |first7=Aurel |last8=Flück |first8=Christa E. |year=2017 |s2cid=22185557 }}</ref><ref name="pmid1968168">{{cite journal|last1=Stewart|first1=P. M.|last2=Shackleton|first2=C. H.|last3=Beastall|first3=G. H.|last4=Edwards|first4=C. R.|title=5 alpha-reductase activity in polycystic ovary syndrome|url=https://pubmed.ncbi.nlm.nih.gov/1968168|journal=Lancet (London, England)|year=1990 |volume=335|issue=8687|pages=431–433|doi=10.1016/0140-6736(90)90664-q|issn=0140-6736|pmid=1968168|s2cid=54422650 }}</ref><ref name="pmid19567518">{{cite journal|title=Increased 5 alpha-reductase activity and adrenocortical drive in women with polycystic ovary syndrome |journal=J Clin Endocrinol Metab |volume=94 |issue=9 |pages=3558–66 |pmid=19567518 |doi=10.1210/jc.2009-0837 |last1=Vassiliadi |first1=Dimitra A. |last2=Barber |first2=Thomas M. |last3=Hughes |first3=Beverly A. |last4=McCarthy |first4=Mark I. |last5=Wass |first5=John A. H. |last6=Franks |first6=Stephen |last7=Nightingale |first7=Peter |last8=Tomlinson |first8=Jeremy W. |last9=Arlt |first9=Wiebke |last10=Stewart |first10=Paul M. |year=2009 }}</ref><ref name="pmid32247282">{{cite journal | last1=Swart | first1=Amanda C. | last2=du Toit | first2=Therina | last3=Gourgari | first3=Evgenia | last4=Kidd | first4=Martin | last5=Keil | first5=Meg | last6=Faucz | first6=Fabio R. | last7=Stratakis | first7=Constantine A. | title=Steroid hormone analysis of adolescents and young women with polycystic ovarian syndrome and adrenocortical dysfunction using UPC2-MS/MS | journal=Pediatric Research | publisher=Springer Science and Business Media LLC | volume=89 | issue=1 | year=2021 | issn=0031-3998 | pmid=32247282 | pmc=7541460 | doi=10.1038/s41390-020-0870-1 | pages=118–126}}</ref> 11-oxygenated androgens may also play an important role in PCOS.<ref name="pmid35611324">{{cite journal |title=11-Oxyandrogens in Adolescents With Polycystic Ovary Syndrome |journal=J Endocr Soc |year=2022 |volume=6 |issue=7 |pages=bvac037|pmid=35611324 |pmc=9123281 |doi=10.1210/jendso/bvac037|last1=Taylor |first1=Anya E. |last2=Ware |first2=Meredith A. |last3=Breslow |first3=Emily |last4=Pyle |first4=Laura |last5=Severn |first5=Cameron |last6=Nadeau |first6=Kristen J. |last7=Chan |first7=Christine L. |last8=Kelsey |first8=Megan M. |last9=Cree-Green |first9=Melanie }}</ref><ref name="pmid32637065">{{cite journal |title=Implicating androgen excess in propagating metabolic disease in polycystic ovary syndrome |journal=Ther Adv Endocrinol Metab |volume=11 |issue= |pages=2042018820934319 |pmid=32637065 |pmc=7315669 |doi=10.1177/2042018820934319|last1=Kempegowda |first1=Punith |last2=Melson |first2=Eka |last3=Manolopoulos |first3=Konstantinos N. |last4=Arlt |first4=Wiebke |last5=o'Reilly |first5=Michael W. |year=2020 }}</ref><ref name="pmid27901631">{{cite journal|title=11-Oxygenated C19 Steroids Are the Predominant Androgens in Polycystic Ovary Syndrome |journal=J Clin Endocrinol Metab |volume=102 |issue=3 |pages=840–848 |pmid=27901631 |pmc=5460696 |doi=10.1210/jc.2016-3285 |last1=o'Reilly |first1=Michael W. |last2=Kempegowda |first2=Punith |last3=Jenkinson |first3=Carl |last4=Taylor |first4=Angela E. |last5=Quanson |first5=Jonathan L. |last6=Storbeck |first6=Karl-Heinz |last7=Arlt |first7=Wiebke |year=2017 }}</ref> In a 2017 study, O'Reilly et al. revealed that 11-oxygenated androgens are the predominant androgens in women with PCOS, while in healthy control subjects, classic androgens constitute the majority of the circulating androgen pool; nevertheless, the levels of 11KT exceeded those of T in both groups, specifically, 3.4 fold in the PCOS group. Recent investigations have reported circulating levels of 11KA4, 11KT and 11OHT levels in PCOS as well as 11-oxygenated pregnanes. Serum 11OHT and 11KT levels have been show to be elevated in PCOS and correlate with body mass index.<ref name="pmid30012903">{{cite journal |title=11-oxygenated C19 steroids as circulating androgens in women with polycystic ovary syndrome |journal=Endocr J |volume=65 |issue=10 |pages=979–990 |pmid=30012903 |doi=10.1507/endocrj.EJ18-0212|last1=Yoshida |first1=Tomoko |last2=Matsuzaki |first2=Toshiya |last3=Miyado |first3=Mami |last4=Saito |first4=Kazuki |last5=Iwasa |first5=Takeshi |last6=Matsubara |first6=Yoichi |last7=Ogata |first7=Tsutomu |last8=Irahara |first8=Minoru |last9=Fukami |first9=Maki |year=2018 }}</ref> Significantly elevated 11KT levels have been detected in the daughters of PCOS mothers and in obese girls while 11OHA4, 11KA4 and 11OHT levels were comparable.<ref name="pmid32797203">{{cite journal |title=11-Oxygenated C19 Steroids Do Not Distinguish the Hyperandrogenic Phenotype of PCOS Daughters from Girls with Obesity |journal=J Clin Endocrinol Metab |volume=105 |issue=11 |pages= e3903–e3909 |pmid=32797203 |pmc=7500474 |doi=10.1210/clinem/dgaa532|last1=Torchen |first1=Laura C. |last2=Sisk |first2=Ryan |last3=Legro |first3=Richard S. |last4=Turcu |first4=Adina F. |last5=Auchus |first5=Richard J. |last6=Dunaif |first6=Andrea |year=2020 }}</ref> 11KT has also been shown to be elevated together with decreased 11KA4 levels in PCOS patients with micronodular adrenocortical hyperplasia. In addition 11OHAST, 11OHEt, DHP4 and 11KDHP4 levels were elevated and 11OHP4, 21dF and 11KDHP4 were elevated in patients with inadequate dexamethasone responses.<ref name="pmid31450227">{{cite journal|last1=Miller|first1=Walter L.|year=2019|title=Congenital Adrenal Hyperplasia: Time to Replace 17OHP with 21-Deoxycortisol|url=https://pubmed.ncbi.nlm.nih.gov/31450227|journal=Hormone Research in Paediatrics|volume=91|issue=6|pages=416–420|doi=10.1159/000501396|issn=1663-2826|pmid=31450227|s2cid=201733086}}</ref> Metformin treatment had no effect on 11-oxygenated androgens in PCOS adolescents in a 2022 study, despite lower levels of T after treatment.<ref name="pmid35611324" /> === Prostate Cancer === High levels of 11KT, 11KDHT and 11OHDHT have also been detected in prostate cancer tissue (~10–20 ng/g) and in circulation, 11KT (~200–350nM) and 11KDHT (~20nM) being the most abundant.<ref name="pmid28939401" /> There is some preliminary evidence that 11-oxygenated androgen pathway may play an important role at the stage of prostatic carcinogenesis.<ref name="pmid34520388" /> Androgen deprivation is a therapeutic approach to prostrate cancer that can be implemented by castration to eliminate gonadal T, but metastatic tumors may then develop into castration-resistant prostate cancer (CRPC). Although castration results in 90-95% reduction of serum T, DHT in the prostate is only decreased by 50%, supporting the notion that the prostrate expresses necessary enzymes to produce DHT without testicular T.{{cn}} The 5α-dione pathway was discovered in the context of CPRC (see History), and is known to mitigate the effects of androgen depravation therapy. 11-Oxygenated androgens contribute significantly to the androgen pool<ref name="pmid23856005" /><ref name="pmid31900912" /> and play a previously overlooked role in the reactivation of androgen signaling in the CRPC patient.<ref name="pmid34520388">{{cite journal |vauthors=Ventura-Bahena A, Hernández-Pérez JG, Torres-Sánchez L, Sierra-Santoyo A, Escobar-Wilches DC, Escamilla-Núñez C, Gómez R, Rodríguez-Covarrubias F, López-González ML, Figueroa M |title=Urinary androgens excretion patterns and prostate cancer in Mexican men |journal=Endocr Relat Cancer |volume=28 |issue=12 |pages=745–756 |date=October 2021 |pmid=34520388 |doi=10.1530/ERC-21-0160 |url=}}</ref><ref name="pmid28939401">{{cite journal |title=Inefficient UGT-conjugation of adrenal 11β-hydroxyandrostenedione metabolites highlights C11-oxy C19 steroids as the predominant androgens in prostate cancer |journal=Mol Cell Endocrinol |volume=461 |issue= |pages=265–276 |pmid=28939401 |doi=10.1016/j.mce.2017.09.026|last1=Du Toit |first1=Therina |last2=Swart |first2=Amanda C. |year=2018 |s2cid=6335125 }}</ref><ref name="pmid30825506" /><ref name="pmid23856005" /><ref name="pmid31900912">{{cite journal | title = Canonical and Noncanonical Androgen Metabolism and Activity | journal = Advances in Experimental Medicine and Biology | volume = 1210 | pages = 239–277 | pmid = 31900912 | doi = 10.1007/978-3-030-32656-2_11 | isbn = 978-3-030-32655-5 | s2cid = 209748543 | last1 = Storbeck | first1 = Karl-Heinz | last2 = Mostaghel | first2 = Elahe A. | year = 2019 }}</ref><ref name="pmid23685396">{{cite journal|title=11β-hydroxyandrostenedione, the product of androstenedione metabolism in the adrenal, is metabolized in LNCaP cells by 5α-reductase yielding 11β-hydroxy-5α-androstanedione |journal=J Steroid Biochem Mol Biol |volume=138 |issue= |pages=132–42 |pmid=23685396 |doi=10.1016/j.jsbmb.2013.04.010 |s2cid=3404940 |last1=Swart |first1=Amanda C. |last2=Schloms |first2=Lindie |last3=Storbeck |first3=Karl-Heinz |last4=Bloem |first4=Liezl M. |last5=Toit |first5=Therina du |last6=Quanson |first6=Jonathan L. |last7=Rainey |first7=William E. |last8=Swart |first8=Pieter |year=2013 }}</ref> Serum 11KT levels are higher than any other androgen in 97% of CRPC patients, accounting for 60% of the total active androgen pool, and are not affected by castration.<ref name="pmid33974560">{{cite journal|title=11-Ketotestosterone is the predominant active androgen in prostate cancer patients after castration |journal=JCI Insight |volume=6 |issue=11 |pmid=33974560 |pmc=8262344 |doi=10.1172/jci.insight.148507 |last1=Snaterse |first1=G. |last2=Van Dessel |first2=L. F. |last3=Van Riet |first3=J. |last4=Taylor |first4=A. E. |last5=Van Der Vlugt-Daane |first5=M. |last6=Hamberg |first6=P. |last7=De Wit |first7=R. |last8=Visser |first8=J. A. |last9=Arlt |first9=W. |last10=Lolkema |first10=M. P. |last11=Hofland |first11=J. |year=2021 }}</ref> === Benign Prostatic Hyperplasia; Chronic Prostatitis/Chronic Pelvic Pain Syndrome === Androgens play a vital role in the development, growth and maintenance of the prostate.<ref name="pmid18471780" /> Therefore, the role of androgens should be seriously considered not only in CRPC, but other prostate-related conditions such as BPH<ref name="pmid18471780"/> and chronic prostatitis/chronic pelvic pain syndrome (CP/CPPS).<ref name="pmid18308097">{{cite journal|title=Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome |journal=Urology |volume=71 |issue=2 |pages=261–6 |pmid=18308097 |pmc=2390769 |doi=10.1016/j.urology.2007.09.025 |last1=Dimitrakov |first1=Jordan |last2=Joffe |first2=Hylton V. |last3=Soldin |first3=Steven J. |last4=Bolus |first4=Roger |last5=Buffington |first5=C.A. Tony |last6=Nickel |first6=J. Curtis |year=2008 }}</ref> 11-oxygenated androgen pathways have been observed in BPH cell models (11OHP4 and 11KP4, a C₂₁ steroid, to 11KDHT), BPH patient tissue biopsy and in their serum.<ref name="pmid31626910">{{cite journal|title = The 11β-hydroxyandrostenedione pathway and C11-oxy C21 backdoor pathway are active in benign prostatic hyperplasia yielding 11keto-testosterone and 11keto-progesterone | journal = The Journal of Steroid Biochemistry and Molecular Biology | volume = 196 | pages = 105497 | pmid = 31626910 | doi = 10.1016/j.jsbmb.2019.105497 | s2cid = 204734045 | url = | last1 = Du Toit | first1 = Therina | last2 = Swart | first2 = Amanda C. |year = 2020 }}</ref> == Future Directions == Relative steroid serum levels in CP/CPPS have suggested that CYP21A2 deficiency may play a role in the disease and that non-classic CAH due to 21-hydroxylase deficiency may be a comorbidity, as described in a study published in 2008 by Dimitrakov et al.<ref name="pmid18308097" /> However, we noticed that the study had a discrepancy in steroid levels reported for the control subjects.<ref name="urology.2022.07.051">{{Cite journal|last1=Masiutin|first1=Maxim G.|last2=Yadav|first2=Maneesh K.|date=2022-08-17||year=2022|title=Letter to the editor regarding the article "Adrenocortical hormone abnormalities in men with chronic prostatitis/chronic pelvic pain syndrome"|url=https://www.goldjournal.net/article/S0090-4295(22)00690-2/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.051|issn=0090-4295}}</ref><ref>{{Cite journal|last1=Dimitrakoff|first1=Jordan|last2=Nickel|first2=J. Curtis|date=2022-08-17|year=2022|title=AUTHOR REPLY|url=https://www.goldjournal.net/article/S0090-4295(22)00691-4/abstract|journal=Urology|language=English|doi=10.1016/j.urology.2022.07.049|issn=0090-4295}}</ref> Given the potential roles that alternative androgen pathways play in the previously described disease areas, it seems that CP/CPPS would seem to be a good candidate to investigate the same way. We are not aware of any work looking at the roles of alternative androgen pathways in CP/CPPS. We, therefore, hypothesize that DHT produced by the "backdoor" pathway and 11-oxygenated androgens may be relevant in CP/CPPS.<ref name="urology.2022.07.051"/> ==PubChem CIDs== In order to unambiguously define all the steroids mentioned in the present review, their respective PubChem IDs are listed below. PubChem is a database of molecules, maintained by the National Center for Biotechnology Information of the United States National Institutes of Health. The IDs given below are intended to eliminate ambiguity caused by the use of different synonyms for the same metabolic intermediate by different authors when describing the androgen backdoor pathways. 11dF: 440707; 11K-5αdione: 11185733; 11KA4: 223997; 11KAST: 102029; 11KDHP4: 968899; 11KDHT: 11197479; 11KP4: 94166; 11KPdiol: 92264183; 11KPdione: 99568471; 11KT: 104796; 11OH-3αdiol: 349754907; 11OH-5αdione: 59087027; 11OHA4: 94141; 11OHAST: 10286365; 11OHDHP4: 11267580; 11OHDHT: 10018051; 11OHEt: 101849; 11OHP4: 101788; 11OHPdiol: 99601857; 11OHPdione: 99572627; 11OHT: 114920; 17OHP5: 3032570; 17-OHP: 6238; 17-OH-DHP: 11889565; 21dE: 102178; 21dF: 92827; 3,11diOH-DHP4: 10125849; 3α-diol: 15818; 3β-diol: 242332; 5α-DHP: 92810; 5α-dione: 222865; 5α-Pdiol: 111243; A4: 6128; A5: 10634; A5-S: 13847309; ALF: 104845; AlloP5: 92786; AST: 5879; DHEA: 5881; DHEA-S: 12594; DHT: 10635; DOC: 6166; P4: 5994; P5: 8955; T: 6013. == Abbreviations == === Steroids === * '''11dF''' 11-deoxycortisol (also known as Reichstein's substance S) * '''11K-3αdiol''' 5α-androstane-3α,17β-diol-11-one * '''11K-5αdione''' 5α-androstane-3,11,17-trione (also known as 11-ketoandrostanedione or 11-keto-5α-androstanedione) * '''11KA4''' 11-ketoandrostenedione (also known as 4-androstene-3,11,17-trione or androst-4-ene-3,11,17-trione or adrenosterone or Reichstein's substance G) * '''11KAST''' 5α-androstan-3α-ol-11,17-dione (also known as 11-ketoandrosterone) * '''11KDHP4''' 5α-pregnane-3,11,20-trione (also known as 11-ketodihydroprogesterone or allopregnanetrione) * '''11KDHT''' 11-ketodihydrotestosterone (also known as "5α-dihydro-11-keto testosterone" or 5α-dihydro-11-keto-testosterone) * '''11KP4''' 4-pregnene-3,11,20-trione (also known as pregn-4-ene-3,11,20-trione or 11-ketoprogesterone) * '''11KPdiol''' 5α-pregnane-3α,17α-diol-11,20-dione * '''11KPdione''' 5α-pregnan-17α-ol-3,11,20-trione * '''11KT''' 11-ketotestosterone (also known as 4-androsten-17β-ol-3,11-dione) * '''11OH-3αdiol''' 5α-androstane-3α,11β,17β-triol * '''11OH-5αdione''' 5α-androstan-11β-ol-3,17-dione (also known as 11β-hydroxy-5α-androstanedione) * '''11OHA4''' 11β-hydroxyandrostenedione (also known as 4-androsten-11β-ol-3,17-dione or androst-4-en-11β-ol-3,17-dione) * '''11OHAST''' 5α-androstane-3α,11β-diol-17-one (also known as 11β-hydroxyandrosterone) * '''11OHDHP4''' 5α-pregnan-11β-ol-3,20-dione (also known as 11β-hydroxydihydroprogesterone) * '''11OHDHT''' 11β-hydroxydihydrotestosterone (also known as 5α-dihydro-11β-hydroxytestosterone or 5α-androstane-11β,17β-diol-3-one or 11β,17β-dihydroxy-5α-androstan-3-one) * '''11OHEt''' 11β-hydroxyetiocholanolone (also known as 3α,11β-dihydroxy-5β-androstan-17-one) * '''11OHP4''' 4-pregnen-11β-ol-3,20-dione (also known as pregn-4-en-11β-ol-3,20-dione or 21-deoxycorticosterone or 11β-hydroxyprogesterone) * '''11OHPdiol''' 5α-pregnane-3α,11β,17α-triol-20-one * '''11OHPdione''' 5α-pregnane-11β,17α-diol-3,20-dione * '''11OHT''' 11β-hydroxytestosterone * '''17OHP5''' 17α-hydroxypregnenolone * '''17-OH-DHP''' 5α-pregnan-17α-ol-3,20-dione (also known as 17α-hydroxydihydroprogesterone) * '''17-OHP''' 17α-hydroxyprogesterone * '''21dE''' 4-pregnen-17α-ol-3,11,20-trione (also known as pregn-4-en-17α-ol-3,11,20-trione or 21-deoxycortisone) * '''21dF''' 4-pregnene-11β,17α-diol-3,20-dione (also known as 11β,17α-dihydroxyprogesterone or pregn-4-ene-11β,17α-diol-3,20-dione or 21-deoxycortisol or 21-desoxyhydrocortisone) * '''3,11diOH-DHP4''' 5α-pregnane-3α,11β-diol-20-one (also known as 3α,11β-dihydroxy-5α-pregnan-20-one) * '''3α-diol''' 5α-androstane-3α,17β-diol (also known by abbreviation "5α-Adiol" or "5α-adiol"), also known as 3α-androstanediol * '''3β-diol''' 5α-androstane-3β,17β-diol (also known as 3β-androstanediol) * '''5α-DHP''' 5α-dihydroprogesterone * '''5α-dione''' androstanedione (also known as 5α-androstane-3,17-dione) * '''5α-Pdiol''' 5α-pregnane-3α,17α-diol-20-one (also known as 17α-hydroxyallopregnanolone) * '''A4''' androstenedione (also known as 4-androstene-3,17-dione or androst-4-ene-3,17-dione) * '''A5''' androstenediol (also known as 5-androstene-3β,17β-diol or androst-5-ene-3β,17β-diol) * '''A5-S''' androstenediol sulfate * '''ALF''' 5α-pregnan-3α-ol-11,20-dione (also known, when used as a medication, as alfaxalone or alphaxalone) * '''AlloP5''' 5α-pregnan-3α-ol-20-one (also known as allopregnanolone) * '''AST''' 5α-androstan-3α-ol-17-one (also known androsterone) * '''DHEA''' dehydroepiandrosterone (also known as 3β-hydroxyandrost-5-en-17-one or androst-5-en-3β-ol-17-one) * '''DHEA-S''' dehydroepiandrosterone sulfate * '''DHT''' 5α-dihydrotestosterone (also known as 5α-androstan-17β-ol-3-one) * '''DOC''' 11-deoxycorticosterone (also known as Reichstein's substance Q) * '''P4''' progesterone * '''P5''' pregnenolone * '''T''' testosterone === Enzymes (Abbreviated by their Gene Names) === * '''AKR1C2''' aldo-keto reductase family 1 member C2 (also known as 3α-hydroxysteroid dehydrogenase type 3) * '''AKR1C3''' aldo-keto reductase family 1 member C3 (also known as 3α-hydroxysteroid dehydrogenase type 2; also known as 17β-hydroxysteroid dehydrogenase type 5 (HSD17B5)) * '''AKR1C4''' aldo-keto reductase family 1 member C4 (also known as 3α-hydroxysteroid dehydrogenase type 1) * '''CYP11A1''' cytochrome P450 cholesterol side-chain cleavage enzyme (also known by abbreviation "P450scc") * '''CYP11B1''' steroid 11β-hydroxylase * '''CYP11B2''' aldosterone synthase * '''CYP17A1''' steroid 17α-hydroxylase/17,20-lyase (also known as cytochrome P450c17) * '''CYP21A2''' steroid 21α-hydroxylase (also known as 21-hydroxylase, or cytochrome P450c21) * '''DHRS9''' dehydrogenase/reductase SDR family member 9 * '''HSD11B1''' 11β-hydroxysteroid dehydrogenase type 1 * '''HSD11B2''' 11β-hydroxysteroid dehydrogenase type 2 * '''HSD17B3''' 17β-hydroxysteroid dehydrogenase type 3 * '''HSD17B6''' 17β-hydroxysteroid dehydrogenase type 6 (also known as retinol dehydrogenase-like hydroxysteroid dehydrogenase, RL-HSD) * '''HSD17B10''' 17β-hydroxysteroid dehydrogenase type 10 * '''POR''' cytochrome P450 oxidoreductase * '''RDH16''' retinol dehydrogenase 16 (also known as RODH4) * '''RDH5''' retinol dehydrogenase 5 * '''SRD5A1''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 1 * '''SRD5A2''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 2 * '''SRD5A3''' 3-oxo-5α-steroid 4-dehydrogenase (also known as steroid 5α-reductase) type 3 === Conditions === * '''BPH''' benign prostatic hyperplasia * '''CAH''' congenital adrenal hyperplasia * '''CP/CPPS''' chronic prostatitis/chronic pelvic pain syndrome * '''CRPC''' castration-resistant prostate cancer * '''DSD''' disorder of sex development * '''PCOS''' polycystic ovary syndrome === Other === * '''ACTH''' adrenocorticotropic hormone * '''STAR''' steroidogenic acute regulatory protein == Additional Information == === Competing Interests === The authors have no competing interest. === Funding === The authors received no financial support for the research, authorship and publication of this article. === Notes on The Use of Abbreviations === The authors sometimes used "full name – abbreviation" pairs repeatedly throughout the article for easier following. === Referencing Convention === {{ordered list |When particular results or conclusions of particular research or review are discussed, it is mentioned by the year when it was published and the last name of the first author with "et al.". The year may not necessarily be mentioned close to the name. |To back up a particular claim which is an exact claim (such as which enzyme catalyzes a particular reaction), the supporting article is cited in the text as a number in square brackets from the numbered list of references, without mentioning the year and the name. The same technique is applied to support a generalization (e.g., "the prevailing dogma", "not always considered", "canonical androgen steroidogenesis") — in such case, there is a reference to one or more supporting reviews without explicitly mentioning these reviews in the text. |When multiple studies that confirm the same finding (or that are on a similar topic) are cited, they are also cited as described in p.2., i.e., giving reference numbers in square brackets and without mentioning the year and the name.}} == References == {{reflist|35em}} cdqikhjcj1ntqlbgpjtvqkpn7pwo57i 0 271287 2416136 2415885 2022-08-18T12:15:40Z Kaexer 2944801 assigned emma chiu to melioidosis task wikitext text/x-wiki <noinclude>{{WikiJ top menu}}__NOTOC__ Tasks for the technical editors can be added at the bottom of the table ([https://en.wikiversity.org/wiki/WikiJournal_User_Group/Technical_editors/tasks?veaction=edit activate editing mode], click bottom row, click chevron that appears on the left, select 'insert below') Tech eds can claim tasks by adding their name to the right ([[WikiJournal User Group/Editorial guidelines/Technical editor summary|process guidelines)]] {| class="wikitable sortable" |+ !item !task !person !time taken !completion !comments |- | |editorial process training |Logan Smith |120 |2021-01-11 |Complete |- | |editorial process training |Joshua Langfus |120 |2021-01-11 |Complete |- | |editorial process training |Wilson Jacobs |120 |2021-01-11 |Complete |- | |editorial process training |Emma Choplin |120 |2021-01-11 |Complete |- | |editorial process training |Jenna Harmon |120 |2021-01-14 |Complete |- | |editorial process training |Cody Naccarato |120 |2021-01-14 |Complete |- |[[WikiJournal Preprints/The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats|The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats]] |create/link author wikidata items |Wilson Jacobs |90 |2021-01-26 |Complete |- |[[WikiJournal of Science/Virtual colony count|Virtual colony count]] |format and upload PDF |Wilson Jacobs |150 |2021-02-13 |Complete |- |Virtual colony count |upload PDF |Wilson Jacobs |300 |2021-02-14 |Complete |- |[[WikiJournal of Science/Evolved human male preferences for female body shape|Evolved human male preferences for female body shape]] |format PDF |Jenna Harmon |150 |2021-01-21 |Complete |- |Evolved human male preferences for female body shape |upload PDF |Jenna Harmon |30 |2021-02-05 |Complete |- |[[WikiJournal of Science/Arabinogalactan-proteins|Arabinogalactan-proteins]] |format PDF and upload |Jenna Harmon |180 |2021-02-13 |Complete |- |[[WikiJournal Preprints/Does the packaging of health information affect the assessment of its reliability? A randomized controlled trial protocol|Does the packaging of health information affect the assessment of its reliability? A randomized controlled trial protocol]] |peer-review processing |Logan Smith |60 |2021-02-01 |Complete |- |[[WikiJournal Preprints/Does the packaging of health information affect the assessment of its reliability? A randomized controlled trial protocol|Does the packaging of health information affect the assessment of its reliability? A randomized controlled trial protocol]] |peer-review processing |Logan Smith |30 |2021-02-03 |Complete |- |[[WikiJournal Preprints/Affine symmetric group|Affine symmetric group]] |peer-review processing |Logan Smith |30 |2021-02-04 |Complete |- |[[WikiJournal Preprints/Affine symmetric group|Affine symmetric group]] |peer-review processing |Logan Smith |30 |2021-02-10 |Complete |- |[[WikiJournal Preprints/The Kivu Ebola epidemic|Kivu Ebola epidemic]] |process newly submitted article |Wilson Jacobs |150 |2021-02-09 |Complete |- |Wikidata items of each WikiJMed article ([https://w.wiki/445Z query]; [[wikidata:Q96317242#P50|example]]) |add email address to corresponding author | | | | |- |Wikidata items of each WikiJSci article ([https://w.wiki/445a query]; [[wikidata:Q96317242#P50|example]]) |add email address to corresponding author | | | | |- |Wikidata items of each WikiJHum article ([https://w.wiki/445b query]; [[wikidata:Q96317242#P50|example]]) |add email address to corresponding author | | | | |- |Authors on this list ([https://w.wiki/4fY2 query], [https://author-disambiguator.toolforge.org/work_item_oauth.php disambiguator]) |Create a wikidata item for each author and disambiguate any of their other publications on wikidata | | | | |- |[[WikiJournal of Science/Structural Model of Bacteriophage T4|Structural Model of Bacteriophage T4]] |format PDF and upload |Jenna Harmon |130 |2022-01-31 |Complete |- |[[WikiJournal of Science/A broad introduction to RNA-Seq|A broad introduction to RNA-Seq]] |format PDF and upload |Jenna Harmon |130 |2022-02-17 |Complete |- |[[WikiJournal Preprints/“Collect, acquire, analyze, report, and disseminate statistical data related to the science and engineering enterprise…”: The National Center for Science and Engineering Statistics|“Collect, acquire, analyze, report, and disseminate statistical data related to the science and engineering enterprise…”: The National Center for Science and Engineering Statistics]] |format PDF and upload |Jenna Harmon |120 |2022-02-11 |Complete |- |Wikidata items of each WikiJMed author ([https://w.wiki/4463 query]) |add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all authors | | | | |- |Wikidata items of each WikiJSci author ([https://w.wiki/4462 query]) |add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all authors | | | | |- |Wikidata items of each WikiJHum author ([https://w.wiki/445q query]) |add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all authors | | | | |- |Wikidata items of each WikiJMed reviewer ([https://w.wiki/445v query]) |add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all peer reviewers | | | | |- |Wikidata items of each WikiJSci reviewer ([https://w.wiki/445x query]) |add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all peer reviewers | | | | |- |Wikidata items of each WikiJHum reviewer ([https://w.wiki/445u query]) |add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all peer reviewers | | | | |- |Edit all titles to sentence case ([[Talk:WikiJournal User Group#Article title format - Why no consistency?|discussion]]) |''on hold until consensus'' Pagemove the articles, update wikidata, update the PDFs | | | | |- |Update 2021 [[WikiJournal of Medicine/Citation metrics|WikiJMed]] citation metrics |Review the number of articles published in the past 2 years (2019-2021) that cite WikiJMed articles to calculate impact factor | | | | |- |Backfill [[WikiJournal of Science/Citation metrics|citation metrics]] for WikiJSci articles from 2019-2021 |Similar to the update for WikiJMed citation metrics, this one will also need to calculate impact factors for 2019, 2020 and 2021 | | | | |- |Calculate citation metrics for WikiJHum articles from 2018-2021 |Similar to the other citation metrics tasks, this one will need to create a page from scratch for WikiJHum and calculate impact factors for 2018, 2019, 2020 and 2021 | | | | |- |[[WikiJournal Preprints/“It’s all about people skills”: Perspectives on the social license of the forest products industry from rural North America]] |Using [[:File:Soc license forestry NA Annotated text and reviewers comments - Ian Thomson.pdf|reviewer's annotated PDF]], extract the original text from the file and transfer onto the bare wiki page. |Emma Chiu |180 |2022-06-20 |Complete |- |[[WikiJournal of Medicine/History of penicillin|History of penicillin]] |format PDF and upload |Ellen Sussman | | | |- |[[WikiJournal of Medicine/Phage Therapy|Phage Therapy]] |format PDF and upload |Andrew Neil | | | |- |[[WikiJournal of Science/“Collect, acquire, analyze, report, and disseminate statistical data related to the science and engineering enterprise…”: The National Center for Science and Engineering Statistics|“Collect, acquire, analyze, report, and disseminate statistical data related to the science and engineering enterprise…”: The National Center for Science and Engineering Statistics]] |format PDF and upload |Jenna Harmon |95 |2022-05-06 |complete |- |[[WikiJournal User Group/Editorial guidelines#Registering article in DOAJ|DOAJ]] |upload WikiJMed and WikiJSci [[WikiJournal User Group/Editorial guidelines#Registering article in DOAJ|article metadata to DOAJ]] | | | | |- |[[WikiJournal Preprints/The Kivu Ebola epidemic|Kivu Ebola epidemic]] |format PDF and upload |Ellen Sussman | | | |- |[[WikiJournal Preprints/Parenting_stress|Parenting stress]] |format PDF and upload |Natalie Charamut | | | |- |[[WikiJournal Preprints/Leptospirosis|Leptospirosis]] |format PDF and upload |Andrew Neil | | | |- |[[WikiJournal Preprints/Melioidosis|Melioidosis]] | Accepted, update article page, generate DOI and PDF please |Emma Chiu | | | |- |[[WikiJournal Preprints/A history of coronaviruses|A history of coronaviruses]] |Almost ready - awaiting final references - no action yet | | | | |- |[[WikiJournal Preprints/The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats|The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats]] |Using [[:File:Kunu and wistar rates after review tracked changes.pdf|updated manuscript's PDF]], copy the text from the file and transfer onto the wiki page |Peter Agan | | | |- |Crossref reference deposit (see [[Talk:WikiJournal User Group#Talk:WikiJournal User Group|discussion]]) |[[WikiJournal User Group/Editorial guidelines#Submitting reference metadata|Submit metadata with references for published articles onto Crossref platform]] |Michelle Fong | | |- | |editorial process training 1 & 2 |Ellen Sussman |120 |2022-07-11 |Complete |- | |editorial process training 1 & 2 |Andrew Neil |120 |2022-07-11 |Complete |- | |editorial process training 1 & 2 |Natalie Charamut |120 |2022-07-11 |Complete |- | |editorial process training 1 & 2 |Emma Chiu |120 |2022-07-11 |Complete |- | |editorial process training 1 & 2 |Crystal Au |120 |2022-07-11 |Complete |- | |editorial process training 1 & 2 |Michelle Fong |120 |2022-07-11 |Complete |} k2gqifn31svoa1dcikslfyx9lxidl1l 2416454 2416136 2022-08-19T07:46:34Z Rwatson1955 2831054 History of coronaviruses ready for acceptance wikitext text/x-wiki <noinclude>{{WikiJ top menu}}__NOTOC__ Tasks for the technical editors can be added at the bottom of the table ([https://en.wikiversity.org/wiki/WikiJournal_User_Group/Technical_editors/tasks?veaction=edit activate editing mode], click bottom row, click chevron that appears on the left, select 'insert below') Tech eds can claim tasks by adding their name to the right ([[WikiJournal User Group/Editorial guidelines/Technical editor summary|process guidelines)]] {| class="wikitable sortable" |+ !item !task !person !time taken !completion !comments |- | |editorial process training |Logan Smith |120 |2021-01-11 |Complete |- | |editorial process training |Joshua Langfus |120 |2021-01-11 |Complete |- | |editorial process training |Wilson Jacobs |120 |2021-01-11 |Complete |- | |editorial process training |Emma Choplin |120 |2021-01-11 |Complete |- | |editorial process training |Jenna Harmon |120 |2021-01-14 |Complete |- | |editorial process training |Cody Naccarato |120 |2021-01-14 |Complete |- |[[WikiJournal Preprints/The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats|The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats]] |create/link author wikidata items |Wilson Jacobs |90 |2021-01-26 |Complete |- |[[WikiJournal of Science/Virtual colony count|Virtual colony count]] |format and upload PDF |Wilson Jacobs |150 |2021-02-13 |Complete |- |Virtual colony count |upload PDF |Wilson Jacobs |300 |2021-02-14 |Complete |- |[[WikiJournal of Science/Evolved human male preferences for female body shape|Evolved human male preferences for female body shape]] |format PDF |Jenna Harmon |150 |2021-01-21 |Complete |- |Evolved human male preferences for female body shape |upload PDF |Jenna Harmon |30 |2021-02-05 |Complete |- |[[WikiJournal of Science/Arabinogalactan-proteins|Arabinogalactan-proteins]] |format PDF and upload |Jenna Harmon |180 |2021-02-13 |Complete |- |[[WikiJournal Preprints/Does the packaging of health information affect the assessment of its reliability? 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A randomized controlled trial protocol]] |peer-review processing |Logan Smith |30 |2021-02-03 |Complete |- |[[WikiJournal Preprints/Affine symmetric group|Affine symmetric group]] |peer-review processing |Logan Smith |30 |2021-02-04 |Complete |- |[[WikiJournal Preprints/Affine symmetric group|Affine symmetric group]] |peer-review processing |Logan Smith |30 |2021-02-10 |Complete |- |[[WikiJournal Preprints/The Kivu Ebola epidemic|Kivu Ebola epidemic]] |process newly submitted article |Wilson Jacobs |150 |2021-02-09 |Complete |- |Wikidata items of each WikiJMed article ([https://w.wiki/445Z query]; [[wikidata:Q96317242#P50|example]]) |add email address to corresponding author | | | | |- |Wikidata items of each WikiJSci article ([https://w.wiki/445a query]; [[wikidata:Q96317242#P50|example]]) |add email address to corresponding author | | | | |- |Wikidata items of each WikiJHum article ([https://w.wiki/445b query]; [[wikidata:Q96317242#P50|example]]) |add email address to corresponding author | | | | |- |Authors on this list ([https://w.wiki/4fY2 query], [https://author-disambiguator.toolforge.org/work_item_oauth.php disambiguator]) |Create a wikidata item for each author and disambiguate any of their other publications on wikidata | | | | |- |[[WikiJournal of Science/Structural Model of Bacteriophage T4|Structural Model of Bacteriophage T4]] |format PDF and upload |Jenna Harmon |130 |2022-01-31 |Complete |- |[[WikiJournal of Science/A broad introduction to RNA-Seq|A broad introduction to RNA-Seq]] |format PDF and upload |Jenna Harmon |130 |2022-02-17 |Complete |- |[[WikiJournal Preprints/“Collect, acquire, analyze, report, and disseminate statistical data related to the science and engineering enterprise…”: The National Center for Science and Engineering Statistics|“Collect, acquire, analyze, report, and disseminate statistical data related to the science and engineering enterprise…”: The National Center for Science and Engineering Statistics]] |format PDF and upload |Jenna Harmon |120 |2022-02-11 |Complete |- |Wikidata items of each WikiJMed author ([https://w.wiki/4463 query]) |add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all authors | | | | |- |Wikidata items of each WikiJSci author ([https://w.wiki/4462 query]) |add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all authors | | | | |- |Wikidata items of each WikiJHum author ([https://w.wiki/445q query]) |add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all authors | | | | |- |Wikidata items of each WikiJMed reviewer ([https://w.wiki/445v query]) |add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all peer reviewers | | | | |- |Wikidata items of each WikiJSci reviewer ([https://w.wiki/445x query]) |add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all peer reviewers | | | | |- |Wikidata items of each WikiJHum reviewer ([https://w.wiki/445u query]) |add [[wikidata:Property:P101|fields of work]], employers, orcid, and official website for all peer reviewers | | | | |- |Edit all titles to sentence case ([[Talk:WikiJournal User Group#Article title format - Why no consistency?|discussion]]) |''on hold until consensus'' Pagemove the articles, update wikidata, update the PDFs | | | | |- |Update 2021 [[WikiJournal of Medicine/Citation metrics|WikiJMed]] citation metrics |Review the number of articles published in the past 2 years (2019-2021) that cite WikiJMed articles to calculate impact factor | | | | |- |Backfill [[WikiJournal of Science/Citation metrics|citation metrics]] for WikiJSci articles from 2019-2021 |Similar to the update for WikiJMed citation metrics, this one will also need to calculate impact factors for 2019, 2020 and 2021 | | | | |- |Calculate citation metrics for WikiJHum articles from 2018-2021 |Similar to the other citation metrics tasks, this one will need to create a page from scratch for WikiJHum and calculate impact factors for 2018, 2019, 2020 and 2021 | | | | |- |[[WikiJournal Preprints/“It’s all about people skills”: Perspectives on the social license of the forest products industry from rural North America]] |Using [[:File:Soc license forestry NA Annotated text and reviewers comments - Ian Thomson.pdf|reviewer's annotated PDF]], extract the original text from the file and transfer onto the bare wiki page. |Emma Chiu |180 |2022-06-20 |Complete |- |[[WikiJournal of Medicine/History of penicillin|History of penicillin]] |format PDF and upload |Ellen Sussman | | | |- |[[WikiJournal of Medicine/Phage Therapy|Phage Therapy]] |format PDF and upload |Andrew Neil | | | |- |[[WikiJournal of Science/“Collect, acquire, analyze, report, and disseminate statistical data related to the science and engineering enterprise…”: The National Center for Science and Engineering Statistics|“Collect, acquire, analyze, report, and disseminate statistical data related to the science and engineering enterprise…”: The National Center for Science and Engineering Statistics]] |format PDF and upload |Jenna Harmon |95 |2022-05-06 |complete |- |[[WikiJournal User Group/Editorial guidelines#Registering article in DOAJ|DOAJ]] |upload WikiJMed and WikiJSci [[WikiJournal User Group/Editorial guidelines#Registering article in DOAJ|article metadata to DOAJ]] | | | | |- |[[WikiJournal Preprints/The Kivu Ebola epidemic|Kivu Ebola epidemic]] |format PDF and upload |Ellen Sussman | | | |- |[[WikiJournal Preprints/Parenting_stress|Parenting stress]] |format PDF and upload |Natalie Charamut | | | |- |[[WikiJournal Preprints/Leptospirosis|Leptospirosis]] |format PDF and upload |Andrew Neil | | | |- |[[WikiJournal Preprints/Melioidosis|Melioidosis]] | Accepted, update article page, generate DOI and PDF please |Emma Chiu | | | |- |[[WikiJournal Preprints/A history of coronaviruses|A history of coronaviruses]] |Accepted, update article page, generate DOI and PDF please | | | | |- |[[WikiJournal Preprints/The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats|The effect of local millet drink (Kunu) on the testis and epididymis of adult male wistar rats]] |Using [[:File:Kunu and wistar rates after review tracked changes.pdf|updated manuscript's PDF]], copy the text from the file and transfer onto the wiki page |Peter Agan | | | |- |Crossref reference deposit (see [[Talk:WikiJournal User Group#Talk:WikiJournal User Group|discussion]]) |[[WikiJournal User Group/Editorial guidelines#Submitting reference metadata|Submit metadata with references for published articles onto Crossref platform]] |Michelle Fong | | |- | |editorial process training 1 & 2 |Ellen Sussman |120 |2022-07-11 |Complete |- | |editorial process training 1 & 2 |Andrew Neil |120 |2022-07-11 |Complete |- | |editorial process training 1 & 2 |Natalie Charamut |120 |2022-07-11 |Complete |- | |editorial process training 1 & 2 |Emma Chiu |120 |2022-07-11 |Complete |- | |editorial process training 1 & 2 |Crystal Au |120 |2022-07-11 |Complete |- | |editorial process training 1 & 2 |Michelle Fong |120 |2022-07-11 |Complete |} gttzdnao689rsr3pt3hgmbvaac2cbyh 0 271409 2416171 2251394 2022-08-18T15:25:22Z 2402:3A80:1CB1:D9D2:4D56:30BA:38AB:8A4F /* Paper 2 */ wikitext text/x-wiki GCSE Biology is a course taken by many students in the United Kingdom. This Wikiversity Course is, primarily, based on the Edexcel GCSE (9-1) specification, however will still be valid for other GCSE courses == Paper 1 == === Key Concepts === [[Cells]] [[Microscopy]] [[Units - Biology|Units]] [[Practical: Investigating biological specimens using microscopes]] [[Enzymes]] [[Practical: Investigating the effect of pH on enzyme activity]] [[Food Tests]] [[Practical: Investigate the use of chemical reagents to identify starch, reducing sugars, proteins, and fats]] [[Osmosis]] [[Practical: Investigating osmosis in potatoes]] === Cells and Control === === Genetics === === Natural Selection and Genetic Modification === === Health, Disease, and the Development of Medicines === == Paper 2 == [https://academicachievement.co.uk/ Academic Achievement] - [https://academicachievement.co.uk/ GCSE Summer Revisions Provider UK] d43k3ck1mu5176c7ef0lmxzr4md9pg0 2416346 2416171 2022-08-18T23:05:03Z Dave Braunschweig 426084 Solicitation wikitext text/x-wiki GCSE Biology is a course taken by many students in the United Kingdom. This Wikiversity Course is, primarily, based on the Edexcel GCSE (9-1) specification, however will still be valid for other GCSE courses == Paper 1 == === Key Concepts === [[Cells]] [[Microscopy]] [[Units - Biology|Units]] [[Practical: Investigating biological specimens using microscopes]] [[Enzymes]] [[Practical: Investigating the effect of pH on enzyme activity]] [[Food Tests]] [[Practical: Investigate the use of chemical reagents to identify starch, reducing sugars, proteins, and fats]] [[Osmosis]] [[Practical: Investigating osmosis in potatoes]] === Cells and Control === === Genetics === === Natural Selection and Genetic Modification === === Health, Disease, and the Development of Medicines === q8zs7umx2hp1k2u0ps8w05plasuhz6u 0 271693 2416235 2415417 2022-08-18T17:46:12Z 2402:7500:901:8CFB:E4FA:2E2:622C:2187 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b''||number of unsolved families in base ''b''||searching limit of length for the unsolved families in base ''b'' (if there are different searching limits for the unsolved families in base ''b'', choose the lowest searching limit) |- |2||1||11||2||3||0||– |- |3||3||111||3||13||0||– |- |4||5||221||3||41||0||– |- |5||22||10₉₃13||96||5⁹⁵+8||0||– |- |6||11||40041||5||5209||0||– |- |7||71||3₁₆1||17||(7¹⁷−5)/2||0||– |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7||0||– |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10||0||– |- |10||77||50₂₈27||31||5×10³⁰+27||0||– |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10||0||– |- |12||106||40₃₉77||42||4×12⁴¹+91||0||– |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183||2||111000 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1||0||– |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2||0||– |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15||1||98000 |- |17^*||10409~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1||21||32074 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221||0||– |- |19^*||31404~31435||1E70₁₂₂₈₉₆1||122900||634×19¹²²⁸⁹⁷+1||31||20000 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13||0||– |- |21^*||13375~13395||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4||20||20000 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21||0||– |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49||0||– |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1||0||– |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7||6||20000 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 19, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁵⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 19, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 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===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777F ====Additional known quasi-minimal primes (not necessarily the next)==== 570₅₁₃₁₀1, 49₁₁₁₃₃₃, 970₁₆₆₀₄₇1, F70₁₈₆₇₆₇1 ===Base 18=== 11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, C0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 19=== ====Additional known quasi-minimal primes (not necessarily the next)==== 90₄₂₉₉₄G, 4F0₄₉₈₄₇6, FG6₁₁₀₉₈₄, 1E70₁₂₂₈₉₆1 ===Base 20=== 13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')) by covering congruence, algebraic factorization, or combine of them. {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||32074||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||32074||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||32074||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||32074||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||32074||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||32074||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||32074||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||32074||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||32074||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}B3||11×17^''n''+2+190||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}DB||11×17^''n''+2+232||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2BE||(11×17^''n''+3−41579)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2E||(11×17^''n''+2−2411)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}E9||(11×17^''n''+2+773)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}EE||(11×17^''n''+2+853)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- 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|21||DH{D}||(5813×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%285813*21%5En-13%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||F{9}D||(309×21^''n''+1+71)/20||20000||[http://factordb.com/index.php?query=%28309*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{F}35||(3×21^''n''+2−1051)/4||20000||[http://factordb.com/index.php?query=%283*21%5E%28n%2B2%29-1051%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||G{0}FK||16×21^''n''+2+335||506722||[http://factordb.com/index.php?query=16*21%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{0}7771||17×21^''n''+4+68062||20000||[http://factordb.com/index.php?query=17*21%5E%28n%2B4%29%2B68062&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{D}||(353×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%28353*21%5En-13%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{J}6J||(19×21^''n''+2−5479)/20||20000||[http://factordb.com/index.php?query=%2819*21%5E%28n%2B2%29-5479%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E For base 19, the smallest prime in family {2}7A may or may not be quasi-minimal prime, since another unsolved family is {2}7, and the smallest prime in family 333{5} may or may not be quasi-minimal prime, since another unsolved family is 3{5}, and the smallest prime in family 5{H}05 may or may not be quasi-minimal prime, since another unsolved family is 5{H}5, and the smallest prime in family FHHH0{H} may or may not be quasi-minimal prime, since another unsolved family is FH0{H} For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D} ===Base 19=== All searched to length 20000 * 1{0}7717 * {2}7 * {2}7A * 333{5} * 3{5} * 4{0}HB5 * 4B5{0}H * {5}3 * 5{H}05 * 5{H}0H * 5{H}5 * 66{B} * 71{0}177 * 7AF{0}H * 97{0}3 * {B}FG * C{7}C * C{H}C * D9{0}9 * D{B} * {D}88 * EE1{6} * F{7}5 * F{B}G * F{D}F * FH0{H} * FHHH0{H} * H0F{0}7A * HB{0}5B5 * {H}6 * II{D} ==Primality certificates for the proven primes > 10²⁹⁹== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003810023889], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003810023889] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] 650j612tlamt0kfgnwhv81zae7pv8mw 2416237 2416235 2022-08-18T17:53:10Z 2402:7500:901:8CFB:E4FA:2E2:622C:2187 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b''||number of unsolved families in base ''b''||searching limit of length for the unsolved families in base ''b'' (if there are different searching limits for the unsolved families in base ''b'', choose the lowest searching limit) |- |2||1||11||2||3||0||– |- |3||3||111||3||13||0||– |- |4||5||221||3||41||0||– |- |5||22||10₉₃13||96||5⁹⁵+8||0||– |- |6||11||40041||5||5209||0||– |- |7||71||3₁₆1||17||(7¹⁷−5)/2||0||– |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7||0||– |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10||0||– |- |10||77||50₂₈27||31||5×10³⁰+27||0||– |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10||0||– |- |12||106||40₃₉77||42||4×12⁴¹+91||0||– |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183||2||111000 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1||0||– |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2||0||– |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15||1||98000 |- |17^*||10409~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1||21||32074 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221||0||– |- |19^*||31404~31435||1E70₁₂₂₈₉₆1||122900||634×19¹²²⁸⁹⁷+1||31||20000 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13||0||– |- |21^*||13375~13395||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4||20||20000 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21||0||– |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49||0||– |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1||0||– |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7||6||20000 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 19, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁵⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 19, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777F ====Additional known quasi-minimal primes (not necessarily the next)==== 570₅₁₃₁₀1, 49₁₁₁₃₃₃, 970₁₆₆₀₄₇1, F70₁₈₆₇₆₇1 ===Base 18=== 11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, C0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 19=== ====Additional known quasi-minimal primes (not necessarily the next)==== 90₄₂₉₉₄G, 4F0₄₉₈₄₇6, FG6₁₁₀₉₈₄, 1E70₁₂₂₈₉₆1 ===Base 20=== 13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')) by covering congruence, algebraic factorization, or combine of them. {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||32074||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||32074||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||32074||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||32074||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||32074||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||32074||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||32074||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||32074||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||32074||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}B3||11×17^''n''+2+190||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}DB||11×17^''n''+2+232||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2BE||(11×17^''n''+3−41579)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2E||(11×17^''n''+2−2411)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}E9||(11×17^''n''+2+773)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}EE||(11×17^''n''+2+853)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- 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|21||B3{0}EB||234×21^''n''+2+305||20000||[http://factordb.com/index.php?query=234*21%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B9{0}E5||240×21^''n''+2+299||20000||[http://factordb.com/index.php?query=240*21%5E%28n%2B2%29%2B299&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B{D}B||(233×21^''n''+1−53)/20||20000||[http://factordb.com/index.php?query=%28233*21%5E%28n%2B1%29-53%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B{H}6H||(237×21^''n''+2−4637)/20||20000||[http://factordb.com/index.php?query=%28237*21%5E%28n%2B2%29-4637%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||DH{D}||(5813×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%285813*21%5En-13%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||F{9}D||(309×21^''n''+1+71)/20||20000||[http://factordb.com/index.php?query=%28309*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{F}35||(3×21^''n''+2−1051)/4||20000||[http://factordb.com/index.php?query=%283*21%5E%28n%2B2%29-1051%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||G{0}FK||16×21^''n''+2+335||506722||[http://factordb.com/index.php?query=16*21%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{0}7771||17×21^''n''+4+68062||20000||[http://factordb.com/index.php?query=17*21%5E%28n%2B4%29%2B68062&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{D}||(353×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%28353*21%5En-13%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{J}6J||(19×21^''n''+2−5479)/20||20000||[http://factordb.com/index.php?query=%2819*21%5E%28n%2B2%29-5479%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E For base 19, the smallest prime in family {2}7A may or may not be quasi-minimal prime, since another unsolved family is {2}7, and the smallest prime in family 333{5} may or may not be quasi-minimal prime, since another unsolved family is 3{5}, and the smallest prime in family 5{H}05 may or may not be quasi-minimal prime, since another unsolved family is 5{H}5, and the smallest prime in family FHHH0{H} may or may not be quasi-minimal prime, since another unsolved family is FH0{H} For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D} ===Base 19=== All searched to length 20000 * 1{0}7717 * {2}7 * {2}7A * 333{5} * 3{5} * 4{0}HB5 * 4B5{0}H * {5}3 * 5{H}05 * 5{H}0H * 5{H}5 * 66{B} * 71{0}177 * 7AF{0}H * 97{0}3 * {B}FG * C{7}C * C{H}C * D9{0}9 * D{B} * {D}88 * EE1{6} * F{7}5 * F{B}G * F{D}F * FH0{H} * FHHH0{H} * H0F{0}7A * HB{0}5B5 * {H}6 * II{D} ==Primality certificates for the proven primes > 10²⁹⁹== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003810023889], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003810023889] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] 00h39v2xzgrc7ft2kzj7k63s9mloo43 2416238 2416237 2022-08-18T17:53:26Z 2402:7500:901:8CFB:E4FA:2E2:622C:2187 /* Base 19 */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b''||number of unsolved families in base ''b''||searching limit of length for the unsolved families in base ''b'' (if there are different searching limits for the unsolved families in base ''b'', choose the lowest searching limit) |- |2||1||11||2||3||0||– |- |3||3||111||3||13||0||– |- |4||5||221||3||41||0||– |- |5||22||10₉₃13||96||5⁹⁵+8||0||– |- |6||11||40041||5||5209||0||– |- |7||71||3₁₆1||17||(7¹⁷−5)/2||0||– |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7||0||– |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10||0||– |- |10||77||50₂₈27||31||5×10³⁰+27||0||– |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10||0||– |- |12||106||40₃₉77||42||4×12⁴¹+91||0||– |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183||2||111000 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1||0||– |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2||0||– |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15||1||98000 |- |17^*||10409~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1||21||32074 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221||0||– |- |19^*||31404~31435||1E70₁₂₂₈₉₆1||122900||634×19¹²²⁸⁹⁷+1||31||20000 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13||0||– |- |21^*||13375~13395||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4||20||20000 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21||0||– |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49||0||– |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1||0||– |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7||6||20000 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 19, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁵⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 19, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 ===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 777070000000047, 777777777771777, 777777777777177, 990000000000799, A00000000444441, 1000000000000073, 1000000000000404, 4700000000000404, 5777777770777777, 6000000000000083, 7077777777777771, 7707000700000047, 7770000000000084, 7770000000007047, 8888888888888306, 8888888888888322, 9707777777777777, 11111111111111111, 14444444444441111, 44444444444444111, 70000000000000004, 70000000000040044, 70000000007477777, 77777777777770044, 77777777777771007, 77777777777777717, A1444444444444444, A5555555555555509, A9999999999999996, 320000000000000002, 597777777777777707, 707700700000000074, 770000000000077704, 805055555555555555, 888888888888888883, 997700000000000007, 1444444444444444444, 5077777777777777077, 7777777777777777771, 7777777777777777793, 8550555555555555555, 8555505555555555555, 9777777777777777773, 4000000000000000A041, 555555555555555550503, 5555555555555555A5552, 55AAAAAAAAAAAAAAAAA58, 855555555055555555555, 45AAAAAAAAAAAAAAAAAAAA, 5307777777777777777777, 7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 5900000000000000000000000000000000000000003, A477777777777777777777777777777777777777777, 90000000000000000000000000000000000000009799, 444444444444444444444444444444444444444444441, 577777777777777777777777777777777777777777707777, 9700000000000000000000000000000000000000000000000007, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA0058, 8055555555555555555555555555555555555555555555555555555555555, A9997777777777777777777777777777777777777777777777777777777777, 44777777777777777777777777777777777777777777777777777777777777777, 99777777777777777777777777777777777777777777777777777777777777777, 577077777777777777777777777777777777777777777777777777777777777777, 835000000000000000000000000000000000000000000000000000000000000000007, 74700000000000000000000000000000000000000000000000000000000000000000000000001, 100000000000000000000000000000000000000000000000000000000000000000000000000035, 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555558A, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000037, 57777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777077, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA058, 55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555053, 3266666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051, 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000057, 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555552A, 5077777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, 8555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555505, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA58, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777F ====Additional known quasi-minimal primes (not necessarily the next)==== 570₅₁₃₁₀1, 49₁₁₁₃₃₃, 970₁₆₆₀₄₇1, F70₁₈₆₇₆₇1 ===Base 18=== 11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, C0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 19=== ====Additional known quasi-minimal primes (not necessarily the next)==== 90₄₂₉₉₄G, 4F0₄₉₈₄₇6, FG6₁₁₀₉₈₄, 1E70₁₂₂₈₉₆1 ===Base 20=== 13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')) by covering congruence, algebraic factorization, or combine of them. {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||32074||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||32074||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||32074||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||32074||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||32074||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||32074||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||32074||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||32074||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||32074||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}B3||11×17^''n''+2+190||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}DB||11×17^''n''+2+232||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2BE||(11×17^''n''+3−41579)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2E||(11×17^''n''+2−2411)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}E9||(11×17^''n''+2+773)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}EE||(11×17^''n''+2+853)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- 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|21||B3{0}EB||234×21^''n''+2+305||20000||[http://factordb.com/index.php?query=234*21%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B9{0}E5||240×21^''n''+2+299||20000||[http://factordb.com/index.php?query=240*21%5E%28n%2B2%29%2B299&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B{D}B||(233×21^''n''+1−53)/20||20000||[http://factordb.com/index.php?query=%28233*21%5E%28n%2B1%29-53%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B{H}6H||(237×21^''n''+2−4637)/20||20000||[http://factordb.com/index.php?query=%28237*21%5E%28n%2B2%29-4637%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||DH{D}||(5813×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%285813*21%5En-13%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||F{9}D||(309×21^''n''+1+71)/20||20000||[http://factordb.com/index.php?query=%28309*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{F}35||(3×21^''n''+2−1051)/4||20000||[http://factordb.com/index.php?query=%283*21%5E%28n%2B2%29-1051%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||G{0}FK||16×21^''n''+2+335||506722||[http://factordb.com/index.php?query=16*21%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{0}7771||17×21^''n''+4+68062||20000||[http://factordb.com/index.php?query=17*21%5E%28n%2B4%29%2B68062&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{D}||(353×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%28353*21%5En-13%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{J}6J||(19×21^''n''+2−5479)/20||20000||[http://factordb.com/index.php?query=%2819*21%5E%28n%2B2%29-5479%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E For base 19, the smallest prime in family {2}7A may or may not be quasi-minimal prime, since another unsolved family is {2}7, and the smallest prime in family 333{5} may or may not be quasi-minimal prime, since another unsolved family is 3{5}, and the smallest prime in family 5{H}05 may or may not be quasi-minimal prime, since another unsolved family is 5{H}5, and the smallest prime in family FHHH0{H} may or may not be quasi-minimal prime, since another unsolved family is FH0{H} For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D} ==Primality certificates for the proven primes > 10²⁹⁹== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003810023889], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003810023889] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] orbyqczkztwdxhoq3auo324iq76y2wv 2416245 2416238 2022-08-18T17:57:59Z 2402:7500:901:8CFB:E4FA:2E2:622C:2187 /* Base 17 */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b''||number of unsolved families in base ''b''||searching limit of length for the unsolved families in base ''b'' (if there are different searching limits for the unsolved families in base ''b'', choose the lowest searching limit) |- |2||1||11||2||3||0||– |- |3||3||111||3||13||0||– |- |4||5||221||3||41||0||– |- |5||22||10₉₃13||96||5⁹⁵+8||0||– |- |6||11||40041||5||5209||0||– |- |7||71||3₁₆1||17||(7¹⁷−5)/2||0||– |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7||0||– |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10||0||– |- |10||77||50₂₈27||31||5×10³⁰+27||0||– |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10||0||– |- |12||106||40₃₉77||42||4×12⁴¹+91||0||– |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183||2||111000 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1||0||– |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2||0||– |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15||1||98000 |- |17^*||10409~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1||21||32074 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221||0||– |- |19^*||31404~31435||1E70₁₂₂₈₉₆1||122900||634×19¹²²⁸⁹⁷+1||31||20000 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13||0||– |- |21^*||13375~13395||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4||20||20000 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21||0||– |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49||0||– |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1||0||– |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7||6||20000 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 19, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁵⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 19, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 ===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 777070000000047, 777777777771777, 777777777777177, 990000000000799, A00000000444441, 1000000000000073, 1000000000000404, 4700000000000404, 5777777770777777, 6000000000000083, 7077777777777771, 7707000700000047, 7770000000000084, 7770000000007047, 8888888888888306, 8888888888888322, 9707777777777777, 11111111111111111, 14444444444441111, 44444444444444111, 70000000000000004, 70000000000040044, 70000000007477777, 77777777777770044, 77777777777771007, 77777777777777717, A1444444444444444, A5555555555555509, A9999999999999996, 320000000000000002, 597777777777777707, 707700700000000074, 770000000000077704, 805055555555555555, 888888888888888883, 997700000000000007, 1444444444444444444, 5077777777777777077, 7777777777777777771, 7777777777777777793, 8550555555555555555, 8555505555555555555, 9777777777777777773, 4000000000000000A041, 555555555555555550503, 5555555555555555A5552, 55AAAAAAAAAAAAAAAAA58, 855555555055555555555, 45AAAAAAAAAAAAAAAAAAAA, 5307777777777777777777, 7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 5900000000000000000000000000000000000000003, A477777777777777777777777777777777777777777, 90000000000000000000000000000000000000009799, 444444444444444444444444444444444444444444441, 577777777777777777777777777777777777777777707777, 9700000000000000000000000000000000000000000000000007, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA0058, 8055555555555555555555555555555555555555555555555555555555555, A9997777777777777777777777777777777777777777777777777777777777, 44777777777777777777777777777777777777777777777777777777777777777, 99777777777777777777777777777777777777777777777777777777777777777, 577077777777777777777777777777777777777777777777777777777777777777, 835000000000000000000000000000000000000000000000000000000000000000007, 74700000000000000000000000000000000000000000000000000000000000000000000000001, 100000000000000000000000000000000000000000000000000000000000000000000000000035, 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555558A, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000037, 57777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777077, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA058, 55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555053, 3266666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051, 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000057, 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555552A, 5077777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, 8555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555505, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA58, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ====Additional known quasi-minimal primes (not necessarily the next)==== 570₅₁₃₁₀1, 49₁₁₁₃₃₃, 970₁₆₆₀₄₇1, F70₁₈₆₇₆₇1 ===Base 18=== 11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, C0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 19=== ====Additional known quasi-minimal primes (not necessarily the next)==== 90₄₂₉₉₄G, 4F0₄₉₈₄₇6, FG6₁₁₀₉₈₄, 1E70₁₂₂₈₉₆1 ===Base 20=== 13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')) by covering congruence, algebraic factorization, or combine of them. {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||32074||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||32074||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||32074||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||32074||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||32074||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||32074||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||32074||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||32074||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||32074||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}B3||11×17^''n''+2+190||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}DB||11×17^''n''+2+232||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2BE||(11×17^''n''+3−41579)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2E||(11×17^''n''+2−2411)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}E9||(11×17^''n''+2+773)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}EE||(11×17^''n''+2+853)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- 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|21||B3{0}EB||234×21^''n''+2+305||20000||[http://factordb.com/index.php?query=234*21%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B9{0}E5||240×21^''n''+2+299||20000||[http://factordb.com/index.php?query=240*21%5E%28n%2B2%29%2B299&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B{D}B||(233×21^''n''+1−53)/20||20000||[http://factordb.com/index.php?query=%28233*21%5E%28n%2B1%29-53%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B{H}6H||(237×21^''n''+2−4637)/20||20000||[http://factordb.com/index.php?query=%28237*21%5E%28n%2B2%29-4637%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||DH{D}||(5813×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%285813*21%5En-13%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||F{9}D||(309×21^''n''+1+71)/20||20000||[http://factordb.com/index.php?query=%28309*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{F}35||(3×21^''n''+2−1051)/4||20000||[http://factordb.com/index.php?query=%283*21%5E%28n%2B2%29-1051%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||G{0}FK||16×21^''n''+2+335||506722||[http://factordb.com/index.php?query=16*21%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{0}7771||17×21^''n''+4+68062||20000||[http://factordb.com/index.php?query=17*21%5E%28n%2B4%29%2B68062&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{D}||(353×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%28353*21%5En-13%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{J}6J||(19×21^''n''+2−5479)/20||20000||[http://factordb.com/index.php?query=%2819*21%5E%28n%2B2%29-5479%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E For base 19, the smallest prime in family {2}7A may or may not be quasi-minimal prime, since another unsolved family is {2}7, and the smallest prime in family 333{5} may or may not be quasi-minimal prime, since another unsolved family is 3{5}, and the smallest prime in family 5{H}05 may or may not be quasi-minimal prime, since another unsolved family is 5{H}5, and the smallest prime in family FHHH0{H} may or may not be quasi-minimal prime, since another unsolved family is FH0{H} For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D} ==Primality certificates for the proven primes > 10²⁹⁹== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003810023889], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003810023889] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] 8k8km15t50nfsu835vvzghlesm4e37w 2416246 2416245 2022-08-18T17:58:29Z 2402:7500:901:8CFB:E4FA:2E2:622C:2187 /* Unsolved families */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b''||number of unsolved families in base ''b''||searching limit of length for the unsolved families in base ''b'' (if there are different searching limits for the unsolved families in base ''b'', choose the lowest searching limit) |- |2||1||11||2||3||0||– |- |3||3||111||3||13||0||– |- |4||5||221||3||41||0||– |- |5||22||10₉₃13||96||5⁹⁵+8||0||– |- |6||11||40041||5||5209||0||– |- |7||71||3₁₆1||17||(7¹⁷−5)/2||0||– |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7||0||– |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10||0||– |- |10||77||50₂₈27||31||5×10³⁰+27||0||– |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10||0||– |- |12||106||40₃₉77||42||4×12⁴¹+91||0||– |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183||2||111000 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1||0||– |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2||0||– |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15||1||98000 |- |17^*||10409~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1||21||32074 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221||0||– |- |19^*||31404~31435||1E70₁₂₂₈₉₆1||122900||634×19¹²²⁸⁹⁷+1||31||20000 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13||0||– |- |21^*||13375~13395||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4||20||20000 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21||0||– |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49||0||– |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1||0||– |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7||6||20000 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 19, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁵⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 19, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 ===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 777070000000047, 777777777771777, 777777777777177, 990000000000799, A00000000444441, 1000000000000073, 1000000000000404, 4700000000000404, 5777777770777777, 6000000000000083, 7077777777777771, 7707000700000047, 7770000000000084, 7770000000007047, 8888888888888306, 8888888888888322, 9707777777777777, 11111111111111111, 14444444444441111, 44444444444444111, 70000000000000004, 70000000000040044, 70000000007477777, 77777777777770044, 77777777777771007, 77777777777777717, A1444444444444444, A5555555555555509, A9999999999999996, 320000000000000002, 597777777777777707, 707700700000000074, 770000000000077704, 805055555555555555, 888888888888888883, 997700000000000007, 1444444444444444444, 5077777777777777077, 7777777777777777771, 7777777777777777793, 8550555555555555555, 8555505555555555555, 9777777777777777773, 4000000000000000A041, 555555555555555550503, 5555555555555555A5552, 55AAAAAAAAAAAAAAAAA58, 855555555055555555555, 45AAAAAAAAAAAAAAAAAAAA, 5307777777777777777777, 7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 5900000000000000000000000000000000000000003, A477777777777777777777777777777777777777777, 90000000000000000000000000000000000000009799, 444444444444444444444444444444444444444444441, 577777777777777777777777777777777777777777707777, 9700000000000000000000000000000000000000000000000007, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA0058, 8055555555555555555555555555555555555555555555555555555555555, A9997777777777777777777777777777777777777777777777777777777777, 44777777777777777777777777777777777777777777777777777777777777777, 99777777777777777777777777777777777777777777777777777777777777777, 577077777777777777777777777777777777777777777777777777777777777777, 835000000000000000000000000000000000000000000000000000000000000000007, 74700000000000000000000000000000000000000000000000000000000000000000000000001, 100000000000000000000000000000000000000000000000000000000000000000000000000035, 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555558A, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000037, 57777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777077, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA058, 55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555053, 3266666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051, 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000057, 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555552A, 5077777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, 8555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555505, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA58, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ====Additional known quasi-minimal primes (not necessarily the next)==== 570₅₁₃₁₀1, 49₁₁₁₃₃₃, 970₁₆₆₀₄₇1, F70₁₈₆₇₆₇1 ===Base 18=== 11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, C0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 19=== ====Additional known quasi-minimal primes (not necessarily the next)==== 90₄₂₉₉₄G, 4F0₄₉₈₄₇6, FG6₁₁₀₉₈₄, 1E70₁₂₂₈₉₆1 ===Base 20=== 13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')) by covering congruence, algebraic factorization, or combine of them. {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||32074||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||32074||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||32074||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||32074||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||32074||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||32074||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||32074||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||32074||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||32074||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}B3||11×17^''n''+2+190||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}DB||11×17^''n''+2+232||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2BE||(11×17^''n''+3−41579)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2E||(11×17^''n''+2−2411)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}E9||(11×17^''n''+2+773)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}EE||(11×17^''n''+2+853)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- 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|21||B9{0}E5||240×21^''n''+2+299||20000||[http://factordb.com/index.php?query=240*21%5E%28n%2B2%29%2B299&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B{D}B||(233×21^''n''+1−53)/20||20000||[http://factordb.com/index.php?query=%28233*21%5E%28n%2B1%29-53%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B{H}6H||(237×21^''n''+2−4637)/20||20000||[http://factordb.com/index.php?query=%28237*21%5E%28n%2B2%29-4637%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||DH{D}||(5813×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%285813*21%5En-13%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||F{9}D||(309×21^''n''+1+71)/20||20000||[http://factordb.com/index.php?query=%28309*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{F}35||(3×21^''n''+2−1051)/4||20000||[http://factordb.com/index.php?query=%283*21%5E%28n%2B2%29-1051%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||G{0}FK||16×21^''n''+2+335||506722||[http://factordb.com/index.php?query=16*21%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{0}7771||17×21^''n''+4+68062||20000||[http://factordb.com/index.php?query=17*21%5E%28n%2B4%29%2B68062&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{D}||(353×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%28353*21%5En-13%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{J}6J||(19×21^''n''+2−5479)/20||20000||[http://factordb.com/index.php?query=%2819*21%5E%28n%2B2%29-5479%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E For base 19, the smallest prime in family {2}7A may or may not be quasi-minimal prime, since another unsolved family is {2}7, and the smallest prime in family 333{5} may or may not be quasi-minimal prime, since another unsolved family is 3{5}, and the smallest prime in family 5{H}05 may or may not be quasi-minimal prime, since another unsolved family is 5{H}5, and the smallest prime in family FHHH0{H} may or may not be quasi-minimal prime, since another unsolved family is FH0{H} For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D} ==Primality certificates for the proven primes > 10²⁹⁹== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003810023889], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003810023889] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] iet3jsu2lzzkjpggyc3b9waj0ad4v89 2416247 2416246 2022-08-18T17:58:58Z 2402:7500:901:8CFB:E4FA:2E2:622C:2187 /* Condensed table */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b''||number of unsolved families in base ''b''||searching limit of length for the unsolved families in base ''b'' (if there are different searching limits for the unsolved families in base ''b'', choose the lowest searching limit) |- |2||1||11||2||3||0||– |- |3||3||111||3||13||0||– |- |4||5||221||3||41||0||– |- |5||22||10₉₃13||96||5⁹⁵+8||0||– |- |6||11||40041||5||5209||0||– |- |7||71||3₁₆1||17||(7¹⁷−5)/2||0||– |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7||0||– |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10||0||– |- |10||77||50₂₈27||31||5×10³⁰+27||0||– |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10||0||– |- |12||106||40₃₉77||42||4×12⁴¹+91||0||– |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183||2||111000 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1||0||– |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2||0||– |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15||1||98000 |- |17^*||10410~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1||20||32074 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221||0||– |- |19^*||31404~31435||1E70₁₂₂₈₉₆1||122900||634×19¹²²⁸⁹⁷+1||31||20000 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13||0||– |- |21^*||13375~13395||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4||20||20000 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21||0||– |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49||0||– |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1||0||– |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7||6||20000 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 19, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁵⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 19, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 ===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 777070000000047, 777777777771777, 777777777777177, 990000000000799, A00000000444441, 1000000000000073, 1000000000000404, 4700000000000404, 5777777770777777, 6000000000000083, 7077777777777771, 7707000700000047, 7770000000000084, 7770000000007047, 8888888888888306, 8888888888888322, 9707777777777777, 11111111111111111, 14444444444441111, 44444444444444111, 70000000000000004, 70000000000040044, 70000000007477777, 77777777777770044, 77777777777771007, 77777777777777717, A1444444444444444, A5555555555555509, A9999999999999996, 320000000000000002, 597777777777777707, 707700700000000074, 770000000000077704, 805055555555555555, 888888888888888883, 997700000000000007, 1444444444444444444, 5077777777777777077, 7777777777777777771, 7777777777777777793, 8550555555555555555, 8555505555555555555, 9777777777777777773, 4000000000000000A041, 555555555555555550503, 5555555555555555A5552, 55AAAAAAAAAAAAAAAAA58, 855555555055555555555, 45AAAAAAAAAAAAAAAAAAAA, 5307777777777777777777, 7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 5900000000000000000000000000000000000000003, A477777777777777777777777777777777777777777, 90000000000000000000000000000000000000009799, 444444444444444444444444444444444444444444441, 577777777777777777777777777777777777777777707777, 9700000000000000000000000000000000000000000000000007, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA0058, 8055555555555555555555555555555555555555555555555555555555555, A9997777777777777777777777777777777777777777777777777777777777, 44777777777777777777777777777777777777777777777777777777777777777, 99777777777777777777777777777777777777777777777777777777777777777, 577077777777777777777777777777777777777777777777777777777777777777, 835000000000000000000000000000000000000000000000000000000000000000007, 74700000000000000000000000000000000000000000000000000000000000000000000000001, 100000000000000000000000000000000000000000000000000000000000000000000000000035, 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555558A, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000037, 57777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777077, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA058, 55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555053, 3266666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051, 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000057, 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555552A, 5077777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, 8555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555505, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA58, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ====Additional known quasi-minimal primes (not necessarily the next)==== 570₅₁₃₁₀1, 49₁₁₁₃₃₃, 970₁₆₆₀₄₇1, F70₁₈₆₇₆₇1 ===Base 18=== 11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, C0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 19=== ====Additional known quasi-minimal primes (not necessarily the next)==== 90₄₂₉₉₄G, 4F0₄₉₈₄₇6, FG6₁₁₀₉₈₄, 1E70₁₂₂₈₉₆1 ===Base 20=== 13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')) by covering congruence, algebraic factorization, or combine of them. {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||32074||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||32074||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||32074||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||32074||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||32074||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||32074||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||32074||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||32074||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||32074||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}B3||11×17^''n''+2+190||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}DB||11×17^''n''+2+232||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2BE||(11×17^''n''+3−41579)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2E||(11×17^''n''+2−2411)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}E9||(11×17^''n''+2+773)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}EE||(11×17^''n''+2+853)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- 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|21||B9{0}E5||240×21^''n''+2+299||20000||[http://factordb.com/index.php?query=240*21%5E%28n%2B2%29%2B299&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B{D}B||(233×21^''n''+1−53)/20||20000||[http://factordb.com/index.php?query=%28233*21%5E%28n%2B1%29-53%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B{H}6H||(237×21^''n''+2−4637)/20||20000||[http://factordb.com/index.php?query=%28237*21%5E%28n%2B2%29-4637%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||DH{D}||(5813×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%285813*21%5En-13%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||F{9}D||(309×21^''n''+1+71)/20||20000||[http://factordb.com/index.php?query=%28309*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{F}35||(3×21^''n''+2−1051)/4||20000||[http://factordb.com/index.php?query=%283*21%5E%28n%2B2%29-1051%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||G{0}FK||16×21^''n''+2+335||506722||[http://factordb.com/index.php?query=16*21%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{0}7771||17×21^''n''+4+68062||20000||[http://factordb.com/index.php?query=17*21%5E%28n%2B4%29%2B68062&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{D}||(353×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%28353*21%5En-13%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{J}6J||(19×21^''n''+2−5479)/20||20000||[http://factordb.com/index.php?query=%2819*21%5E%28n%2B2%29-5479%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E For base 19, the smallest prime in family {2}7A may or may not be quasi-minimal prime, since another unsolved family is {2}7, and the smallest prime in family 333{5} may or may not be quasi-minimal prime, since another unsolved family is 3{5}, and the smallest prime in family 5{H}05 may or may not be quasi-minimal prime, since another unsolved family is 5{H}5, and the smallest prime in family FHHH0{H} may or may not be quasi-minimal prime, since another unsolved family is FH0{H} For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D} ==Primality certificates for the proven primes > 10²⁹⁹== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003810023889], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003810023889] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] k70p4fab4f20ju9z2kxhb6ouvfsdiib 2416249 2416247 2022-08-18T17:59:30Z 2402:7500:901:8CFB:E4FA:2E2:622C:2187 /* Condensed table */ wikitext text/x-wiki A '''quasi-minimal prime''' is a [[w:Prime number|prime number]] for which there is no shorter [[w:Subsequence|subsequence]] of its [[w:Numerical digit|digit]]s in a given [[w:Radix|base]] ''b'' that form a prime > ''b''. For example, 857 is a quasi-minimal prime in [[w:Decimal|decimal]] because there is no prime > 10 among the shorter subsequences of the digits: 8, 5, 7, 85, 87, 57. The subsequence does not have to consist of consecutive digits, so 149 is not a quasi-minimal prime in decimal (because 19 is prime and 19 > 10). But it does have to be in the same order; so, for example, 991 is still a quasi-minimal prime in decimal even though a subset of the digits can form the shorter prime 19 > 10 by changing the order. (using A−Z to represent digit values 10 to 35) For the quasi-minimal primes in bases up to 36, I have only solved (found all quasi-minimal primes and proved that these are all such primes) bases 2~12, 14~15, 18, 20, 22, 24, 30 (bases 11, 22, 30 need primality proving of the probable primes). For the remain bases 13, 16~17, 19, 21, 23, 25~29, 31~36, there are some ''x''{''d''}''y'' (with ''x'', ''y'' strings (may be [[w:Empty string|empty]]) with digits in base ''b'', ''d'' digit in base ''b'') families which are not solved (not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')). I left as a challenge to readers the task of solving (finding all quasi-minimal primes and proving that these are all such primes) bases 13, 16~17, 19, 21, 23, 25~29, 31~36 (this will be a hard problem, e.g. base 23 has a quasi-minimal prime 9E₈₀₀₈₇₃, and base 36 has quasi-minimal prime P₈₁₉₉₃SZ). Proving the set of the quasi-minimal primes in base ''b'' is ''S'', is equivalent to: * Prove that all elements in ''S'', when read as base ''b'' representation, are primes > ''b''. * Prove that all [[w:Proper subset|proper]] subsequence of all elements in ''S'', when read as base ''b'' representation, which are > ''b'', are composite. * Prove that all primes > ''b'', when written in base ''b'', contain at least one element in ''S'' as subsequence (equivalently, prove that all strings not containing any element in ''S'' as subsequence, when read as base ''b'' representation, which are > ''b'', are composite). e.g. proving the set of the quasi-minimal primes in base ''b'' = 10 is {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027}, is equivalent to: * Prove that all of 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 are primes > 10. * Prove that all proper subsequence of all elements in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} which are > 10 are composite. * Prove that all primes > 10 contain at least one element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence (equivalently, prove that all numbers > 10 not containing any element in {11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027} as subsequence are composite). ==Condensed table== {|class=wikitable |''b''||number of quasi-minimal primes base ''b''||base-''b'' form of largest known quasi-minimal prime base ''b''||length of largest known quasi-minimal prime base ''b''||algebraic ((''a''×''b''^''n''+''c'')/''d'') form of largest known quasi-minimal prime base ''b''||number of unsolved families in base ''b''||searching limit of length for the unsolved families in base ''b'' (if there are different searching limits for the unsolved families in base ''b'', choose the lowest searching limit) |- |2||1||11||2||3||0||– |- |3||3||111||3||13||0||– |- |4||5||221||3||41||0||– |- |5||22||10₉₃13||96||5⁹⁵+8||0||– |- |6||11||40041||5||5209||0||– |- |7||71||3₁₆1||17||(7¹⁷−5)/2||0||– |- |8||75||4₂₂₀7||221||(4×8²²¹+17)/7||0||– |- |9||151||30₁₁₅₈11||1161||3×9¹¹⁶⁰+10||0||– |- |10||77||50₂₈27||31||5×10³⁰+27||0||– |- |11^*||1068||57₆₂₆₆₈||62669||(57×11⁶²⁶⁶⁸−7)/10||0||– |- |12||106||40₃₉77||42||4×12⁴¹+91||0||– |- |13^*||3195~3197||80₃₂₀₁₇111||32021||8×13³²⁰²⁰+183||2||111000 |- |14||650||4D₁₉₆₉₈||19699||5×14¹⁹⁶⁹⁸−1||0||– |- |15||1284||7₁₅₅97||157||(15¹⁵⁷+59)/2||0||– |- |16^*||2346~2347||4₇₂₇₈₅DD||72787||(4×16⁷²⁷⁸⁷+2291)/15||1||98000 |- |17^*||10410~10428||F70₁₈₆₇₆₇1||186770||262×17¹⁸⁶⁷⁶⁸+1||18||32074 |- |18||549||C0₆₂₆₈C5||6271||12×18⁶²⁷⁰+221||0||– |- |19^*||31404~31435||1E70₁₂₂₈₉₆1||122900||634×19¹²²⁸⁹⁷+1||31||20000 |- |20||3314||G0₆₂₆₉D||6271||16×20⁶²⁷⁰+13||0||– |- |21^*||13375~13395||CF₄₇₉₁₄₇0K||479150||(51×21⁴⁷⁹¹⁴⁹−1243)/4||20||20000 |- |22^*||8003||BK₂₂₀₀₁5||22003||(251×22²²⁰⁰²−335)/21||0||– |- |24||3409||N00N₈₁₂₉LN||8134||13249×24⁸¹³¹−49||0||– |- |30^*||2619||OT₃₄₂₀₅||34206||25×30³⁴²⁰⁵−1||0||– |- |36^*||35257~35263||P₈₁₉₉₃SZ||81995||(5×36⁸¹⁹⁹⁵+821)/7||6||20000 |} ^* Data assumes the primality of the [[w:probable prime|probable prime]]s. Except bases ''b'' = 13, 16, 17, 19, 21, all bases in this table are completely solved (if we allow strong probable primes > 10²⁵⁰⁰⁰), also, except bases ''b'' = 11, 13, 16, 17, 19, 21, 22, 30, 36, all bases in this table are completely solved even if we only allow definitely primes (thus, we can complete the classification of the quasi-minimal primes in these bases, i.e. the “quasi-minimal problems” in these bases are now theorems), for the quasi-minimal primes see the data below. Base ''b'' = 13 has 3195 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 13, then they must be of the form 9{5} or A{3}A (we are unable to determine if these two families contain a prime or not, i.e. these two families have no known prime members, nor can these two families be ruled out as only containing composites), and must have at least 111000 digits in base 13, besides, since these two families can contain at most one quasi-minimal prime, there are at most 3197 quasi-minimal primes in base 13. (i.e. the quasi-minimal primes in base 13 are the 3195 known quasi-minimal primes in base 13 (they are given in the data section) plus the smallest prime in the family 9{5} in base 13 (if exists) plus the smallest prime in the family A{3}A in base 13 (if exists)) Base ''b'' = 16 has 2346 known quasi-minimal primes (or PRPs), see the data below, and if there are more quasi-minimal primes in base 16, then they must be of the form {3}AF (we are unable to determine if this family contains a prime or not, i.e. this family have no known prime members, nor can this family be ruled out as only containing composites), and must have at least 98000 digits in base 16, besides, since this family can contain at most one quasi-minimal prime, there are at most 2347 quasi-minimal primes in base 16. (i.e. the quasi-minimal primes in base 16 are the 2346 known quasi-minimal primes in base 16 (they are given in the data section) plus the smallest prime in the family {3}AF in base 16 (if exists)) ==Data for quasi-minimal primes== ===Base 2=== 11 ===Base 3=== 12, 21, 111 ===Base 4=== 11, 13, 23, 31, 221 ===Base 5=== 12, 21, 23, 32, 34, 43, 104, 111, 131, 133, 313, 401, 414, 3101, 10103, 14444, 30301, 33001, 33331, 44441, 300031, 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 ===Base 6=== 11, 15, 21, 25, 31, 35, 45, 51, 4401, 4441, 40041 ===Base 7=== 14, 16, 23, 25, 32, 41, 43, 52, 56, 61, 65, 113, 115, 131, 133, 155, 212, 221, 304, 313, 335, 344, 346, 364, 445, 515, 533, 535, 544, 551, 553, 1022, 1051, 1112, 1202, 1211, 1222, 2111, 3031, 3055, 3334, 3503, 3505, 3545, 4504, 4555, 5011, 5455, 5545, 5554, 6034, 6634, 11111, 11201, 30011, 30101, 31001, 31111, 33001, 33311, 35555, 40054, 100121, 150001, 300053, 351101, 531101, 1100021, 33333301, 5100000001, 33333333333333331 ===Base 8=== 13, 15, 21, 23, 27, 35, 37, 45, 51, 53, 57, 65, 73, 75, 107, 111, 117, 141, 147, 161, 177, 225, 255, 301, 343, 361, 401, 407, 417, 431, 433, 463, 467, 471, 631, 643, 661, 667, 701, 711, 717, 747, 767, 3331, 3411, 4043, 4443, 4611, 5205, 6007, 6101, 6441, 6477, 6707, 6777, 7461, 7641, 47777, 60171, 60411, 60741, 444641, 500025, 505525, 3344441, 4444477, 5500525, 5550525, 55555025, 444444441, 744444441, 77774444441, 7777777777771, 555555555555525, 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447 ===Base 9=== 12, 14, 18, 21, 25, 32, 34, 41, 45, 47, 52, 58, 65, 67, 74, 78, 81, 87, 117, 131, 135, 151, 155, 175, 177, 238, 272, 308, 315, 331, 337, 355, 371, 375, 377, 438, 504, 515, 517, 531, 537, 557, 564, 601, 638, 661, 702, 711, 722, 735, 737, 751, 755, 757, 771, 805, 838, 1011, 1015, 1101, 1701, 2027, 2207, 3017, 3057, 3101, 3501, 3561, 3611, 3688, 3868, 5035, 5051, 5071, 5101, 5501, 5554, 5705, 5707, 7017, 7075, 7105, 7301, 8535, 8544, 8555, 8854, 20777, 22227, 22777, 30161, 33388, 50161, 50611, 53335, 55111, 55535, 55551, 57061, 57775, 70631, 71007, 77207, 100037, 100071, 100761, 105007, 270707, 301111, 305111, 333035, 333385, 333835, 338885, 350007, 500075, 530005, 555611, 631111, 720707, 2770007, 3030335, 7776662, 30300005, 30333335, 38333335, 51116111, 70000361, 300030005, 300033305, 351111111, 1300000007, 5161111111, 8333333335, 300000000035, 311111111161, 544444444444, 2000000000007, 5700000000001, 7270000000007, 88888888833335, 100000000000507, 5111111111111161, 7277777777777777707, 8888888888888888888335, 30000000000000000000051, 1000000000000000000000000057, 56111111111111111111111111111111111111, 7666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666662, 27777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777707, 300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000011 ===Base 10=== 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 227, 251, 257, 277, 281, 349, 409, 449, 499, 521, 557, 577, 587, 727, 757, 787, 821, 827, 857, 877, 881, 887, 991, 2087, 2221, 5051, 5081, 5501, 5581, 5801, 5851, 6469, 6949, 8501, 9001, 9049, 9221, 9551, 9649, 9851, 9949, 20021, 20201, 50207, 60649, 80051, 666649, 946669, 5200007, 22000001, 60000049, 66000049, 66600049, 80555551, 555555555551, 5000000000000000000000000000027 ===Base 11=== 12, 16, 18, 21, 27, 29, 34, 38, 3A, 43, 49, 54, 56, 61, 65, 67, 72, 76, 81, 89, 92, 94, 98, 9A, A3, 10A, 115, 117, 133, 139, 153, 155, 171, 193, 197, 199, 1AA, 225, 232, 236, 25A, 263, 315, 319, 331, 335, 351, 353, 362, 373, 379, 391, 395, 407, 414, 452, 458, 478, 47A, 485, 4A5, 4A7, 502, 508, 511, 513, 533, 535, 539, 551, 571, 579, 588, 595, 623, 632, 70A, 711, 715, 731, 733, 737, 755, 759, 775, 791, 797, 7AA, 803, 847, 858, 85A, 874, 885, 887, 913, 919, 931, 937, 957, 959, 975, 995, A07, A1A, A25, A45, A74, A7A, A85, AA1, AA7, 1101, 11A9, 1305, 1451, 1457, 15A7, 175A, 17A5, 17A9, 2023, 2045, 2052, 2083, 20A5, 2333, 2A05, 2A52, 3013, 3026, 3059, 3097, 3206, 3222, 3233, 3307, 3332, 3505, 4025, 4151, 4157, 4175, 4405, 4445, 4487, 450A, 4575, 5017, 5031, 5059, 5075, 5097, 5099, 5105, 515A, 517A, 520A, 5301, 5583, 5705, 577A, 5853, 5873, 5909, 5A17, 5A57, 5A77, 5A8A, 6683, 66A9, 7019, 7073, 7079, 7088, 7093, 7095, 7309, 7451, 7501, 7507, 7578, 757A, 75A7, 7787, 7804, 7844, 7848, 7853, 7877, 78A4, 7A04, 7A57, 7A79, 7A95, 8078, 8245, 8333, 8355, 8366, 8375, 8425, 8553, 8663, 8708, 8777, 878A, 8A05, 9053, 9101, 9107, 9305, 9505, 9703, A052, A119, A151, A175, A515, A517, A575, A577, A5A8, A719, A779, A911, AAA9, 10011, 10075, 10091, 10109, 10411, 10444, 10705, 10709, 10774, 10901, 11104, 11131, 11144, 11191, 1141A, 114A1, 13757, 1411A, 14477, 144A4, 14A04, 14A11, 17045, 17704, 1774A, 17777, 177A4, 17A47, 1A091, 1A109, 1A114, 1A404, 1A411, 1A709, 20005, 20555, 22203, 25228, 25282, 25552, 25822, 28522, 30037, 30701, 30707, 31113, 33777, 35009, 35757, 39997, 40045, 4041A, 40441, 4045A, 404A1, 4111A, 411A1, 42005, 44401, 44474, 444A1, 44555, 44577, 445AA, 44744, 44A01, 47471, 47477, 47701, 5057A, 50903, 5228A, 52A22, 52A55, 52A82, 55007, 550A9, 55205, 55522, 55557, 55593, 55805, 57007, 57573, 57773, 57807, 5822A, 58307, 58505, 58A22, 59773, 59917, 59973, 59977, 59999, 5A015, 5A2A2, 5AA99, 60836, 60863, 68636, 6A609, 6A669, 6A696, 6A906, 6A966, 70048, 70103, 70471, 70583, 70714, 71474, 717A4, 71A09, 74084, 74444, 74448, 74477, 744A8, 74747, 74774, 7488A, 74A48, 75773, 77144, 77401, 77447, 77799, 77A09, 78008, 78783, 7884A, 78888, 788A8, 79939, 79993, 79999, 7A051, 7A444, 7A471, 80005, 80252, 80405, 80522, 80757, 80AA5, 83002, 84045, 85307, 86883, 88863, 8A788, 90073, 90707, 90901, 95003, 97779, 97939, 99111, 99177, 99973, A0111, A0669, A0966, A0999, A0A09, A1404, A4177, A4401, A4717, A5228, A52AA, A5558, A580A, A5822, A58AA, A5A59, A5AA2, A6096, A6966, A6999, A7051, A7778, A7808, A9055, A9091, A9699, A9969, AA52A, AA58A, 100019, 100079, 101113, 101119, 101911, 107003, 140004, 144011, 144404, 1A0019, 1A0141, 1A5001, 1A7005, 1A9001, 222223, 222823, 300107, 300202, 300323, 303203, 307577, 310007, 332003, 370777, 400555, 401A11, 404001, 404111, 405AAA, 41A011, 440A41, 441011, 451777, 455555, 470051, 470444, 474404, 4A0401, 4A4041, 500015, 500053, 500077, 500507, 505577, 522A2A, 525223, 528A2A, 531707, 550777, 553707, 5555A9, 555A99, 557707, 55A559, 5807A7, 580A0A, 580A55, 58A0AA, 590007, 599907, 5A2228, 5A2822, 5A2AAA, 5A552A, 5AA22A, 5AAA22, 60A069, 683006, 6A0096, 6A0A96, 6A9099, 6A9909, 700778, 701074, 701777, 704408, 704417, 704457, 704484, 707041, 707441, 707708, 707744, 707784, 710777, 717044, 717077, 740008, 74484A, 770441, 770744, 770748, 770771, 777017, 777071, 777448, 777484, 777701, 7778A8, 777A19, 777A48, 778883, 78A808, 790003, 7A1009, 7A4408, 7A7708, 80A555, 828283, 828883, 840555, 850505, 868306, 873005, 883202, 900701, 909739, 909979, 909991, 970771, 977701, 979909, 990739, 990777, 990793, 997099, 999709, 999901, A00009, A00599, A01901, A05509, A0A058, A0A955, A10114, A555A2, A55999, A59991, A5A222, A5A22A, A60609, A66069, A66906, A69006, A79005, A87888, A90099, A90996, A96006, A96666, A97177, A97771, AA0A58, AA5A22, AAA522, 1000501, 1011141, 1030007, 1070047, 111114A, 1111A14, 1111A41, 1144441, 14A4444, 1700005, 1700474, 1A44444, 2555505, 2845055, 3030023, 3100003, 3333397, 4000111, 4011111, 41A1111, 4411111, 444441A, 4444771, 4470004, 4505005, 4744417, 4774441, 4777404, 4777417, 4777747, 4A11111, 4A40001, 5000093, 50005A7, 5005777, 5050553, 5055503, 5070777, 5222222, 5222AAA, 52AAAA2, 52AAAAA, 5505053, 5552AAA, 5555599, 5555A58, 5558A0A, 5558A55, 5558AAA, 55A0009, 55AAA52, 580000A, 5822222, 58AAAAA, 5A2222A, 5AA2222, 6000A69, 6000A96, 6A00069, 7000417, 7000741, 7000835, 7000857, 7007177, 7008305, 7014447, 7017444, 7074177, 7077477, 7077741, 7077747, 7100447, 7174404, 717444A, 7400404, 7700717, 7701077, 7701707, 7707778, 7774004, 7777104, 777741A, 7777441, 777774A, 7777A47, 7779003, 777A008, 777A778, 777A808, 77A4777, 7900399, 8305007, 8500707, 8555707, 8883022, 8AA5222, 9000035, 9007999, 9009717, 9009777, 9009997, 9090997, 9099907, 9355555, 9790099, 9900991, 9900997, 9907909, 9909079, 9979009, 9990079, 9990091, 9990907, 9999771, 9999799, 9999979, A000696, A000991, A001091, A006906, A010044, A040041, A0AAA58, A141111, A5222A2, A600A69, A906606, A909009, A990009, A997701, AA55A52, AAA5552, AAAAA52, 10004747, 10005007, 17000744, 22888823, 28888223, 30010111, 30555777, 31011111, 33000023, 40A00041, 45000055, 47040004, 50377777, 50555553, 5282AAA2, 55505003, 555A5A52, 555AAA2A, 55A5A552, 5AAAAA2A, 60A99999, 70000057, 70070474, 70074704, 70174004, 70700078, 70700474, 70704704, 70710707, 70771007, 70777177, 71074004, 74470001, 77000177, 77070477, 77100077, 77470004, 77700404, 77710007, 77717707, 77748808, 7774A888, 77770078, 77770474, 77774704, 77777008, 77777404, 77777778, 80555055, 88828823, 88888326, 88888823, 8A522222, 90097909, 90700999, 90977777, 97000001, 97000717, 97770007, 99000001, 99000771, 99077001, 99090097, 99777707, 99900097, 99970717, 99999097, 99999707, A0000058, A0004041, A00055A9, A000A559, A1900001, A5555009, A5A55552, A9700001, A9909006, A9990006, A9990606, A9999917, A9999966, 100000507, 100035077, 100050777, 100057707, 101111114, 15A000001, 170000447, 300577777, 40000A401, 447771777, 44A444441, 474000004, 477700004, 477777774, 505000003, 55555AA2A, 5555A5A2A, 700000147, 700017004, 700044004, 700077774, 700170004, 701000047, 701700004, 704000044, 704040004, 707070774, 707077704, 707770704, 707777004, 717000004, 717700007, 770000078, 770004704, 770070747, 770070774, 770700008, 770700084, 770707074, 777000044, 777000774, 777717007, 777770477, 777770747, 7777777A4, 77A700008, 888888302, 900000091, 900090799, 970009099, 990990007, 997000077, 999999997, A0000AA58, A00990001, A05555559, A44444111, A44444777, A44477777, A66666669, A90000606, A99999006, A99999099, 1000007447, 1005000007, 1500000001, 2888882883, 2888888883, 3555555509, 3577777077, 3700000001, 4000000005, 40000005AA, 5377777707, 5555505553, 555555580A, 600000A999, 7000100047, 7000704777, 7007777107, 7057777777, 7070007774, 7077707774, 7077777074, 7100000704, 7470000041, 7701000004, 7707077774, 7770707774, 7777707074, 8888822883, 9555555503, 9900000979, 9999770007, A000144444, A900000066, A999999971, 10000000477, 33333333337, 44444444447, 44444444777, 55A55555552, 60000008883, 68888888306, 68888888883, 70000003999, 70000007717, 70004777777, 70477777777, 77007770004, 77700000477, 77707000704, 77707770074, 77707777774, 77777077774, 77777770004, 83000000006, 97000000999, A0000000001, A0014444444, A4777777771, 100000000057, 305007777777, 305777777777, 370000000007, 377777770007, 377777777107, 700000007474, 707077000074, 707077777774, 707777777717, 770000010004, 771007000007, 777070700004, 777700000704, A95555555555, A99999777777, 1000000003007, 40000000A0041, 58A5555555555, 7004400000004, 7700000000104, 7707000007047, 7707707000004, 7777007000004, 7777700000004, 7777770077704, 7777777710077, 9977777777717, A000000014444, A044444444441, A144444444411, 40000000000401, 45557777777777, 4555AAAAAAAAAA, 59077777777777, 70007777777771, 70077070000074, 70700000004777, 77000007700704, 77700000700047, 77777777770704, 88888888830006, 90900000000799, A0000044444441, 300000000005777, 302000000000002, 55555555A555552, 700000000000174, 770000000000474, 771700000000007, 777070000000047, 777777777771777, 777777777777177, 990000000000799, A00000000444441, 1000000000000073, 1000000000000404, 4700000000000404, 5777777770777777, 6000000000000083, 7077777777777771, 7707000700000047, 7770000000000084, 7770000000007047, 8888888888888306, 8888888888888322, 9707777777777777, 11111111111111111, 14444444444441111, 44444444444444111, 70000000000000004, 70000000000040044, 70000000007477777, 77777777777770044, 77777777777771007, 77777777777777717, A1444444444444444, A5555555555555509, A9999999999999996, 320000000000000002, 597777777777777707, 707700700000000074, 770000000000077704, 805055555555555555, 888888888888888883, 997700000000000007, 1444444444444444444, 5077777777777777077, 7777777777777777771, 7777777777777777793, 8550555555555555555, 8555505555555555555, 9777777777777777773, 4000000000000000A041, 555555555555555550503, 5555555555555555A5552, 55AAAAAAAAAAAAAAAAA58, 855555555055555555555, 45AAAAAAAAAAAAAAAAAAAA, 5307777777777777777777, 7707777777777777777704, 7900000000000000000005, 9777777777777777777707, A999999999999999999999, 10000000000000000000747, 970000000000000000000777, 999900000000000000000007, 3577777777777777777777777, 5555555555555555555555A52, 7000000000000000000777771, 7000000000000000077777771, AAAAAAAAAAAAAAAAAAA000058, 10000000000000000000000044, 77700000000000000000000008, 500777777777777777777777777, 777777777777777777777770774, 855555555555555555555055555, A44444444444444444444444441, 1500000000000000000000000007, 40000000000000000000000000041, 440000000000000000000000000001, 70000000000000000000000000007771, 999999999999999999999999999999991, 95555555555555555555555555555555553, 1900000000000000000000000000000000001, 7777777777777777777777777777777777474, 7777777777777777777777777777777777704, 10000000000000000000000000000000000000307, 50777777777777777777777777777777777777707, 475777777777777777777777777777777777777777, 555555555555555555555555555555555555555A2A, 5900000000000000000000000000000000000000003, A477777777777777777777777777777777777777777, 90000000000000000000000000000000000000009799, 444444444444444444444444444444444444444444441, 577777777777777777777777777777777777777777707777, 9700000000000000000000000000000000000000000000000007, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA0058, 8055555555555555555555555555555555555555555555555555555555555, A9997777777777777777777777777777777777777777777777777777777777, 44777777777777777777777777777777777777777777777777777777777777777, 99777777777777777777777777777777777777777777777777777777777777777, 577077777777777777777777777777777777777777777777777777777777777777, 835000000000000000000000000000000000000000000000000000000000000000007, 74700000000000000000000000000000000000000000000000000000000000000000000000001, 100000000000000000000000000000000000000000000000000000000000000000000000000035, 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555558A, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000037, 57777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777077, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA058, 55555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555053, 3266666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666, 10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000051, 500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000057, 555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555552A, 5077777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, 8555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555505, AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA58, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ===Base 15=== 12, 14, 18, 1E, 21, 27, 2B, 2D, 32, 38, 3E, 41, 47, 4B, 4D, 54, 58, 5E, 67, 6B, 6D, 72, 74, 78, 87, 8B, 92, 94, 9E, A1, A7, AD, B2, B8, BE, C1, CB, CD, D2, D4, E1, ED, 111, 11B, 131, 137, 13B, 13D, 157, 15B, 15D, 171, 177, 197, 19D, 1B7, 1BB, 1D1, 1DB, 1DD, 234, 298, 311, 31B, 337, 33D, 344, 351, 357, 35B, 364, 377, 391, 39B, 39D, 3A4, 3BD, 3C4, 3D7, 3DB, 3DD, 452, 51B, 51D, 531, 53B, 551, 55D, 562, 571, 577, 5A2, 5B1, 5B7, 5BB, 5BD, 5C2, 5D1, 5D7, 634, 652, 681, 698, 717, 71B, 731, 737, 757, 75D, 77D, 79B, 79D, 7B1, 7B7, 7BD, 7D7, 7DD, 801, 852, 88D, 8D8, 91D, 93B, 93D, 95B, 95D, 971, 977, 97B, 97D, 988, 991, 9BD, 9C8, 9D1, A98, AAB, B1D, B31, B3B, B44, B51, B57, B7B, B7D, B97, B9B, BB7, BC4, BD1, BD7, BDD, C07, C34, C52, C7E, C98, CC7, CE7, D0E, D1D, D31, D51, D5B, D68, D77, D7B, D91, D97, DA8, DAE, DCE, DD1, EB4, EEB, 107B, 1091, 10B1, 1107, 110D, 1561, 1651, 1691, 1B01, 2052, 2502, 2522, 303B, 307D, 3097, 30BB, 30D1, 3107, 3361, 3701, 3907, 3B01, 3B0B, 3C97, 4434, 4498, 4834, 4898, 49A8, 4E34, 5037, 507D, 5091, 509B, 5107, 5161, 5202, 53C7, 5552, 570B, 590B, 590D, 59C7, 5A5B, 5C97, 5D0D, 5DAB, 6061, 6151, 6191, 6511, 6601, 6911, 707B, 7091, 7097, 70AE, 70BB, 70CE, 70DB, 7561, 760E, 7691, 76CE, 7907, 7961, 7A0E, 7A3B, 7AEE, 7B0B, 7BAB, 7C0E, 7C77, 7CAE, 7D0B, 7D61, 7DAB, 7E5B, 7E6E, 7E7B, 7EBB, 8098, 811D, 8191, 835D, 853D, 8881, 8908, 8951, 8968, 899D, 8D3D, 8D5D, 8D6E, 8DDD, 8E98, 9011, 9037, 9097, 90D7, 9301, 93C7, 95C7, 9611, 9631, 96A8, 9811, 9851, 989D, 990B, 990D, 998D, 99AB, 99C7, 99D8, 9A08, 9A9B, 9AA8, 9ABB, 9B61, 9BC7, 9D0B, 9DAB, 9DC7, 9DD8, A052, A304, A502, A55B, A9BB, AB04, AB64, B09D, B107, B10B, B161, B1AB, B1C7, B30D, B3C7, B50B, B664, B691, B6A4, B707, B761, B90D, B961, BA5B, BABB, BBAB, BBB4, BC37, BC77, C777, C937, C997, D011, D03D, D05D, D09B, D0B1, D0BD, D101, D10B, D30D, D3AB, D507, D50D, D66E, D761, D7DE, D811, D85D, D86E, D89D, D8C8, D8E8, D9AB, D9D8, DA3B, DA9B, DABB, DB01, DB61, DBAB, DC88, DD07, DD0B, DD7E, DD8D, DDE7, DE6E, E252, E33B, E522, E57B, E7AE, E7CE, E898, E997, E9A8, E9BB, EA34, EB5B, EE98, EEC7, 10017, 10B0D, 170AB, 17A0B, 19001, 19601, 1A09B, 1D0C7, 22E52, 2EA52, 30017, 3001D, 300B1, 301C7, 30334, 30631, 307AB, 3300B, 3333B, 36031, 36301, 37A0B, 37BBB, 39997, 3A30B, 3B0C7, 3D001, 3D601, 40034, 40968, 43334, 49668, 49998, 50022, 5009D, 501C7, 50222, 50507, 505C7, 50611, 50C57, 53007, 53997, 55537, 5555B, 5557B, 5599B, 56101, 56691, 56961, 5700D, 5755B, 59001, 59557, 59997, 5999D, 599DB, 59DDD, 5D99B, 5DD3D, 5DD9D, 60931, 63031, 65691, 66951, 69031, 69361, 69561, 70011, 70051, 7005B, 7006E, 7030D, 703AB, 70501, 70701, 707C7, 71601, 71951, 7300D, 7333B, 75001, 7555B, 75911, 76011, 76051, 766EE, 76EEE, 7700B, 77191, 77661, 7776E, 77771, 777BB, 77911, 77BBB, 79001, 7A05B, 7A66E, 7AA6E, 7AAAE, 7ACCE, 7C6EE, 7CCEE, 7CECE, 7CEEE, 7D3BB, 7E7C7, 7EECE, 80034, 80304, 80434, 809DD, 80A34, 84A34, 850DD, 85961, 86661, 88151, 88331, 88511, 88591, 88898, 890DD, 89998, 89D0D, 8D90D, 8E434, 90017, 90051, 900A8, 900DB, 901C7, 90C57, 90D8D, 91007, 91061, 9199B, 95997, 96068, 96561, 99397, 99537, 9999B, 999B7, 999D7, 999DB, 999DD, 99BBB, 99DBB, 99DD7, 99DDD, 9B007, 9B00B, 9B0AB, 9BB11, 9BBBB, 9D007, 9D08D, 9D537, 9D9BB, 9D9DB, 9DD57, 9DDB7, 9DDDB, 9DDDD, A0A34, A0B5B, A0BBB, A0E34, A2E52, A330B, A8434, A8834, A8E34, A909B, AAA34, AAE52, AB0BB, AB334, ABB34, AE034, AE834, AE99B, AEA52, AEE52, B0011, B0071, B0077, B00B1, B0611, B0A64, B500D, B599D, B6101, B7771, B7911, BA064, BAAA4, BAB34, BB061, BB304, BB53D, BB601, BBB91, BBB9D, BBBBD, BDA0B, BDBBB, D0088, D00D7, D0307, D05C7, D070D, D0888, D0B07, D0BC7, D0C08, D0DC7, D0DD8, D1661, D59DD, D5D3D, D5DDD, D6611, D700D, D8D0D, D900B, D9908, D999D, D9BBB, D9D9D, D9DDB, DB007, DB00D, DB1B1, DB53D, DB59D, DB99D, DBBB1, DD0D8, DD33B, DD3B7, DD3BB, DD57D, DD898, DD9DD, DDB37, DDBDB, DDD08, DDD3D, DDD5D, DDD7D, DDD88, DDD9D, DDDB7, DDDC8, DDDD7, DDE98, DE037, DE998, DEB07, E0098, E00C7, E0537, E0557, E077B, E0834, E0968, E3334, E37AB, E39C7, E4034, E5307, E55AB, E705B, E750B, E766E, E76EE, E8304, E8434, E9608, E9C37, EAE52, EBB0B, EC557, EC597, EC957, 1000BD, 1009AB, 10A90B, 1900AB, 190661, 19099B, 190A0B, 1A900B, 222A52, 2AAA52, 31000D, 330331, 333334, 3733AB, 373ABB, 3BBB61, 430004, 490068, 490608, 5000DB, 500D0B, 505557, 505A0B, 50D00B, 50DDDB, 50DDDD, 522222, 5500AB, 5500C7, 550957, 550A0B, 555A9B, 559057, 560011, 590661, 633331, 666331, 666591, 666661, 7050AB, 705A0B, 706101, 70A50B, 7300AB, 761661, 76666E, 777011, 777101, 77750B, 777A5B, 777CEE, 779051, 791501, 7E7797, 7ECCCE, 7EEE97, 800D9D, 808834, 836631, 83D661, 843004, 856611, 884034, 884304, 888E34, 88A434, 88AE34, 8A4034, 8AEE34, 8E8034, 8E8E34, 8EEE34, 9000BB, 9001AB, 900B07, 900D98, 903661, 905661, 906651, 9080DD, 9099A8, 909D9B, 90A668, 90DD9B, 90DDBB, 910001, 9100AB, 91A00B, 930007, 950001, 956661, 9909A8, 995907, 999068, 999507, 999907, 9B0B1B, 9B0BB1, 9BB01B, 9C5597, 9C5957, 9D09DD, 9D0D9D, 9D800D, 9DB307, 9DD09D, A00034, A0033B, A033B4, A2A252, AAAA52, ABBBBB, B00004, B0001B, B0003D, B00A04, B0555B, B07191, B07711, B07777, B0B911, B0BDBB, B77011, B777C7, BB0001, BB0034, BB035D, BB055B, BB0BDB, BB9101, BBB0DB, BBB50D, BBBB01, BBD0BB, C55397, C55557, C55597, D0003B, D00057, D0007D, D000B7, D000C8, D008DD, D00DAB, D0333B, D05537, D099DD, D09DDD, D0DDBB, D555C7, D5C537, D88008, D88088, D888EE, D909DD, D9D0DD, D9DD0D, DB0BBB, DBBB0B, DBBB0D, DC0008, DC5537, DDDDD8, DDDEBB, DDE99B, DE0808, DE0C57, DE300B, DE5537, DE8888, DEE088, DEE307, DEE888, DEEE37, DEEE57, DEEEC8, E0000B, E007BB, E00A52, E03BC7, E07ABB, E09B07, E0A99B, E0C397, E0E76E, E50057, E55007, E55597, E55937, E730AB, E73A0B, E80E34, E88834, E8E034, E90008, E95557, EA099B, EE4304, EE5057, EE5507, EE8E34, EE9307, EEE434, 100001D, 1000A9B, 1000DC7, 22AA252, 3000BC7, 3033301, 3076661, 333B304, 33B3034, 3B33304, 3D66661, 50007AB, 5005957, 5500597, 5550057, 5559007, 5559597, 5595007, 5966661, 5DDDDDB, 6366631, 7010001, 7066651, 7100061, 733BBBB, 766A6AE, 77505AB, 7776501, 777775B, 777AACE, 777ECCE, 777EEAE, 7CCCCCE, 7E30A0B, 7EEEEAE, 8300004, 8363331, 8693331, 880E834, 8833304, 8888034, 8888434, 888A034, 88A3334, 88E8834, 88EE034, 88EE304, 8AA3334, 8D0009D, 8EE8834, 9000361, 9000668, 9003331, 9005557, 9006008, 9008D0D, 9083331, 9090968, 90BBB01, 90D0908, 9500661, 9555597, 9555957, 9660008, 9900968, 9995597, 9996008, 9999557, 9999597, 9999908, 9A66668, A003B34, A003BB4, AA22252, B00B034, B00B35D, B033334, B0B6661, B0BB01B, B100001, B333304, B777777, B99999D, BA60004, BAA0334, BBB001B, BBB6611, BBBBB11, BBBD00B, BD000AB, D0000DB, D009098, D00CCC8, D00D908, D00D99D, D03000B, D0BB0BB, D0D9008, D0D9998, D1000C7, D800008, D8DDEEE, D90080D, DBBBBBB, DD09998, DDD5557, DDDDBBB, DDDDDBD, DDDE8EE, DECC008, DECCCC8, DEE0CC8, DEEC0C8, E000397, E0003BB, E000434, E00076E, E000937, E007A5B, E00909B, E0090B7, E009307, E00B077, E00E434, E00E797, E00E937, E05999B, E09009B, E0900B7, E0E0937, E0E7E97, E0EAA52, E0EEA52, E555057, E5555C7, E7777C7, E77E797, E88EE34, E999998, EA5999B, EB000BB, EB0BBBB, EE00434, EE0E797, EEE076E, EEE706E, EEE8834, EEEE557, EEEE797, 30333331, 30B66661, 33000034, 33030004, 33B33004, 500575AB, 55000007, 5500075B, 55500907, 55555057, 55555907, 55559507, 60003301, 60033001, 60330001, 7000003D, 70106661, 70666611, 77000001, 7777770B, 777777C7, 77777ACE, 77777EAE, 777E30AB, 777E3A0B, 7CCCC66E, 800005DD, 88AA0834, 90000008, 900008DD, 90099668, 90500557, 90555007, 90666668, 90909998, 90990998, 90996668, 9099999D, 90D00098, 90D90998, 95500057, 99099098, 99555057, 99900998, 99966608, 99966668, 99999668, 99999998, 9D009008, 9D090998, A0803334, A2222252, AAA52222, B00005AB, B000B55B, B0BBBB5B, B3330034, BB0BBB1B, BBAA3334, BBB0BB1B, BBB0BB5B, BBDB000B, D000BBBB, D00100C7, D8888888, D900008D, D9000098, DBB000BB, DC0CCCC8, DCC0CCC8, DCCCC008, DD000908, DD09009D, DDDDDDAB, DDDDDEEE, DDDEEE8E, DDDEEEE8, DEE80008, E0777E97, E0E0E397, E0E77797, E0EE0397, E7777797, E9066668, EE00E397, EE077797, EE0E0397, EEE00797, EEE07E97, EEE0AA52, EEE55397, EEE55557, EEEAAA52, EEEEE834, EEEEEA52, 300003331, 300007661, 300330031, 333000004, 333300001, 333B00034, 3700000AB, 3B3300034, 500000057, 555555007, 555555557, 5DDDDDDDD, 600000331, 7500000AB, 75000A00B, 75A00000B, 761000001, 77000E0C7, 777700EC7, 7777730AB, 7777777AE, 77777EE97, 7777E7E97, 777999997, 7A500000B, 7BBBBBB5B, 88888A834, 900000031, 900666608, 909990098, 90D009998, 950000557, 966666008, 990000007, 990555507, 999999997, A000000B4, A0005999B, AAEEEEE34, B000AA334, BBBBB005B, BBBBBBB5B, D09999998, D0D90009D, D800000DD, D90009998, DCCCC0CC8, DE88EEEEE, DEEEEEE88, E000B7777, E000BBBBB, E003ABBBB, EE0000797, EE0EEE397, EE5555557, EE777EE97, EEEEEE537, EEEEEE937, 2222222252, 3000000071, 3330030001, 3333303001, 3333330001, 500000007B, 5555555097, 7000000071, 77000000C7, 8333333331, 8888883334, 8888888834, 888888AA34, 900000009B, 900000009D, 900000DD9D, 9000099998, 9955555507, 9D0000099D, 9D05555557, AB0000005B, B000000DAB, B00000BBDB, BB00BB0B5B, BB0BB00B5B, D000099998, D00090008D, D0D000909D, D0DDDDDDDB, D300000007, D88EEEEEEE, D900999998, DD00900008, DDD6EEEEEE, DDDDDDD6EE, DDDDDDDDDE, DDDEEEEEEE, DEEEEE8008, E000000797, 7777777CCCE, 88888830004, 90000009D9D, 99955555557, 9999999999D, B00000D00AB, BB000BBB05B, BBBB0000B5B, D000009080D, D000090800D, D090800000D, DDDDDDD999B, DDDDDDDDD9B, EEEEEE00397, EEEEEEE0397, 333000000301, 5000000000DD, 73A00000000B, 9000000000B7, 903333333331, ABB00000000B, D000000001C7, DCCCCCCCCCC8, E0EEEEEEE397, 19A000000000B, 3333333333331, 3BBBBBBBBBBBB, 9333333333331, A00000000099B, B00000000050D, EEEEEEEEEE76E, 1000000000999B, 71000000000001, 908D000000000D, BBBBBBBBBB6661, 77777777777777B, BB00000000BBB5B, DEEEEEEEEEEEEEE, 7777777777777E97, B0BBBBBBBBBBBB1B, BB0000000000DB0B, D000000000000998, D908000000000000D, DDDDDDDDDDDDDDDDB, E9666666666666668, 3330000000000000031, D00000000000000908D, E0BBBBBBBBBBBBBBBBB, 2EEEEEEEEEEEEEEEEE52, 77777777777777777ECE, 5000000000000000005AB, 777777777777777777997, 7BBBBBBBBBBBBBBBBBBBB, BB0000000000000000DBB, DD000000000000000909D, D900000000000000000DDD, DD0000000000000000099D, BBBBBBBBBBBBBBBBBBBBBB1, B00000000000000000000005B, B0700000000000000000000001, B70000000000000000000000001, 705000000000000000000000000B, 633000000000000000000000000001, EBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB, 500000000000000000000000000000000017, 77777777777777777777777777777777777777777777777777777777777CCE, 7777777777777777777777777777777777777777777777777777777777777777777777777CE, 96666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666608, EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE397, 7777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777797 ===Base 16=== 11, 13, 17, 1D, 1F, 25, 29, 2B, 2F, 35, 3B, 3D, 43, 47, 49, 4F, 53, 59, 61, 65, 67, 6B, 6D, 71, 7F, 83, 89, 8B, 95, 97, 9D, A3, A7, AD, B3, B5, BF, C1, C5, C7, D3, DF, E3, E5, E9, EF, F1, FB, 14B, 15B, 185, 199, 1A5, 1BB, 1C9, 1EB, 223, 22D, 233, 241, 277, 281, 287, 28D, 2A1, 2D7, 2DD, 2E7, 301, 337, 373, 377, 38F, 3A1, 3A9, 41B, 42D, 445, 455, 45D, 481, 4B1, 4BD, 4CD, 4D5, 4E1, 4EB, 50B, 515, 51B, 527, 551, 557, 55D, 577, 581, 58F, 5AB, 5CB, 5CF, 5D1, 5D5, 5DB, 5E7, 623, 709, 727, 737, 745, 74B, 755, 757, 773, 779, 78D, 7BB, 7C3, 7C9, 7CD, 7DB, 7EB, 7ED, 805, 80F, 815, 821, 827, 841, 851, 85D, 85F, 8A5, 8DD, 8E1, 8F5, 923, 98F, 99B, 9A9, 9EB, A21, A6F, A81, A85, A99, A9F, AA9, AAB, ACF, B1B, B2D, B7B, B8D, B99, B9B, BB7, BB9, BCB, BDD, BE1, C0B, CB9, CBB, CEB, D01, D21, D2D, D55, D69, D79, D81, D85, D87, D8D, DAB, DB7, DBD, DC9, DCD, DD5, DDB, DE7, E21, E27, E4B, E7D, E87, EB1, EB7, ED1, EDB, EED, F07, F0D, F4D, FD9, FFD, 1069, 1505, 1609, 1669, 16A9, 19AB, 1A69, 1AB9, 2027, 204D, 2063, 207D, 20C3, 20ED, 2221, 22E1, 2327, 244D, 26C3, 274D, 2E01, 2E0D, 2ECD, 3023, 3079, 3109, 3263, 3341, 36AF, 3941, 3991, 39AF, 3E41, 3E81, 3EE1, 3EE7, 3F79, 4021, 40DB, 440B, 444B, 44A1, 44AB, 44DB, 4541, 45BB, 4A41, 4B0B, 4BBB, 4C4B, 4D41, 4DED, 5045, 50A1, 50ED, 540D, 5441, 555B, 556F, 5585, 560F, 56FF, 5705, 574D, 580D, 582D, 5855, 588D, 5A01, 5AA1, 5B01, 5B4B, 5B87, 5BB1, 5BEB, 5C4D, 5CDD, 5CED, 5DD7, 5DDD, 5E0D, 5E2D, 5EBB, 68FF, 6A69, 6AC9, 6C8F, 6CA9, 6CAF, 6F8F, 6FAF, 7033, 7063, 7075, 7087, 70A5, 70AB, 7303, 7393, 74DD, 754D, 7603, 7633, 7663, 7669, 7705, 772D, 775D, 77D5, 7807, 7877, 7885, 7939, 7969, 7993, 79AB, 7A05, 7A69, 7A9B, 7AA5, 7B77, 7BA9, 7D4D, 7D75, 7D77, 8077, 808D, 80D7, 80E7, 8587, 86CF, 8777, 8785, 8885, 88CF, 88ED, 88FD, 8C6F, 8C8F, 8E8D, 8EE7, 8F2D, 8F8D, 9031, 9041, 90AF, 90B9, 9221, 9319, 9401, 944B, 9881, 9931, 9941, 9991, 99AF, 9A0F, 9A1B, 9A4B, 9AFF, 9BA1, 9BB1, 9CAF, 9E81, 9EA1, 9FAF, A001, A05B, A0C9, A105, A10B, A4CB, A55B, A6C9, A88F, A91B, A9B1, A9BB, AA15, AB01, AB0B, AB19, ABBB, AC09, AF09, B041, B04B, B069, B07D, B087, B0B1, B0ED, B1A9, B201, B40B, B40D, B609, B70D, B7A9, B807, B9A1, BA41, BAA1, BB4B, BBB1, BBDB, BBED, BD19, BD41, BDBB, BDEB, BE07, BEE7, C0D9, C203, C24D, C6A9, C88D, C88F, C8CF, C8ED, C9AF, C9CB, CA09, CA4B, CA69, CAC9, CC0D, CC23, CC4D, CC9B, CD09, CDD9, CE4D, CEDD, CFA9, CFCD, D04B, D099, D405, D415, D44B, D4A5, D4DD, D50D, D70B, D74D, D77B, D7CB, D91B, D991, DA05, DA09, DA15, DA51, DB91, DBEB, DD7D, DDA1, DDED, DE0B, DE41, DE4D, DEA1, E02D, E07B, E0D7, E1CB, E2CD, E401, E801, EABB, EACB, EAEB, EBAB, EC4D, ECDD, ED07, EDD7, EE7B, EE81, EEAB, EEE1, F08F, F0A9, F227, F2ED, F3AF, F485, F58D, F72D, F763, F769, F787, F7A5, F7E7, F82D, F86F, F877, F88D, F8D7, F8E7, F8FF, FCCD, FED7, FF85, FF8F, FFA9, 100AB, 10BA9, 1A0CB, 1BA09, 200E1, 2C603, 2CC03, 30227, 303AF, 30AAF, 32003, 32207, 32CC3, 330AF, 33169, 33221, 33391, 33881, 33AFF, 38807, 38887, 3AFFF, 3F203, 3F887, 3FAFF, 400BB, 4084D, 40A4B, 42001, 44221, 44401, 444D1, 4480D, 4488D, 44CCB, 44D4D, 44E8D, 4804D, 4840D, 4A0CB, 4A54B, 4CACB, 4D0DD, 4D40D, 4D44D, 5004D, 50075, 502CD, 5044D, 50887, 50EE1, 5448D, 548ED, 55A45, 55F45, 5844D, 5A4A5, 5AE41, 5B0CD, 5B44D, 5BBCD, 5D4ED, 5E0E1, 5EB4D, 5EC8D, 5ECCD, 5EE41, 5F06F, 5F7DD, 5F885, 5F8CD, 5FC8D, 5FF75, 6088F, 60AFF, 630AF, 633AF, 660A9, 668CF, 669AF, 66A09, 66A0F, 66FA9, 6886F, 6A00F, 6A0FF, 6A8AF, 6AFFF, 7002D, 7024D, 70B0D, 70B7D, 7200D, 73363, 73999, 7444D, 770B7, 777D7, 77B07, 77D7D, 77DD7, 79003, 79999, 7B00D, 7D05D, 7D7DD, 8007D, 800D1, 8074D, 82CCD, 82E4D, 8448D, 8484D, 8704D, 8724D, 87887, 88001, 8800D, 880CD, 88507, 88555, 8866F, 8872D, 8877D, 888D1, 888D7, 88AA1, 88C2D, 88D57, 88D75, 88D77, 8AFAF, 8C2CD, 8C40D, 8C8CD, 8CCED, 8CE2D, 8CFED, 8E007, 8E20D, 8E24D, 8F6FF, 8FAAF, 900CB, 901AB, 90901, 909A1, 90AB1, 90AE1, 90EE1, 910AB, 93331, 940AB, 963AF, 966AF, 99019, 99109, 99A01, 9AAE1, 9B00B, 9B0AB, 9B441, 9BABB, 9BBBB, 9E441, A00BB, A0405, A044B, A08AF, A0A51, A0B91, A0C4B, A1B09, A54A5, A5B41, A6609, A904B, A94A1, A9C4B, A9E01, A9E41, AA0A1, AA441, AA501, AA8AF, AAEE1, AAF45, AAF8F, ABBA1, ACC69, AE0BB, AE0EB, AEAE1, AEE0B, AEEA1, AEECB, AF045, AF4A5, AFA8F, B00A1, B00D7, B044D, B0777, B0A0B, B0A91, B0BBD, B0BCD, B0C09, B0DA9, B0EAB, B2207, B4001, B6669, B7707, B7D07, B8081, B9021, BA091, BA109, BA4BB, BB001, BB0EB, BB8A1, BBBEB, BBE0B, BBEBB, BC009, BCECD, BD0A9, BE44D, BEB0D, BEBBB, BEEBB, C0263, C02C3, C02ED, C040D, C0CA9, C0CCD, C2663, C2CED, C32C3, C3323, C400D, C40ED, C44CB, C44ED, C480D, C484D, C4CAB, C60AF, C686F, C6A0F, C86FF, C8C2D, CAA0F, CAFAF, CBCED, CC0AF, CC44B, CC82D, CC8FF, CCAF9, CCAFF, CCCFD, CCFAF, CD00D, CD4CB, CD4ED, CDDDD, CF2C3, CFC8F, CFE8D, D0045, D07DD, D09BB, D0D4D, D0DD7, D0EBB, D0EEB, D1009, D1045, D10B9, D1BA9, D54BB, D54ED, D5AE1, D5D07, D5EE1, D70DD, D7707, D7777, D77DD, D7DD7, D9441, D9AE1, D9B0B, DA9A1, DA9E1, DAA41, DAAA1, DBB0B, DBBA1, DC4CB, DD227, DD44D, DDDD7, E0081, E00E1, E010B, E088D, E08CD, E0B0D, E0BBD, E100B, E4D0D, E777B, E77AB, E7CCB, E844D, E848D, E884D, E88A1, EB0BB, EBB4D, EBBEB, EBEEB, EC8CD, ECBCD, ECC8D, ED04D, EE001, EE0EB, EE4A1, EEEBB, F0085, F09AF, F0C23, F0CAF, F2663, F2C03, F3799, F3887, F4A05, F4AA5, F506F, F5845, F5885, F5C2D, F5ECD, F5F45, F66A9, F688F, F6AFF, F7399, F777D, F8545, F8555, F8AAF, F8F87, F9AAF, FA0F9, FA405, FA669, FAFF9, FC263, FCA0F, FCAFF, FCE8D, FCF23, FD777, FDDDD, FDEDD, FEC2D, FEC8D, FF545, FF6AF, FF739, FF775, FF9AF, FFC23, 100055, 100555, 10A9CB, 1A090B, 1A900B, 1CACCB, 1CCACB, 20DEE1, 266003, 3000AF, 300A0F, 300AFF, 308087, 308E07, 3323E1, 333A0F, 339331, 33CA0F, 33CF23, 33CFAF, 33F323, 380087, 3A00AF, 3A0F0F, 3AA0FF, 3AAF0F, 3C33AF, 3C3A0F, 3C3FAF, 3CCAAF, 3F0FAF, 3F32C3, 3FF0AF, 3FFAAF, 4004CB, 400A05, 4048ED, 404DDD, 40AA05, 40D04D, 40DD4D, 40E0DD, 40E48D, 440041, 44008D, 44044D, 4404DD, 44440D, 4448ED, 4484ED, 448E4D, 44E44D, 48888D, 4AA005, 4DD00D, 4DD04D, 4DDD0D, 4E048D, 4E448D, 4E880D, 5000DD, 500201, 50066F, 5008CD, 500C2D, 500D7D, 50C20D, 520C0D, 544EDD, 54AA05, 54AAA5, 54ED4D, 566AAF, 57D00D, 580087, 5A5545, 5C20CD, 5C8CCD, 5CC2CD, 5D000D, 5D070D, 5F666F, 5FAA45, 5FFF45, 60008F, 600A0F, 603AAF, 6060AF, 6066AF, 60A0AF, 63AA0F, 6663AF, 66668F, 666AAF, 668A8F, 66AFF9, 68888F, 693AAF, 7007B7, 70404D, 70770B, 70770D, 707BE7, 70DD0D, 733339, 733699, 74004D, 74040D, 77007B, 770CCB, 777B4D, 777BE7, 777CCB, 77ACCB, 77B74D, 77D0DD, 7A0CCB, 7B744D, 7CACCB, 7DDD99, 80044D, 800807, 80200D, 8044ED, 80C04D, 80CC2D, 80E44D, 8404ED, 84888D, 84E04D, 84E40D, 86686F, 8668AF, 8686AF, 86F66F, 86FFFF, 87000D, 87744D, 880807, 886AFF, 88824D, 88870D, 888787, 88884D, 88886F, 88887D, 88888D, 888C4D, 888FAF, 88AA8F, 88CC8D, 88F6AF, 88F8AF, 88FA8F, 88FF6F, 88FF87, 88FFAF, 8A8FFF, 8C0C2D, 8C802D, 8CCFFF, 8CE00D, 8CE0CD, 8CFCCF, 8E00CD, 8E044D, 8E0CCD, 8EC0CD, 8F68AF, 8F88F7, 8FCFCF, 8FF887, 8FFCCF, 8FFF6F, 9002E1, 9004AB, 9008A1, 900919, 900ABB, 900B21, 90B801, 90CCCB, 9332E1, 944441, 94ACCB, 990001, 9900A1, 9A4441, 9A4AA1, 9AA4A1, 9AAA41, 9AAAAF, 9B66C9, 9BBA0B, 9BC0C9, 9BC669, 9BC6C9, 9C4ACB, A0094B, A00ECB, A09441, A0A08F, A0E0CB, A0ECCB, A0F669, A40A05, A4AAA5, A50E41, A5AA45, A60069, A8FAFF, A9AA41, AA5E41, AAA4A5, AAA545, AC6669, ACCC4B, ACCCC9, AEAA41, AFF405, AFF669, AFFA45, AFFFF9, B00921, B00BEB, B00CC9, B00D91, B08801, B0D077, B70077, B70E77, B77E77, B88877, B88881, B94421, BAE00B, BB00AB, BB0DA1, BB444D, BB44D1, BB8881, BBBBBD, BBBC4D, BBCCCD, BC0CC9, BC66C9, BCC669, BCC6C9, BCCC09, BE000D, BE00BD, BE0B4D, BE0CCD, BEA00B, BECCCD, C0084D, C00A0F, C0608F, C0668F, C0844D, C0A0FF, C0AFF9, C0C3AF, C0C68F, C0CAAF, C0CDED, C0D0ED, C0E80D, C0EC2D, C0EC8D, C0FA0F, C0FAAF, C2CC63, C30CAF, C333AF, C3CAAF, C3CCAF, C4048D, C40D4D, C4404D, C4408D, C4440D, C44DDD, C4ACCB, C4DCCB, C4DD4D, C6068F, C66AAF, C68AAF, C6AA8F, C8044D, C8440D, C8666F, CA00FF, CA0FFF, CAAAAF, CAAFFF, CAFF0F, CBE0CD, CC008F, CC0C8F, CC3CAF, CC4ACB, CC608F, CC66AF, CCBECD, CCC4AB, CCCA0F, CCCC8F, CCCE8D, CE0C8D, CF0F23, CF0FAF, CFAFFF, CFCAAF, CFFAFF, D0005D, D00BA9, D05EDD, D077D7, D10CCB, D22207, D4000B, D4040D, D4044D, D40CCB, D70077, D7D00D, D90009, D900BB, DB00BB, DB4441, DD400D, DDD109, DDD1A9, DDD919, DDD941, DED00D, E00D4D, E00EEB, E0AAE1, E0AE41, E0AEA1, E0B44D, E0BCCD, E0BEBB, E0D0DD, E0E441, E4048D, E4448D, E800CD, E8200D, EA0E41, EAA0E1, EBB00B, ECCCAB, EDDDDD, EEBE0B, F00263, F0056F, F00A45, F02C63, F03F23, F05405, F060AF, F08585, F0A4A5, F0F2C3, F0F323, F2CCC3, F33203, F33C23, F5F66F, F5FF6F, F68CCF, F6AA8F, F888AF, FA0F45, FAA045, FAA545, FAFC69, FC0AAF, FC66AF, FCCCAF, FCFFAF, FF0323, FF056F, FF3203, FF7903, FFA045, FFA4A5, FFAA45, FFC0AF, FFF4A5, FFF575, FFFA45, FFFCAF, 10A009B, 20000D1, 2CCC663, 30A00FF, 30CCCAF, 30FA00F, 30FCCAF, 3333C23, 333C2C3, 33C3AAF, 33FCAAF, 33FFFAF, 3A0A00F, 3AAAA0F, 3AF000F, 3AFAAAF, 3C0CA0F, 3CCC3AF, 3CFF323, 3F33F23, 3FAA00F, 3FF3323, 4004441, 400DDD1, 400E00D, 400ED0D, 404404D, 404448D, 404E4DD, 440EDDD, 4440EDD, 44444ED, 4444E4D, 44DDDDD, 4A000A5, 4CCCCAB, 4D0CCCB, 4E4404D, 4E4444D, 4E4DDDD, 5000021, 5004221, 5006AAF, 500FF6F, 5042201, 508CCCD, 5400005, 5400AA5, 5555405, 5808007, 5AA4005, 5C0008D, 5CCC8CD, 5D4444D, 5EEEEEB, 5F40005, 5F554A5, 5F6AAAF, 60000AF, 60006A9, 600866F, 6008AAF, 600AA8F, 600F6A9, 606608F, 606686F, 608666F, 60AA08F, 60AAA8F, 66000AF, 66666A9, 6666AF9, 6866A8F, 6AAAAAF, 70070D7, 70077DD, 700DDDD, 707077D, 707D007, 70D00DD, 770077D, 770400D, 770740D, 7777775, 77777B7, 77777DD, 7777ACB, 77B88E7, 77DD00D, 77DDDDD, 7D0D00D, 7DD0D07, 7DDD00D, 800002D, 8000CED, 80C0E0D, 80CECCD, 840400D, 844000D, 844E00D, 868688F, 880444D, 884404D, 887D007, 8888801, 8888881, 8888E07, 8888F77, 8888FE7, 88A8AFF, 88AAAFF, 88FAFFF, 8A8AAAF, 8A8AAFF, 8AAA8FF, 8C00ECD, 8C8444D, 8E4400D, 8FCCCCF, 900BBAB, 90CC4AB, 9908AA1, 99E0E01, 9B00801, 9B6CCC9, A000FF9, A006069, A00A8FF, A01CCCB, A05F545, A0BEEEB, A0E4AA1, AA0008F, AA08FFF, AA40AA5, AA8FFFF, AAAA405, AE04AA1, AE44441, AE4AAA1, AECCCCB, AF40005, AFA5A45, AFFFC69, B000BAB, B000EBB, B0D0007, B222227, B6CCCC9, B8880A1, BA000EB, BA0BEEB, BAEEEEB, BB000CD, BB00C0D, BB0B00D, BC6CC69, BC6CCC9, BCCCC69, BCCCCED, C0000A9, 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4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444DD ===Base 17=== 12, 16, 1C, 1E, 23, 27, 29, 2D, 32, 38, 3A, 3G, 43, 45, 4B, 4F, 54, 5C, 5G, 61, 65, 67, 6B, 78, 7C, 81, 83, 8D, 8F, 94, 9A, 9E, A3, A9, AB, B4, B6, BA, BC, C7, D2, D6, D8, DC, E1, E3, ED, F2, F8, FE, FG, G5, G9, GB, 104, 111, 115, 117, 11B, 137, 139, 13D, 14A, 14G, 155, 159, 15F, 171, 17B, 17D, 188, 191, 197, 19F, 1A4, 1A8, 1B3, 1BB, 1BF, 1DB, 1DD, 1F3, 1FD, 1G8, 1GA, 1GG, 20F, 214, 221, 225, 241, 25A, 25E, 285, 2B8, 2C5, 2CF, 2E5, 2EB, 2F6, 30E, 313, 331, 33B, 346, 34C, 351, 35F, 36E, 375, 37B, 391, 39B, 39D, 3B7, 3B9, 3BF, 3D3, 3D5, 3D9, 3DF, 3E4, 3EC, 3F1, 3F7, 407, 418, 447, 44D, 472, 474, 47E, 47G, 489, 49C, 4A1, 4C1, 4CD, 4D4, 4G1, 502, 506, 508, 50E, 519, 522, 528, 52A, 52E, 533, 53F, 551, 55D, 562, 566, 573, 577, 57F, 582, 593, 599, 59B, 59F, 5A6, 5B5, 5D1, 5D3, 5EA, 5EE, 5F9, 60D, 62F, 634, 649, 689, 692, 6CD, 6EF, 6F4, 6FA, 704, 706, 70G, 71D, 726, 737, 739, 73D, 73F, 753, 755, 764, 766, 76G, 771, 77B, 793, 7AA, 7AE, 7B3, 7BB, 7D7, 7E6, 7F3, 7F9, 7FF, 7G2, 7GE, 7GG, 825, 82B, 849, 852, 85E, 869, 876, 87A, 87G, 88B, 892, 898, 89C, 8C5, 8E7, 8G7, 908, 90G, 913, 91F, 92C, 935, 937, 93B, 951, 953, 957, 95D, 968, 96G, 979, 97B, 98C, 98G, 99D, 9B1, 9B3, 9B9, 9BD, 9BF, 9DB, 9DF, 9F1, 9F5, 9G6, A07, A0D, A1A, A2F, A4D, A72, A7A, A7E, AA1, AA7, ACF, ADA, AG1, AG7, B02, B08, B17, B1D, B28, B2G, B57, B71, B73, B79, B7F, B88, B8E, B8G, B9B, B9F, BB5, BB7, BD7, BDD, BEG, BFF, BGG, C01, C2F, C3E, C56, C6D, C89, C92, C9G, CA5, CBG, CC1, CC5, CF4, CFA, D04, D0A, D15, D3D, D3F, D55, D59, D5B, D71, D75, D7D, D91, D97, D99, D9D, DA4, DAG, DB3, DDB, DF1, DF7, DF9, DFF, E05, E0B, E2B, E52, E58, E69, E92, E9C, EAF, EB8, EC9, ECB, EE5, F04, F15, F1B, F35, F3B, F46, F51, F53, F64, F6A, F73, F79, F95, FAC, FB1, FCA, FD5, FDB, FF1, FF7, FFD, G0D, G0F, G18, G1A, G1G, G2F, G34, G63, G7G, GA7, GC3, GDG, GEF, GFA, GG7, GGD, 1013, 101D, 1033, 1035, 1051, 105B, 105D, 1077, 108A, 109B, 10AG, 10B1, 10B7, 10BD, 10FB, 1149, 1189, 11AF, 11G3, 1303, 130B, 1314, 1341, 1479, 14D9, 1501, 1503, 15A1, 15B8, 1734, 1749, 17AF, 17G3, 1844, 185B, 1875, 1877, 18AG, 18B5, 1903, 1909, 1958, 19BG, 19G3, 1A5D, 1A75, 1A7F, 1ADF, 1AF1, 1B01, 1B09, 1B18, 1B85, 1B89, 1BDG, 1BGD, 1D07, 1D49, 1D9G, 1DF4, 1F09, 1F47, 1F5A, 1F74, 1F7A, 1FA1, 1FAF, 2018, 201G, 202B, 208B, 20G1, 215B, 218G, 21AG, 21B1, 222F, 22AF, 22BG, 22EF, 22F4, 22GF, 251B, 2526, 25F1, 266F, 26FC, 280B, 2A05, 2A58, 2AFC, 2AGF, 2B1B, 2B1F, 2BGE, 2C1G, 2C2B, 2C8B, 2CG1, 2E2F, 2EGF, 2F0C, 2F55, 2FAA, 2FC4, 2FFF, 2GA1, 2GFC, 2GG1, 2GGF, 301B, 301F, 3037, 3053, 3057, 3079, 3095, 30B3, 30BD, 30C4, 31F4, 330D, 3334, 333E, 3349, 3376, 337E, 33CD, 33EF, 3411, 3417, 3499, 3503, 3505, 3509, 353E, 35E5, 35EB, 3604, 36FD, 3701, 3741, 374D, 376F, 3796, 37D4, 37F4, 3956, 3B03, 3B05, 3B0B, 3BBE, 3C04, 3C15, 3C19, 3C4E, 3C59, 3C64, 3CB3, 3CDB, 3CE6, 3D07, 3D14, 3DDE, 3E77, 3E79, 3E7F, 3E99, 3EEE, 3EFB, 3F05, 3F0D, 3FCB, 3FF4, 4009, 4021, 4069, 4098, 40DG, 40GD, 419D, 4201, 4401, 4492, 46AD, 46C9, 46DA, 4719, 476A, 4779, 479D, 47A6, 4906, 4911, 4917, 4919, 491D, 492G, 4982, 4988, 49D7, 49D9, 49GG, 4ADE, 4AE7, 4C49, 4C96, 4CC9, 4D79, 4DAE, 4DEG, 4E7A, 4E96, 4EG7, 4G6D, 4G87, 501B, 5037, 5059, 507D, 50BB, 50BF, 50D7, 50DD, 50F1, 5105, 51A7, 51AD, 521B, 525F, 52FB, 5307, 5356, 53BE, 53DE, 53E9, 5507, 550B, 5587, 5598, 55EF, 560A, 568E, 56AA, 56F3, 5709, 5725, 572B, 575A, 575E, 5769, 57A1, 57B2, 5868, 586E, 58AE, 58B9, 590D, 5918, 5952, 5958, 596D, 5A17, 5A1F, 5ADD, 5ADF, 5AE8, 5B07, 5B21, 5B2F, 5B3E, 5BEF, 5DA7, 5DEB, 5E57, 5E5F, 5E86, 5E97, 5EB9, 5EBF, 5EF5, 5F01, 5F1A, 5F6F, 5FA7, 5FDA, 60AF, 60G3, 64AD, 64DE, 64DG, 663E, 666D, 66AF, 693D, 69CG, 69D3, 69D9, 69G8, 69GC, 6ADE, 6AGD, 6C98, 6D33, 6D4E, 6D93, 6D9F, 6DDD, 6DEE, 6DF3, 6DFD, 6DGE, 6E09, 6G36, 6G4D, 6G6D, 6GD4, 6GDE, 6GFC, 702E, 7057, 705B, 7073, 7079, 7095, 70B5, 70BD, 70D1, 70E2, 70F5, 7107, 7149, 719G, 71BG, 71F4, 724E, 724G, 725F, 72A2, 72BF, 72EE, 72GA, 7314, 733E, 7341, 7363, 73EB, 7419, 742A, 742G, 7442, 74EG, 7501, 750F, 751A, 756D, 757E, 75A1, 75A7, 75BE, 75DA, 75E9, 75F6, 7622, 769F, 76EA, 7734, 773E, 776D, 779G, 77AF, 7905, 790B, 7976, 79B2, 79F6, 79GD, 7A1F, 7A5D, 7AD5, 7ADF, 7AF1, 7AFD, 7B01, 7B09, 7B2F, 7B52, 7B72, 7BE5, 7D01, 7D05, 7D9G, 7DAF, 7DBG, 7E0A, 7E75, 7EA2, 7EA4, 7EB5, 7EBF, 7EE2, 7EF7, 7EG4, 7F0B, 7F14, 7F5A, 7F76, 7FA7, 7G1F, 7G46, 7GA6, 7GD3, 7GDF, 8009, 8058, 80B8, 80E9, 84A7, 850A, 8557, 857B, 85A8, 870E, 8744, 8777, 879B, 87B5, 87B7, 87EE, 8805, 8872, 8887, 8889, 88E9, 8906, 8959, 8966, 89GG, 8A87, 8AE5, 8B0G, 8B59, 8B95, 8B97, 8CB2, 8CB8, 8CE9, 8E56, 8EE9, 9026, 9031, 903D, 907F, 9091, 909B, 90FB, 9101, 910D, 9118, 917G, 9185, 9189, 91B8, 9202, 9288, 92B5, 92FB, 92GG, 93C1, 9505, 950B, 950F, 952B, 956F, 9592, 9596, 9598, 9602, 96D9, 96FD, 971G, 9725, 9752, 97DG, 9855, 9862, 9895, 9899, 98B7, 98BB, 9901, 990B, 9921, 992F, 99G3, 9B0B, 9B2B, 9B8B, 9BB8, 9BBG, 9C19, 9C1B, 9C31, 9C59, 9C95, 9CD5, 9CFB, 9CGC, 9D03, 9D07, 9D7G, 9DG1, 9DGD, 9F0B, 9F76, 9FCB, 9G11, 9G1D, 9G28, 9G3F, 9G7D, 9GCC, 9GD7, 9GF7, 9GFD, 9GG8, A025, A041, A058, A0C5, A0F6, A0GF, A11F, A184, A1F7, A21G, A258, A401, A421, A476, A511, A517, A57D, A5A8, A5E8, A6AD, A6FC, A6GF, A751, A77F, A7F5, A7FD, A7G6, A847, AACD, AC1G, AC41, AC58, AC5E, ACGD, AD0E, AD0G, AD1F, AD51, ADD5, ADE4, ADF5, ADGE, AE56, AE74, AEF6, AEFA, AF77, AF7D, AFA4, AFCC, AFD7, AFDD, AGAF, AGF4, B00G, B037, B055, B05B, B075, B0D5, B0FD, B10F, B198, B25F, B2F1, B2F5, B307, B309, B35E, B3EF, B50D, B589, B7BE, B7BG, B7E7, B875, B952, B958, B97G, B99G, B9G7, B9GD, BB01, BB2F, BB3E, BB89, BB98, BBDE, BD03, BD09, BD5E, BDE5, BDEB, BDG1, BE5F, BF01, BF0D, BG13, BG1F, BG3F, BGD1, BGE2, BGE8, C00B, C034, C05A, C0AF, C0EF, C0GF, C153, C15B, C199, C1B9, C1D1, C1D5, C1F9, C205, C21A, C21G, C252, C258, C2B2, C335, C33D, C35D, C364, C395, C3B3, C3F5, C3FB, C3FD, C414, C41A, C469, C496, C4DA, C4GD, C535, C55B, C5B1, C5BD, C5D9, C5DF, C5E8, C5F3, C5F5, C6E9, C85A, C885, C8B8, C8BE, C8CB, C8E5, C919, C931, C959, C95F, C9D3, CA0F, CA18, CA1G, CAD4, CADE, CAEF, CAGD, CB22, CB33, CB35, CB3F, CB5D, CB82, CB99, CBB1, CBFB, CC49, CCCB, CCDE, CD11, CD1D, CD39, CD4A, CD53, CD93, CDAE, CDD5, CDF3, CDFD, CDG4, CE49, CE5A, CE8B, CF13, CF19, CF5D, CF5F, CFB9, CFBF, CFD9, CFDF, CG14, CG41, CG6F, CGCF, CGF6, CGG1, D01F, D039, D079, D09B, D09F, D0B7, D0BB, D0D1, D0EG, D0GG, D10D, D19G, D1G3, D30B, D347, D3BE, D4E4, D50D, D57E, D5AD, D5FA, D707, D73E, D7E7, D7GF, DA1F, DA57, DAAE, DB01, DB09, DB0D, DB7E, DB9G, DD05, DD7E, DDA5, DDFA, DDG3, DE0G, DE44, DE4A, DE77, DEAE, DEB9, DEBB, DF03, DF05, DG0E, DGDF, E009, E06F, E072, E07G, E089, E0CF, E0E9, E0G7, E47A, E498, E4E7, E50A, E559, E55F, E575, E5B9, E5BF, E5F5, E5F7, E6FC, E722, E724, E72A, E72E, E744, E746, E75B, E76E, E79B, E7A4, E7A6, E7AG, E7B5, E7B7, E7EG, E7G4, E887, E89G, E8E9, E906, E955, E95B, E95F, E988, E99F, E9F9, E9G8, E9GG, EA25, EA7G, EAC5, EAE7, EB7B, EBF5, EBF7, EBFB, EC6F, ECCF, ECEF, EE72, EE76, EE89, EE9G, EF0A, EF44, EF77, EF97, EFA4, EFB5, EFC6, EFFF, EG6F, EG74, EGE7, EGFC, EGGF, F019, F01F, F075, F091, F09B, F0BF, F0FB, F10F, F1A7, F1AD, F1D4, F376, F3CD, F3F4, F40A, F411, F444, F44A, F497, F499, F49D, F4D7, F509, F57A, F5AD, F5F6, F6D3, F6D9, F70D, F741, F747, F76D, F7F6, F7FA, F907, F976, F9CB, FA11, FA7D, FADD, FB09, FC4C, FC5D, FC5F, FC91, FCB9, FD1A, FD41, FD47, FDF4, FF0B, FF56, G021, G07A, G0A1, G0E7, G11F, G17F, G1DF, G1F1, G1F7, G201, G2A1, G306, G311, G36C, G377, G37F, G3CC, G3CE, G3D1, G476, G487, G4DE, G6AF, G6D4, G6F6, G6GF, G713, G724, G731, G742, G74E, G76E, G7A2, G872, G874, GA21, GAC1, GC6F, GCAF, GCD4, GCDA, GCG1, GD73, GD7F, GDAE, GDDF, GDEA, GDFD, GE47, GE7E, GF13, GF33, GF3F, GF4C, GF71, GF7F, GFDD, GG01, GG21, GGAF, GGC1, 1000G, 10053, 100AA, 100B9, 100F1, 100FF, 10301, 10587, 10705, 1075A, 107GF, 10895, 108B9, 10985, 1099G, 10B98, 10B9G, 10D03, 10D0F, 10D7A, 10DG3, 10DG7, 10G1F, 10G3F, 110GF, 1140D, 11D93, 11DG4, 11F0A, 11G4D, 11GD4, 13333, 133FF, 13F44, 14109, 14499, 150A7, 153B1, 1570A, 17005, 17799, 177AG, 17995, 17A7G, 17G47, 18079, 18507, 185A7, 18B07, 18B9G, 19333, 199B5, 1A00A, 1A00G, 1A0F5, 1AAAA, 1AAAG, 1AF05, 1AFFA, 1B07G, 1B10G, 1B807, 1D001, 1D1AA, 1D7G4, 1DG03, 1DG41, 1F001, 1F00F, 1F01A, 1F0A7, 1F199, 1F1F9, 1F414, 1F449, 1F7F5, 1F999, 1FF0A, 1FFAA, 1FFB5, 1G073, 1G14D, 1G1F4, 1G301, 1G477, 1GD01, 1GD47, 1GF07, 1GFF4, 20005, 200A1, 2010A, 20586, 20588, 20A01, 20B11, 20B15, 20BEE, 20C1A, 20CBE, 210B5, 21A1F, 21A51, 21F1A, 21G1F, 21GFF, 222BE, 228B2, 228BE, 22BE2, 22C0B, 22F0A, 252BB, 25505, 25552, 26GAF, 2A001, 2A1FF, 2A55F, 2AEEF, 2AF44, 2B051, 2B20E, 2BB2B, 2BBBG, 2BE22, 2BEE2, 2BEEE, 2BF0B, 2C0BE, 2C18A, 2F101, 2F1FA, 2F44C, 2FCBB, 2G1FF, 2GA6F, 2GF44, 30035, 300B1, 300FB, 30101, 303C5, 30444, 30497, 304D1, 304D7, 30703, 30714, 30734, 30763, 30774, 30CF5, 30CFD, 30D41, 30FC5, 3100B, 31779, 31F5B, 31FB5, 31FFF, 330C5, 330F4, 33357, 33373, 33379, 33555, 33557, 33777, 3379F, 337FD, 33997, 33D44, 33D4E, 33F3D, 33FF5, 34019, 34044, 340D1, 353DD, 35535, 355B3, 355E6, 35BB3, 35DDD, 3636D, 364DD, 3663D, 36DD4, 37003, 3700F, 3717F, 373EE, 37609, 3774E, 37773, 37797, 37977, 3797F, 37EEF, 39007, 390C5, 39777, 39973, 3B355, 3B553, 3BBDB, 3BDB1, 3C03D, 3C0F5, 3C10F, 3C141, 3C444, 3CBE5, 3CD0D, 3CE5B, 3CEBB, 3CEF9, 3D401, 3DEBE, 3E006, 3E066, 3E57E, 3E5E9, 3E666, 3E90F, 3EF6F, 3F33D, 3F3C4, 3F5BB, 3FB33, 3FDDD, 3FF59, 4006D, 400DE, 4011D, 401D9, 40414, 4041G, 404C9, 40966, 40D11, 40D19, 40D1D, 40E49, 41019, 411DA, 41AAG, 4210A, 44049, 4410G, 44144, 441G4, 44441, 444E9, 446E9, 44986, 44E49, 4609G, 460E9, 466DE, 469DD, 46E9G, 4711A, 476D9, 4770D, 47A77, 47D09, 49099, 490D1, 49226, 49622, 49699, 496DD, 49996, 4999G, 499G7, 49G22, 49G77, 4A7DD, 4AA6D, 4ADD7, 4C0E9, 4C999, 4D1DA, 4DADD, 4DD01, 4DD1G, 4DD7A, 4DDA7, 4DDE9, 4DG0G, 4DGAA, 4DGGA, 4DGGE, 4E049, 4E449, 4E49G, 4E4E9, 4E797, 4G7DD, 4GDAA, 4GDD7, 50011, 50079, 50095, 500B1, 500F3, 501A5, 501AF, 50503, 507A5, 50AF7, 50F03, 50F7A, 510A1, 510DA, 511AA, 511DF, 5135B, 515B7, 5180B, 51A0F, 51F0A, 520B1, 53005, 531BD, 53559, 53609, 53B11, 55205, 55357, 553E6, 5555B, 5556E, 55588, 5558A, 555F3, 555FB, 556AF, 556E9, 55759, 5575B, 55805, 55885, 55896, 558B8, 55926, 55BE2, 55E8B, 55F57, 560FF, 5700D, 570A5, 570DA, 575B9, 576AD, 576DA, 579D5, 57A05, 57A52, 57B9D, 57DBD, 58057, 58509, 5855A, 585A7, 587EB, 58857, 588E8, 58A75, 58B0B, 58B87, 58BBE, 58BEB, 58E5B, 591D5, 59201, 59256, 59715, 59807, 5A88A, 5AA88, 5AFAD, 5B001, 5B00B, 5B1F1, 5B31B, 5B7E2, 5B80B, 5BB13, 5BBE8, 5BBFB, 5BE87, 5BE8B, 5BF37, 5BFBD, 5D00F, 5DA05, 5DA5A, 5DAE5, 5DBBD, 5DD95, 5DDAA, 5DFDD, 5E879, 5E8B7, 5E8BB, 5F07A, 5F0AD, 5F37D, 5F70A, 5F7BD, 5FB7B, 5FBBB, 5FBF3, 5FFF3, 6003E, 60098, 603E6, 606GF, 60986, 609C8, 60G6F, 60GCF, 6336D, 633E9, 63CCE, 63E06, 63E66, 6609G, 660E9, 66AD4, 66D4A, 66DG4, 66DGG, 66E98, 66FD9, 66GF6, 69806, 69866, 69C86, 69CC8, 6A66F, 6AAGF, 6AF06, 6AF66, 6AGGF, 6C6G3, 6C6GF, 6CCGF, 6CG03, 6DA0E, 6DAEA, 6DD9G, 6DDE9, 6DEGA, 6DGD3, 6E986, 6EEE9, 6F69D, 6F6DF, 6F96D, 6FD03, 6FD09, 6G003, 6G3F3, 6G3FF, 6G6CF, 6GAAF, 6GCCF, 70031, 70099, 700BF, 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DDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD ====Additional known quasi-minimal primes (not necessarily the next)==== 570₅₁₃₁₀1, 49₁₁₁₃₃₃, 970₁₆₆₀₄₇1, F70₁₈₆₇₆₇1 ===Base 18=== 11, 15, 1B, 1D, 21, 25, 27, 2B, 2H, 35, 37, 3D, 3H, 41, 47, 4B, 4H, 57, 5B, 5D, 5H, 61, 65, 71, 75, 7B, 7D, 85, 87, 8D, 91, 95, 9B, 9H, A1, AB, AD, AH, B1, BD, C7, CB, CD, CH, D5, D7, DH, E5, EB, EH, F1, F7, FB, FD, G5, H1, H5, H7, HB, 107, 167, 16H, 177, 17H, 1G7, 1HH, 20D, 24D, 26D, 29D, 30B, 36B, 381, 3BB, 405, 445, 44D, 49D, 4A5, 4DD, 4F5, 4GD, 501, 545, 5E1, 607, 62D, 64D, 66B, 66H, 67H, 68B, 697, 6A7, 6BB, 6E7, 6G7, 6GB, 6HH, 767, 76H, 77H, 797, 7HH, 801, 80H, 831, 83B, 86B, 88H, 8BB, 8FH, 8GH, 94D, 96D, 977, 9DD, 9ED, 9GD, A77, AC5, AE7, B07, B0H, B55, B77, B8B, B97, BB5, BB7, BBH, BE7, BFH, BGB, C01, C31, CA5, CG1, D2D, D4D, D81, DBB, DD1, DDB, DGD, E0D, E17, E31, E4D, E67, E6D, EA7, EDD, EE1, EED, EG7, F0H, F45, F8H, FC5, FFH, G0D, G17, G2D, G6B, G6H, GBB, GBH, GD1, GDD, GE1, GE7, GED, GFH, GG7, GGB, GHD, GHH, H0D, H2D, H8H, H9D, HGH, HHD, 100H, 19E7, 1A97, 1EE7, 1G8H, 1GGH, 22ED, 22GD, 2DED, 2E2D, 3001, 3031, 30C1, 30E1, 3331, 33G1, 3CC1, 40ED, 45C5, 46ED, 4CC5, 5331, 5551, 55G1, 5C05, 608H, 60ED, 60FH, 60HD, 666D, 66ED, 699D, 6B67, 6BGH, 6D0D, 6DDD, 6E9D, 6EGD, 6G0H, 6G9D, 6HGD, 700H, 70A7, 7A07, 7FGH, 7G77, 808B, 8881, 88G1, 88GB, 8BHH, 8EG1, 8GC1, 8H6H, 900D, 90E7, 90G7, 9667, 9907, 999D, 99E7, 9A67, 9A97, 9E97, 9EE7, 9G07, 9G67, 9GA7, AA45, AA97, AGA7, B005, B03B, B06B, B0C5, B60B, B63B, BAA5, BAA7, BCC5, BFA5, BG8H, C045, C055, C555, C5C1, C5F5, CC05, CC81, CCC5, D06D, D09D, D0ED, D38B, D3E1, D60D, D6DD, D8GB, DD6D, DE9D, DG01, E001, E097, E0G1, E8C1, EDC1, EE97, EGC1, EGG1, EGGD, FH6H, G007, G00B, G00H, G03B, G067, G097, G0C1, G0G1, G1GH, G33B, G38B, G3G1, G70H, G777, G88B, GA67, GAA7, GG81, GGC1, GGGH, H0FH, H66D, HEGD, HFHH, 1AAA7, 222DD, 30GG1, 3388B, 33E01, 38G8B, 3G3C1, 3GGG1, 4002D, 500C5, 50C55, 50CF5, 53GG1, 558C1, 55CC5, 55CF5, 58GG1, 5C8C1, 5CFF5, 5G881, 5GG31, 6000H, 6003B, 6006D, 600DB, 6033B, 606GD, 60D0B, 66GGD, 6D03B, 6D33B, 6H6DD, 6HD6D, 6HDED, 70G07, 70GGH, 777A7, 7AAG7, 7G0GH, 80G0B, 8888B, 8CCE1, 90067, 90097, 9022D, 99967, 99997, 9A007, 9A0A7, 9AA07, 9AAA7, 9E007, A0045, A0455, A0667, A09G7, A0A07, A0G07, A0G97, A9997, AA0A7, AAG67, B0AF5, B6GGH, B7GGH, B8HHH, BA045, BAF05, BG667, C0F05, C5005, C5581, C88C1, C8CC1, C8CE1, CCF55, D03C1, D060B, D080B, D0CC1, D0G0B, D0G8B, D3G3B, D600B, DDDED, DG331, DG80B, E8G81, E9007, F6GGH, G018H, G0301, G0331, G466D, G6667, G66GD, GD08B, GG18H, GG6GD, GGG4D, H060H, HGGGD, HHH6H, 199AA7, 40006D, 40600D, 46600D, 5055C5, 5505C5, 55CCC1, 588CC1, 58CCC1, 60000D, 60009D, 7077G7, 7707G7, 777G07, 88000B, 9099A7, A000A7, A009A7, A09067, A099A7, A0AAA7, A90AA7, A99AA7, AA0007, AA6667, AAAG07, BFFF05, BFFFF5, C0FFF5, CCECC1, CECCC1, CF0FF5, CFF005, D0008B, D0033B, D0088B, D0333B, D033GB, D03G31, D0633B, DD990D, DGGG31, FHHHHH, G00081, G6GGGD, G8GGG1, GGG001, GGG331, GGGGG1, GGGGGD, H0006H, H00H6H, HH600H, 222222D, 22DDDDD, 333333B, 5CCCCC1, 70007G7, 88CCCC1, 9000007, 9000A07, A000G67, AAAA667, BBBB33B, C000CF5, C000FF5, CCCCCE1, CCCCEC1, D00063B, D00GG31, D63333B, DCCCCC1, DDDDD9D, DGCCCC1, GCCCCC1, GG00031, 4022222D, 6000GGGD, 66666667, 770000G7, AAAAA007, B6666667, BBBBBB3B, CFFFFF55, D00000C1, D0000EC1, 455555555, 5555550C5, 667777777, A00000967, A00009097, A00009967, A45555555, AAAAAAA07, BHHHHHHHH, CCCCCCCC1, CF0000005, CFFFFFF05, D00000G3B, E0CCCCCC1, G00000031, 70000000G7, A000000097, D000003301, 777777700G7, A0000900007, D0000000001, D000000GGG1, 677777777777, 8HHHHHHHHHHH, 2DDDDDDDDDDDD, 55555555555C5, 77AAAAAAAAAA7, D00000000006B, D0000000003GB, AAAAAAAAAAAAAA7, D0000000000000B, 77777777777777G7, CCFFFFFFFFFFFFFF5, BBBBBBBBBBBBBBBBBBB6B, CFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF5, GG0000000000000000000000000000001, HDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, C00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000F5, 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000B, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHFH, C0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000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===Base 19=== ====Additional known quasi-minimal primes (not necessarily the next)==== 90₄₂₉₉₄G, 4F0₄₉₈₄₇6, FG6₁₁₀₉₈₄, 1E70₁₂₂₈₉₆1 ===Base 20=== 13, 19, 1B, 1H, 21, 23, 27, 2D, 2J, 31, 37, 3B, 3D, 3J, 43, 49, 4H, 51, 53, 57, 59, 5D, 67, 6B, 6H, 6J, 79, 7B, 7H, 83, 87, 8D, 8J, 91, 9B, 9D, 9H, 9J, AB, B3, B7, B9, BD, BJ, C1, CB, CH, D3, D9, DB, DH, E1, E3, ED, F7, FB, FD, FH, GB, GH, H7, H9, HD, HJ, I7, ID, IJ, J3, J9, JH, 101, 10J, 111, 11D, 11J, 147, 14J, 161, 171, 177, 1A1, 1A7, 1AD, 1AJ, 1C7, 1CD, 1CJ, 1D1, 1DD, 1F1, 1FJ, 1G7, 1GD, 1GJ, 1I1, 1J7, 209, 20B, 22H, 25B, 269, 28B, 28H, 2A9, 2BB, 2C9, 2EB, 2EH, 2F9, 2G9, 2HB, 2IB, 30H, 329, 33H, 3A9, 3E9, 3G3, 3H3, 401, 407, 40D, 40J, 411, 417, 44D, 44J, 461, 46D, 471, 477, 47D, 47J, 4A1, 4BB, 4C7, 4D1, 4D7, 4DD, 4DJ, 4F1, 4GD, 4J7, 4JD, 4JJ, 50B, 50H, 54J, 55B, 5BH, 5EH, 5GJ, 5HB, 5HH, 5IB, 5IH, 5JJ, 661, 6A9, 6E9, 6G9, 701, 703, 70J, 71D, 747, 77D, 7A1, 7AJ, 7D1, 7D7, 7DJ, 7FJ, 7G1, 7I3, 7J1, 7J7, 809, 80H, 811, 82B, 82H, 869, 881, 88B, 899, 8C9, 8EB, 8G9, 8H1, 8HH, 8IB, 907, 989, 9A3, 9C7, 9E9, 9G3, 9G9, A01, A03, A07, A0D, A0J, A11, A17, A29, A2H, A4D, A4J, A69, A6D, A7D, A7J, A8H, AA1, AAH, AAJ, AC3, ACD, ACJ, AD1, ADD, AE9, AEH, AG7, AGJ, AHH, AI3, B11, B2B, B2H, B41, B5H, B81, BB1, BBH, BEB, BG1, BHB, C0D, C5J, C6D, C73, C89, C97, CA3, CA9, CCJ, CE7, CEJ, CFJ, D17, D1D, D41, D6D, D77, DA7, DAD, DAJ, DDJ, DF1, DFJ, DG1, DG7, DJ1, E2B, E2H, E5B, E5H, EA7, EC9, EEH, EG7, EGJ, EJ7, F61, FA3, FEJ, FF1, FG3, FG9, FI1, G11, G17, G29, G39, G41, G61, G69, G77, G7D, G89, GA7, GAJ, GCD, GCJ, GD1, GDD, GDJ, GE9, GF1, GF3, GF9, GFJ, GGD, GI1, GI3, GJ1, GJD, H03, H2H, H33, H5B, H5H, H81, H8B, H8H, HA1, HC3, HF3, HG1, HHB, HIH, I0B, I61, I89, IAH, IE9, IG3, IG9, IH1, II1, IIH, J07, J11, J1J, J41, J47, J4B, J4J, J71, J7D, J7J, JCD, JD7, JDD, JDJ, JF1, JFJ, JG7, JGD, JJD, 104D, 10E7, 1DE7, 1DEJ, 1E7J, 1EJJ, 1G81, 1J6D, 1J81, 20AH, 25AH, 2829, 28E9, 2A5H, 2E29, 2H0H, 2HAH, 2IHH, 3089, 30A3, 30G9, 325H, 358H, 38F9, 3A63, 3CG9, 3F89, 3GC9, 402B, 40IB, 44I1, 458B, 45CJ, 45FJ, 4841, 484B, 485B, 48G1, 4AEJ, 4AFJ, 4BI1, 4CAD, 4CAJ, 4CGJ, 4E4B, 4EJB, 4F5J, 4FAJ, 4G81, 4GEJ, 4I2B, 4I8B, 4IG1, 4J81, 4JB1, 4JIB, 52AH, 542B, 548B, 550J, 55EJ, 584B, 5A5J, 5B4B, 5C0J, 5E4B, 5FAJ, 6029, 610D, 6141, 616D, 6299, 62I9, 6389, 641D, 6441, 64CD, 64G1, 66G3, 68G1, 6A41, 6AF1, 6AG1, 6AI1, 6D01, 6DA1, 6DCD, 6F01, 6F29, 6G01, 6G03, 6G0D, 6G4D, 6GA1, 6GG1, 6I01, 6I29, 6IF1, 704D, 70A7, 70GD, 715J, 71E7, 73F3, 745J, 74CD, 74CJ, 74EJ, 7641, 76A3, 76AD, 76GD, 7761, 7773, 77G3, 7841, 78I1, 7C4J, 7C63, 7CA7, 7CC3, 7F41, 7FF3, 7G6D, 7GA3, 7GE7, 7GG3, 7I41, 7I81, 7J5J, 8041, 804B, 80BB, 80F1, 8229, 8289, 82E9, 84G1, 86A1, 86F1, 86G1, 8889, 88A9, 88E9, 88IH, 8AA9, 8B4B, 8B61, 8BIH, 8EA9, 8F01, 8FA1, 8FE9, 8FF9, 8FG1, 8H4B, 8I29, 8I5H, 8II9, 9629, 9763, 9973, 9997, 9A77, 9AA7, 9AC9, 9AI9, 9E47, 9E77, 9F29, 9G47, A0A9, A0F9, A0G9, A0I9, A3F9, A3I9, A481, A633, A681, A6G1, A6G3, A7A3, A7C7, A7F1, A8I1, A909, A933, A9F3, A9I9, AA73, AAC7, AC09, AC77, ACC9, ACF9, ADC7, ADE7, AEC7, AEJJ, AF39, AF81, AF93, AFA9, AFC9, AFI9, AFJ1, AFJJ, AG81, AGG9, AH63, AI41, AI5H, AIF9, AJ5J, AJ61, AJE7, AJI1, B001, B08H, B0F1, B40B, B601, B84B, B8IH, BAIH, BFA1, BHF1, BI5B, BIA1, C0E9, C0G3, C0G9, C299, C2I9, C447, C4AD, C4G7, C707, C74D, C74J, C777, C7AD, C7CD, C7GD, C7GJ, CAA7, CAAD, CAD7, CADJ, CAGD, CAJD, CCE9, CD07, CD47, CD4D, CD7J, CDD7, CDGD, CDJJ, CE99, CEG9, CG07, CG09, CG4J, CG63, CG7J, CGC3, CGC7, CGD7, CGI9, CJ0J, CJAD, D011, D047, D05J, D081, D0E7, D0JD, D0JJ, D181, D4EJ, D50J, D761, D781, D7CD, D7EJ, D801, DA81, DC47, DC4D, DC7J, DCDD, DCGD, DCGJ, DCJJ, DD01, DD61, DDCD, DE0J, DEC7, DECJ, DG0J, DJC7, E00J, E047, E069, E0BH, E0C7, E0E9, E0EB, E2E9, E45J, E4AJ, E4EB, E4EJ, E5CJ, E5EJ, E5FJ, E6I9, E7EJ, E80B, E829, EA09, EA99, EAG9, EB0B, EB4B, EC0J, EC7J, EE0J, EE97, EEA9, EEE9, EEEJ, EEJB, EF89, EF99, EFAJ, EFCJ, EFI9, EFJJ, EG09, EG99, EH4B, EI4B, EI99, EIHB, EIHH, EII9, EJ0B, EJ8B, EJBB, EJEB, EJIB, F029, F0A9, F0FJ, F1G1, F2I9, F389, F4G1, F5AJ, F629, F8A1, FAC9, FAF9, FC0J, FE99, FF0J, FG4J, FGA1, FGGJ, FI29, FJ01, FJAJ, FJCJ, FJG1, G01D, G04J, G05J, G07J, G099, G0A1, G0A3, G0AD, G0E7, G0G1, G0G7, G0GJ, G0JJ, G10D, G15J, G333, G3A3, G3C3, G45J, G4E7, G663, G6C3, G947, G973, G993, G9C9, G9G7, G9I9, GAG9, GC33, GC47, GC99, GCI9, GDC7, GEJJ, GG01, GG97, GGA9, GGEJ, GI09, GI99, GIA9, GIC9, GJ5J, GJE7, GJEJ, H0AH, H0BH, H0I1, H141, H601, H6I3, HA63, HB01, HB0B, HB0H, HB61, HBAH, HBH1, HBI1, HEIB, HHH1, HI41, HI4B, HIF1, I081, I0A3, I141, I20H, I25H, I2BH, I441, I48B, I52B, I52H, I55H, I5EB, I629, I6A3, I85B, I88H, I8A1, I8HB, IA33, IA63, IAC9, IAF1, IAF3, IE8B, IEBH, IEIB, IF01, IFA9, IG01, IGG1, IHEB, IHHH, IHI3, IHIB, J04D, J05B, J0AD, J0AJ, J0BB, J0J1, J16D, J22B, J5EB, J64D, J6AD, J7C7, J7E7, J801, J8G1, JA5J, JAI1, JB5B, JB61, JBA1, JBBB, JCA7, JCAJ, JD61, JDI1, JE77, JE8B, JEBB, JEJB, JEJJ, JG0J, JG5J, JGEJ, JI5B, JI81, JIB1, JIBB, JIEB, JIG1, JIIB, JJ61, JJEJ, 1060D, 1666D, 1706D, 17E5J, 17JJJ, 1D007, 1D7JJ, 1J5EJ, 1JJJ1, 200IH, 20I5H, 22299, 2242B, 2244B, 22929, 29229, 29I99, 2E8I9, 2HHHH, 2I2I9, 2II99, 33389, 33G99, 366A3, 368I9, 38A5H, 38EAH, 38EIH, 3E8IH, 3G0I9, 3GGG9, 3HHAH, 404EB, 40E0B, 41EEJ, 4224B, 444EB, 444G1, 44E47, 44EEB, 44GG1, 455AJ, 45EAJ, 4A447, 4A55J, 4AE47, 4CCCD, 4EEAJ, 4EIEB, 4EIIB, 4G447, 4G4G7, 4GG1J, 4II4B, 4II5B, 4J80B, 4JE0B, 5005J, 50CAJ, 50ECJ, 5588H, 55A5H, 55FCJ, 5AEFJ, 5E5AJ, 5EAFJ, 5EB8B, 5EE8B, 5EEBB, 5EF0J, 5EFFJ, 6014D, 604AD, 6060D, 60689, 606A3, 606CD, 60AAD, 60AF3, 60AGD, 60DGD, 60G33, 60GAD, 60I81, 62229, 62889, 633A3, 6600D, 668F9, 66929, 66AAD, 66CCD, 66DGD, 66IA3, 68FI9, 68I41, 69929, 6A663, 6A6F3, 6D0GD, 6DDI1, 6G6AD, 6GGA3, 7066D, 707G7, 70C07, 70CAD, 70CCD, 70CG7, 70DDD, 71JJJ, 73363, 74441, 7606D, 76363, 76663, 76C4D, 76F11, 76G33, 77107, 77441, 7777J, 777C7, 777G7, 77AC7, 77AF3, 77C07, 77E4J, 77E7J, 77GGJ, 77JGJ, 7A733, 7AAA7, 7ACC7, 7C077, 7CC4D, 7CF33, 7CG4D, 7CJ4D, 7CJGJ, 7DD0D, 7ECJJ, 7EJEJ, 7F333, 7F6C3, 7FC33, 7G007, 7G4GJ, 7G733, 7G763, 7G7C3, 7GCC7, 7GGC7, 7GGGJ, 7GJGJ, 7J06D, 7JAAD, 7JGJJ, 800B1, 80BA1, 80IA1, 84I41, 8555H, 8558H, 85A5H, 8855H, 8888H, 88F29, 8A6I1, 8AGG1, 8AIF1, 8AIG1, 8BB0B, 8BE8H, 8EEF9, 8EF29, 8F829, 8F8I9, 8FIA9, 8GAG1, 8H00B, 8HBBB, 8IE8H, 900A9, 90AF9, 90IA9, 92II9, 97333, 97F33, 990A9, 994A7, 994G7, 999A9, 99A47, 99A99, 9A009, 9A999, 9C029, 9C929, 9CC29, 9FFA9, 9FIA9, 9I9A9, 9IA99, 9IAF9, A3009, A3309, A3333, A3393, A3939, A3963, A3993, A39C9, A3A33, A3AA3, A3C99, A3FF3, A4E47, A4EE7, A555H, A66F3, A6F63, A7771, A77F3, A7AA7, A7EE7, A8641, A88F9, A9399, A94A7, A9663, A9777, A97A7, A97E7, A9977, A9999, A9EE7, AA3A3, AA4A7, AA7A7, AA9E7, AAA33, AAA89, AAA97, AAAF3, AAF89, AAG09, AAG93, ACA47, ACCC7, AEE47, AEE77, AF099, AF363, AF5FJ, AF889, AFF09, AFF99, AFFF3, AGAI9, AGG33, AGGG1, AHGG3, AI009, AIA09, AIA99, AIII9, AJAA7, AJAAD, AJJC7, AJJG1, AJJJ7, B00IB, B044B, B06A1, B08BB, B0EAH, B0EHH, B44IB, B544B, B5BBB, B8E8H, BAH61, BB44B, BB45B, BBB5B, BBBIB, BE0AH, BH00H, BH0H1, BH6I1, BI0EH, BI44B, BI8BB, BIBBB, BIE8H, C0029, C04AJ, C07G7, C0A77, C0C29, C0CC7, C0G47, C0GGJ, C0I29, C2EE9, C6C29, C7AC7, C9029, C9929, C9C29, CC0C7, CC3G9, CC7C7, CCA77, CCAC7, CCCCD, CCF29, CCG03, CCG47, CCG93, CCGAD, CCGC9, CD0GJ, CE0I9, CE629, CE6F9, CEIF9, CFC29, CFE09, CFEF9, CFF29, CG003, CG033, CGGG9, D0061, D00A1, D00D1, D00GJ, D01EJ, D074D, D07DD, D07GJ, D0C4J, D0CCD, D0D7D, D0EEJ, D0G4D, D0GGJ, D155J, D4CCD, D4EE7, D55CJ, D6001, D60A1, D6IA1, D7D0D, D7DGD, D7G4J, D7GGJ, D7JCJ, DC0J7, DCC07, DD7I1, DDC07, DDD07, DDD11, DDD47, DDDD1, DDDGD, DDG0D, DE4E7, DEE7J, DEJ5J, DG4GJ, DGE4J, DGJJJ, DJ55J, DJEEJ, DJGJJ, DJJJ7, E00HB, E00I9, E00IH, E044B, E04FJ, E08BB, E08HB, E0999, E09A9, E09E7, E09F9, E0AIH, E0BBB, E0C4J, E0E7J, E0F4J, E0GA9, E0I09, E0IA9, E0JEJ, E0JJB, E22I9, E2I29, E448B, E6009, E6229, E6889, E69F9, E6F09, E6FF9, E755J, E7CJJ, E7EC7, E7JJJ, E844B, E888H, E8A89, E8EI9, E8IA9, E90A9, E90I9, E9699, E96F9, E9IA9, E9IF9, EA55J, EA889, EAEFJ, EAFF9, EAFFJ, EB0AH, EB0IH, EB88H, EBI0H, ECC47, EE00B, EE299, EE4FJ, EE74J, EE7C7, EEBIB, EEF29, EF229, EFF4J, EFFA9, EGAI9, EH0IB, EHBIB, EHEEB, EHH0H, EHIIB, EI00H, EI229, EI8BB, EIEBB, EIF09, EIFF9, EIIEB, EJAJJ, EJE5J, F0001, F000J, F0081, F0089, F00CJ, F0141, F041J, F04GJ, F0841, F08F9, F08G1, F0AJJ, F0CE9, F0E69, F0F89, F0FE9, F0GG1, F0GJJ, F1J5J, F2229, F2289, F22E9, F4FGJ, F500J, F50CJ, F5FFJ, F8EE9, F9A09, F9A99, F9IA9, FA099, FA8G1, FC4GJ, FCJJJ, FE0I9, FE669, FEAA9, FEAI9, FEF69, FEFE9, FEFF9, FEI69, FEIF9, FF089, FF4FJ, FF55J, FF8I9, FFA09, FFAI9, FFE69, FFF29, FFF5J, FFF89, FFFE9, FFIA9, FFJGJ, FG081, FGG81, FJ05J, FJA81, FJJ0J, G001J, G0047, G004D, G0063, G00C7, G0363, G0603, G0633, G066D, G0963, G0AC9, G0AI9, G0CC7, G0CC9, G0II9, G4AAD, G4EEJ, G600D, G64AD, G666D, G66AD, G6G33, G7E4J, G7GJJ, G7JGJ, G9009, G9303, G9603, G9A09, G9CC3, GA6A3, GAA33, GAA93, GAG33, GC009, GC093, GC0C3, GCC03, GCC09, GD447, GDE47, GEE07, GEEC7, GG00J, GG073, GG1JJ, GG763, GG7C3, GG8A1, GGG07, GGG4J, GGG71, GGGC7, GGGGJ, GGGJ7, 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CDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDDD, 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71D, 73D, 73H, 73J, 75B, 75H, 768, 77H, 77J, 7B1, 7B5, 7BB, 7BD, 7C8, 7D1, 7DB, 7DD, 7E8, 7F5, 7FB, 7H5, 7HD, 7HH, 7HJ, 7J5, 7JD, 801, 80J, 832, 841, 86H, 86J, 881, 89G, 8E1, 8F4, 8F8, 8FA, 8FK, 8H4, 8IH, 8J4, 8JG, 90K, 91B, 91D, 91H, 93H, 93J, 955, 95H, 95J, 96G, 97B, 97D, 97H, 991, 99J, 9B1, 9BB, 9BH, 9BJ, 9CK, 9D1, 9DB, 9DH, 9DJ, 9EK, 9F5, 9FD, 9H1, 9HB, 9HD, 9IG, 9J5, 9KK, A0B, ABG, AF4, AF8, AG5, B15, B1H, B35, B3H, B3J, B4G, B51, B5B, B5D, B5H, B6G, B71, B75, B7B, B7D, B9B, B9J, BB5, BBH, BBJ, BC4, BCG, BF1, BF5, BFD, BH1, BHJ, BI4, BJB, C2D, C9K, CCD, CDG, CID, D0G, D2G, D35, D3B, D3H, D51, D55, D5B, D5D, D5J, D62, D71, D7H, D91, D95, D9H, DBH, DC2, DD1, DD5, DE2, DEG, DF5, DFJ, DH1, DHB, DI2, DJ1, DJB, DJJ, E14, E4B, E5K, E78, E81, E9G, EA5, EBG, EC1, EE1, EE5, EK5, F04, F11, F1H, F31, F3B, F3D, F44, F4A, F4K, F5D, F5H, F6K, F71, F7H, F7J, F88, F8A, F8K, F9J, FA4, FA8, FBB, FBH, FC4, FDB, FDJ, FE8, FFH, FFJ, FH5, FHB, FHJ, FI4, FI8, FIK, FJ5, FJD, FK4, FK8, G32, G5G, G78, G8J, 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====Additional known quasi-minimal primes (not necessarily the next)==== 40₄₇₃₃₃9G, CF₄₇₉₁₄₇0K ===Base 22=== 11, 17, 19, 1F, 1J, 1L, 23, 29, 2F, 2H, 31, 35, 37, 3D, 3H, 41, 49, 4D, 4F, 4J, 4L, 53, 5H, 5L, 65, 67, 6H, 6J, 73, 79, 7D, 7J, 83, 85, 8F, 8H, 8L, 91, 9D, A3, A7, A9, AD, AJ, AL, B9, BF, BL, C5, C7, CD, CH, CJ, D7, DL, E3, E5, E9, F1, F7, FH, FJ, G1, G7, GF, GL, H5, H9, HF, I1, I5, ID, J1, J3, JD, JF, JL, K3, K9, KH, KL, L1, L5, LH, 103, 12D, 145, 155, 15D, 163, 18D, 1A5, 1BD, 1BH, 1C3, 1D3, 1DH, 1EH, 1G3, 1GH, 1I3, 1K5, 1KD, 221, 227, 22J, 22L, 245, 247, 25D, 25J, 271, 277, 287, 28J, 2A5, 2B7, 2BD, 2BJ, 2D5, 2E1, 2E7, 2ED, 2EL, 2K1, 2KJ, 2LL, 30J, 343, 389, 39J, 3B3, 3GJ, 3IJ, 3J9, 3JJ, 3KF, 3LJ, 427, 443, 445, 457, 4A5, 4C3, 4E7, 4G5, 4I7, 4K5, 4K7, 515, 52D, 551, 559, 55D, 55J, 575, 58D, 59F, 5B1, 5C9, 5CF, 5D1, 5D5, 5DD, 5E1, 5ED, 5G5, 5GJ, 5J5, 5JJ, 5K1, 5KJ, 60D, 61D, 62L, 661, 66D, 689, 6C1, 6D1, 6DD, 6G3, 6KF, 701, 721, 751, 76L, 775, 77F, 77H, 787, 7A5, 7AF, 7B1, 7B7, 7CL, 7E1, 7FF, 7FL, 7H7, 7HH, 7K5, 809, 81D, 821, 827, 82D, 847, 869, 871, 881, 889, 899, 8A1, 8BD, 8BJ, 8D1, 8DJ, 8GJ, 8J7, 907, 90H, 92L, 93J, 943, 947, 95F, 977, 997, 9AF, 9B5, 9EF, 9F5, 9H3, 9HL, 9I7, 9J9, 9JJ, 9K5, A25, A45, A51, A61, A6F, AAH, AB5, ABH, ACF, AG5, AGH, AHH, AK1, B15, B2D, B2J, B33, B45, B47, B57, B71, B75, B81, B87, B8J, BB3, BB7, BC3, BDD, BE7, BEJ, BGD, BGH, BH3, BHD, BHJ, BIH, BJ7, BKJ, CA1, CAF, CB3, CC1, CEF, CG3, CKF, D09, D0J, D13, D21, D33, D39, D3F, D4H, D5J, D63, D81, D8D, DAH, DBD, DBH, DBJ, DCF, DD3, DEJ, DFF, DG9, DGH, DHD, DI3, DIF, DJ9, DK1, DK5, E0F, E0H, E27, E2D, E2L, E47, E7H, E87, E8J, EA1, EAH, EB1, EDH, EEJ, EFF, EFL, EH1, EIF, EIL, EJH, EJJ, EKD, ELF, F25, F43, FB5, FD3, FDD, FDF, FEF, FEL, FFD, FG9, G09, G0D, G25, G3J, G5D, G5J, G63, G8D, G99, GC3, GC9, GD9, GEH, GG5, GJ5, GJ9, GJH, H03, H1D, H21, H2J, H2L, H33, H63, H77, H8J, HCL, HDD, HE1, HE7, HGH, HGJ, HH7, HHL, HI3, HIH, HJH, HK1, HKD, HL3, I07, I0J, I43, I47, I7L, I9J, IBH, IEL, IG3, 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7KC1, 7L27, 7LCF, 7LK7, 802J, 80B1, 80GD, 80JJ, 80KJ, 820J, 86E1, 880D, 882J, 88E7, 88I7, 892J, 89EJ, 8B61, 8BI7, 8CI9, 8CK1, 8D6D, 8DD9, 8DGD, 8E01, 8E07, 8EE1, 8EI7, 8EK1, 8EKJ, 8I77, 8I87, 8IC9, 8IK7, 8J0J, 8K6D, 8KI7, 8KIJ, 8KK7, 9025, 9055, 9089, 9275, 92EJ, 92G5, 92GJ, 93EL, 9455, 9505, 9557, 9599, 96B3, 970L, 976F, 97G5, 982J, 98E7, 98I9, 98K7, 9905, 990J, 9925, 9995, 999H, 99BH, 99EL, 99G9, 99GJ, 99LJ, 9A55, 9B03, 9B27, 9BGJ, 9BH7, 9BK7, 9CG9, 9E6L, 9EHJ, 9EKJ, 9ELL, 9G39, 9G45, 9G4H, 9GA5, 9GAH, 9GBJ, 9GEJ, 9GHH, 9GHJ, 9GI9, 9GIJ, 9H7H, 9H87, 9HAH, 9HB7, 9I89, 9I9H, 9IB3, 9IEH, 9IKF, 9J7H, 9J87, 9JB7, 9JGH, 9K0F, 9K27, 9KGJ, 9KKF, 9L89, 9L9J, 9LG9, 9LGJ, 9LIJ, 9LKF, A081, A14H, A1IH, A201, A2C1, AA15, AA81, AAIF, ABC1, AC01, AHC1, AIIF, AKEF, B005, B01D, B01H, B051, B05D, B0A5, B0B1, B0BJ, B0DH, B0E1, B0GJ, B0H1, B0JH, B143, B16D, B255, B2B5, B2C1, B2G5, B2K7, B4EH, B4G3, B501, B5A5, B5BJ, B5C1, B5IJ, B5K5, B621, B663, B6I3, B6K1, B777, B7I7, B80D, B88D, B8ED, B8KD, BB01, BB0J, BB21, BB25, BB4H, BB6D, BBD5, BBE1, BBEH, BBH1, BBIJ, BBJH, BCK1, BD61, BDB5, BDGJ, BEE1, BG55, BGB5, BGGJ, BGIJ, BH27, BH7H, BHC1, BHH1, BHK7, BI77, BIGJ, BJ0H, BJBH, BJIJ, BJJJ, BK25, BK27, BK55, BKA5, BKK1, C043, C143, C2B1, C32L, C3G9, C601, C6EL, C6G9, C8B1, CBK1, CC2L, CC89, CE21, CECL, CF2L, CF89, CG69, CIG9, D003, D015, D045, D05D, D06D, D06F, D0DH, D0H1, D1D5, D1GD, D205, D26D, D2DJ, D32J, D38J, D403, D525, D561, D56D, D5AF, D5F5, D5GD, D6C9, D6D9, D6DF, D6E1, D6ED, D6F9, D6FD, D8IJ, D8JJ, D945, D993, D99F, D9C9, D9F3, D9G5, D9KJ, DA15, DAA5, DAE1, DAKF, DB25, DB61, DBC1, DBG3, DC43, DC99, DCE1, DD0F, DD1D, DD2D, DD2J, DD5D, DD9H, DDA5, DDD5, DDDJ, DDED, DDH1, DDHJ, DDI9, DDJ5, DE0D, DED1, DEDF, DEFD, DF03, DF45, DF55, DF5D, DF6D, DG05, DGGJ, DH0H, DH43, DHC1, DHEH, DHH3, DHJJ, DI8J, DIC9, DII9, DIJH, DJG5, DJHJ, DJKJ, DK6D, DKAF, DKDD, DKGD, E081, E0BJ, E0DD, E0DJ, E0E1, E0ED, E0J7, E0K7, E0KJ, E1GD, E1IH, E201, E281, E66F, E6BD, E6ED, E6K1, E6LL, E7EF, E861, E86D, E88D, EB0D, EB4H, EBHH, EBI7, EC6F, ECEL, ED0D, ED1D, ED6D, EDAF, EDDD, EDFD, EE01, EE0D, EE1H, EE21, EECF, EEE1, EEGH, EEH7, EF8D, EGGD, EGHH, EGIH, EGIJ, EH07, EH0L, EH8D, EHBH, EHBJ, EHK7, EHL7, EI4H, EIK7, EIKJ, EJ77, EK07, EK0J, EKC1, EKIJ, EKK1, EKK7, EL6L, ELBD, ELCL, ELDJ, ELK7, F059, F20L, F26L, F28D, F2GD, F2KD, F32L, F3AF, F3G3, F455, F56F, F595, F5KD, F6AF, F6GD, F88D, F8I9, F955, F995, F9B3, F9G3, F9KF, FB03, FB8D, FBI3, FC89, FD99, FDA5, FDI9, FEGD, FF59, FG2D, FGI3, FGKD, FI89, FIB3, FIKF, FK2D, FK45, FKD5, FL6D, FLG3, G003, G04H, G055, G0BH, G0HH, G0HJ, G0I3, G0K5, G0KJ, G26D, G2DJ, G2GJ, G303, G333, G393, G3G9, G3I3, G403, G40H, G433, G4AH, G4GH, G4H3, G4IH, G589, G5G9, G80J, G89J, G94H, G9G3, GAA5, GAIH, GB0H, GB43, GB6D, GBB5, GBG3, GBGJ, GBJJ, GD0H, GD45, GDD5, GDKD, GG2D, GG39, GG8J, GG9H, GGAH, GGDJ, GGEJ, GGGH, GGH3, GGKJ, GH0H, GHG3, GHH3, GHHH, GHJJ, GI03, GI2J, GI93, GIAH, GIEJ, GIG9, GII3, GJJJ, GJKJ, GK05, GKDD, GKGD, GKGJ, GKIJ, H081, H087, H0AH, H0BJ, 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5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BB5, J000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000BIJ, C4IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII3, F0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000066B3, G0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000A5, D5KKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKKF, HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHBH, L0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000IKF, 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LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLKE7, 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80BN, 80FH, 80LN, 80MN, 840N, 848N, 866B, 86FB, 880B, 880N, 884N, 88CB, 88FB, 88LN, 88MN, 8BBB, 8BLB, 8C6B, 8CCH, 8CFH, 8F0B, 8FHH, 8FLB, 8H0H, 8HCH, 8IGN, 8ILN, 8KKH, 8L8B, 8LBB, 8LFB, 8LIN, 8M8N, 8MLN, 8N0N, 8NGN, 8NLN, 9061, 9091, 90EJ, 90F1, 90GJ, 90K1, 940J, 9501, 95F1, 9CC1, 9E0J, 9E95, 9F61, 9FI5, 9G3J, 9II5, 9K01, 9KK1, 9M01, A007, A05D, A0AD, A33D, A3AD, A3F5, A44J, A727, A9EJ, AA0D, AAAD, AAAJ, ACM7, AD8D, ADKD, AE27, AE9J, AEAJ, AEE7, AIAD, AIDD, AIID, AJ9J, AK5D, AM07, AM27, AM35, AMK5, B08B, B0CB, B0GB, B18B, B1CB, B80B, B8CB, BB21, BB4N, BBCB, BBF1, BBFB, BBK1, BC8B, BCF1, BCLB, BF1B, BF8B, BFB1, BFM1, BGC1, BGF1, BK21, BL8B, BLFB, BM1B, BM3N, BMB1, BMMN, BNF1, C00D, C06B, C077, C0D7, C0H7, C0L7, C0LB, C0M1, C60H, C6LB, C76H, C7E7, CAID, CC01, CCFH, CCKH, CDLB, CGE7, CH07, CHE7, CI8D, CIAD, CK0D, CL8B, CLDB, CLE7, CM01, CM07, CME7, CMM1, D007, D08D, D0C7, D0HH, D0LN, D0M7, D0NN, D207, D2KH, D2M7, D777, D7E7, D80H, D8LD, DA27, DAC7, DAM7, DBFB, DBMB, DC77, DCLB, DDL7, DE77, DF0H, DF2H, DFFH, DFMB, DH27, DH8H, DHC7, DHHH, DILN, DK0H, DK2H, DK8H, DKHH, DLIN, DLL7, DLM7, DLMB, DM07, DMH7, DMMB, DNGN, E07J, E09J, E335, E355, E555, E5A5, E5K5, E79J, E93J, E995, EA35, EE95, EKE5, F00B, F00H, F06H, F08B, F0I5, F11B, F18B, F1L1, F20H, F26H, F2FH, F355, F661, F6K1, F80B, F86B, F8BB, FBGB, FBK1, FBLB, FC0B, FC6H, FCLB, FEK5, FGB1, FH05, FH0H, FH35, FH6H, FHCH, FHF5, FHHH, FI05, FK91, FKK1, FL1B, FLB1, FLBB, FM61, FMBB, FMK1, G00J, G021, G027, G0EJ, G0JJ, G0M1, G0M7, G1CB, G2E7, G40J, G4AJ, G4IJ, G4JJ, G6C1, G701, G94J, G9IJ, GAEJ, GAJJ, GB21, GBM1, GC01, GCF1, GCLB, GE0J, GEAJ, GEE7, GEG7, GEIJ, GEL7, GFM1, GGE7, GGMB, GI0J, GIIJ, GIIN, GJ9J, GM27, GMB1, GNM7, H005, H0K5, H0M5, H207, H2E7, H335, H3I5, H595, H5K5, H60H, H68H, H76H, H80H, H8HH, HAAJ, HE7J, HEC7, HGE7, HGM7, HH35, HI55, HIM5, I00J, I035, I08D, I0CD, I4JJ, IC0D, ICID, II0D, II0J, II35, IIAD, IILD, IIM5, IIMN, IJ9J, ILCD, IM05, IM35, IMNN, INLD, J03J, J0HJ, J0JH, J2CH, J39J, J3ID, J60H, J62H, J8CH, J9IJ, JGAJ, JGJJ, JH9J, JI0J, JIDD, JJ0H, JJCD, JJJD, JJLD, JL3D, JLCD, K0E5, K0I5, K0K1, K0KH, K191, K211, K2F1, K2G1, K591, K5AD, K6F1, K6G1, K9I5, KA0D, KAAD, KAM5, KCCH, KCHH, KD8D, KDDD, KFI5, KG01, KG61, KH0H, KHHH, KI55, KIDD, KK21, KK8H, KKF1, KKK1, KKKD, KM8H, KMHH, KMK5, L027, L0M7, L1C1, L1MB, L211, L727, L8BB, L8BN, L8FB, L8LN, L9C1, L9M1, LB8B, LBC1, LBM1, LCAD, LD77, LDIN, LDL7, LDM7, LF8B, LG21, LIIN, LLLN, LLN7, LM07, LM1B, LM77, LMG1, LN77, LNM1, M00N, M01B, M03N, M055, M077, M08B, M0B1, M0C1, M0GB, M0K1, M0M7, M0N7, M18B, M1BB, M1MB, M26H, M335, M3GN, M3M5, M3MN, M3NN, M501, M53N, M5M1, M5NN, M6BB, M6C1, M6G1, M6KH, M88B, M88N, M8BB, M8NN, MBB1, MC01, MCC1, MCKH, MCM1, ME07, ME35, MEK5, MGGB, MGMB, MH35, MH8H, MHE7, MHM5, MK8H, MKC1, MKG1, MKHH, MKK5, ML8N, MM01, MM8B, MMC1, MMLN, MMM5, MMMN, MMN1, MN0N, MN27, MNGN, MNLN, N007, N027, N077, N0C7, N0DN, N0IN, N1M1, N227, N2M7, N661, N707, N727, N8LD, NA27, NA3D, NC07, ND0D, ND0N, NDLD, NF11, NF61, NGG7, NILN, 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5M0M1, 5MNNN, 5N03D, 5N3LD, 5NA8D, 5NADD, 5NDGN, 5NNGN, 600KH, 61661, 616L1, 61MM1, 66161, 66611, 666B1, 666L1, 666LB, 66BB1, 66BG1, 66BLB, 66G61, 66KF1, 66LBB, 66LG1, 6BBBB, 6BFCB, 6CC11, 6F1M1, 6F66B, 6F6B1, 6F6L1, 6FBCB, 6FLMB, 6FMCB, 6FMM1, 6FMMB, 6GCM1, 6GM61, 6GMC1, 6K1C1, 6K1K1, 6KK11, 6KKL1, 6KL11, 6L1G1, 6LBFB, 6LCC1, 6LFMB, 6MM61, 6MM6B, 70291, 702C1, 702G1, 72CF1, 72EE7, 7433J, 7443J, 77A07, 79901, 799F1, 7AAEJ, 7EE27, 7H9EJ, 7K2KH, 7KK2H, 7KKKH, 7KKMH, 7L2C1, 800NN, 806BB, 808BB, 808LB, 80F8B, 80IIN, 833ID, 860KH, 8886B, 888NN, 8BG4N, 8CH6H, 8CKHH, 8FC0H, 8FFCH, 8HHHH, 8IIIN, 8K0HH, 8LL4N, 8M0NN, 8MNNN, 8NNND, 9000J, 900M1, 9034J, 90IIJ, 94IIJ, 96CM1, 96KF1, 96MM1, 990C1, 990M1, 99591, 99961, 999C1, 99F91, 99FM1, 99KF1, 99M61, 99MK1, 9AAA5, 9FEE5, 9FFA5, 9FFF5, 9II4J, 9K6C1, 9K9C1, 9K9F1, 9KF91, 9M6M1, 9MK61, A02M7, A0A35, A0AM5, A0C77, A0D0D, A0DDD, A0EC7, A0M55, A0MM7, A2ME7, A3335, A33M5, A3555, A3MM5, A550D, A58ID, A5D0D, A5DDD, A74AJ, A7E07, AA0M5, AA3ID, AA3M5, AA83D, AA8ID, AAAM5, AACID, AAICD, AAM05, AC08D, AC0ID, AC0KD, AC8ID, ACA8D, AFA35, AIC8D, AJAJJ, AK0KD, AKI0D, AKKM5, AKM05, AM505, B00FB, B00LB, B0LBB, B1MBB, B1MMB, B2GM1, B3MGN, B88BB, BB3MN, BBB3N, BBB8B, BBBBN, BBBMN, BBCC1, BBGB1, BBGLB, BBMC1, BBMGB, BBNM1, BFB0B, BFBMB, BFGLB, BFGMB, BGGLB, BGMM1, BLMMB, BMC0B, BMGM1, BMMK1, BMNM1, C006H, C00CH, C00M7, C0C11, C0CC1, C0E07, C0F6H, C0GG7, C0K8D, C0KDH, C0KHH, C1CC1, C1CM1, C1MC1, C70F1, C886B, CC06H, CCCCH, CCCF1, CCCM1, CDHH7, CE707, CE7L7, CEE77, CEEG7, CEL07, CFFHH, CGLL7, CHC6H, CHCCH, CHH7H, CHHH7, CHHM7, CI0LD, CK0HH, CKHCH, CL007, CLD07, CLLL7, CM777, CMMM7, D008H, D07L7, D0DE7, D0EL7, D0L77, D0M8H, D0NA7, D22E7, D70A7, D7227, DBLBB, DCL07, DD7A7, DDAE7, DDD8D, DDE07, DDK0D, DDM77, DEE27, DKK8D, DL227, DL707, DLBBB, DLDFB, DLE27, DM2E7, DMM77, DMME7, DN0A7, DNIIN, DNLLN, E000J, E0227, E02E7, E0AAJ, E0GL7, E0I0J, E2E07, E7AAJ, E7E27, EAA55, EAK55, EC777, ECEE7, ECEL7, ECGL7, ECL77, EEC77, EECL7, EEEC7, EEG27, EEGG7, EG0E7, EG0L7, EG207, EGE07, EGG07, EGG27, EGL07, EI9IJ, F02HH, F06LB, F0C6B, F0CCH, F0CHH, F0E55, F0EA5, F0FH5, F0GGB, F0M2H, F0MGB, F0MMB, F1BMB, F3FF5, F3I35, F3II5, F6BCB, FA035, FB1MB, FBBM1, FBBMB, FBM8B, FC0HH, FC88B, FCFHH, FFC0H, FFCFH, FFCHH, FFF0H, FFF6H, FFFH5, FFHI5, FFI55, FG1MB, FGGCB, FGM1B, FHK55, FI5I5, FKEA5, FKKI5, FKL11, FM16B, FM1CB, FM62H, FM6CB, FMBG1, FMM0B, G0001, G00E7, G039J, G06F1, G07C1, G07F1, G0A9J, G0G07, G0GG7, G0I4J, G0I9J, G0LL7, G22M7, G2M07, G339J, G433J, G62M1, G6M61, G6MM1, G903J, G933J, GAA9J, GBCC1, GE007, GGGGB, GGGL7, GJ0IJ, GL007, GL2M7, GLL07, GLLM7, GMGCB, GMM07, GMM1B, H007H, H05I5, H0CCH, H0CM7, H0GG7, H0H27, H0H8H, H0HC7, H0HH5, H0I35, H0MM7, H3555, H35F5, H3F55, H3FF5, H5055, H50F5, H7HEJ, H9995, H9GEJ, HCC0H, HCC6H, HCGG7, HCHM7, HE027, HE0G7, HEE07, HEEG7, HEG27, HF0F5, HF505, HF555, HFF05, HFKF5, HG0G7, HH06H, HH08H, HH0H5, HH5I5, HH7HJ, HH9EJ, HH9I5, HHCHH, HHE07, HHGG7, HHH05, HHH7H, HHH9J, HHKI5, HHM05, HHM07, HHM27, HI0I5, HJ86H, HJC0H, HK055, HK9F5, HKF05, HKFF5, HKFK5, HKII5, HKK95, HM2M7, HME27, HMKM5, HMM05, HMM27, HMM55, HMMM7, I00DD, I00M5, I044J, I0505, I09IJ, I0AAD, I0D0D, I0DDD, I0I05, I0IDD, I0II5, I0IJJ, I0JIJ, I33M5, I4I4J, I5INN, I888N, I8NND, I8NNN, I904J, I94IJ, IA8ID, IAADD, IAI8D, IDDDD, IDINN, II88N, II8NN, IID8D, IIDIN, III8N, IIIDD, IIIID, IIIIN, IIIND, IIN8D, IINDD, IJJ0J, IJJJJ, ILILN, ILLIN, IMM8N, INA0D, INDNN, INGIN, INNDN, INNND, J000H, J002H, J00AJ, J00GJ, J00KH, J02KH, J068H, J080H, J090J, J0A9J, J0AAJ, J0C0H, J0G0J, J0IIJ, J0JGJ, J0JIJ, J0K8H, J2K0H, J2KKH, J6K8H, J86KH, JC00H, JC0KH, JCCCH, JCK0H, JCKCH, JDDLD, JG93J, JIIJJ, JJ0IJ, JJ2KH, JJ9GJ, JJCCH, JJG9J, JJGIJ, JJJ9J, JJJJH, JJK8H, JK08H, JK0CH, JK8KH, JKC0H, JKKKH, K0001, K0091, K020H, K02C1, K03ID, K0611, K06L1, K083D, K08HH, K0961, K09C1, K0CF1, K0F91, K0KM5, K0LG1, K1G21, K20HH, K29K1, K2KHH, K5001, K500D, K58ID, K5D0D, K5L11, K6621, K6C11, K6LC1, K8CKH, K8KCH, K96C1, K99E5, K9F91, K9FA5, K9FE5, K9K91, KA55D, KC011, KCF11, KD02H, KD0MH, KD20H, KDM2H, KEA55, KEAA5, KEK95, KEKK5, KF1G1, KF1K1, KF611, KF6L1, KFEA5, KH8CH, KI005, KIMM5, KK05D, KK0AD, KK0DH, KK2CH, KK2KH, KK33D, KK961, KK9C1, KKA5D, KKD0D, KKE55, KKI0D, KKIID, KKIM5, KKK0H, KKKM5, KLGC1, KMK2H, L188B, L1991, L2007, L22M7, L2EE7, L2MM7, L333D, L3LIN, L7291, L72G1, L88IN, L8C8B, L9991, LBB1B, LBBBB, LBBBN, LD0E7, LDBBN, LE207, LFMCB, LGCC1, LL227, LL3IN, LL48N, LLM27, LLMM7, LMBCB, LME27, LMMBB, LMMM7, LN33D, LN3AD, LNAAD, LNACD, M0007, M0061, M00K5, M0207, M066B, M0BCB, M0EE5, M0G01, M0GM1, M0M8N, M27KH, M2E27, M2M07, M2M27, M5005, M5555, M66CB, M66K1, M6K61, M6MCB, M7007, M7EE7, M8C0B, M8KCH, M8MGN, MBBGB, MBGM1, MCCCH, MCHCH, MEE55, MGBC1, MGMM1, MH227, MH2M7, MHH7H, MKM55, ML3LN, MM0CB, MM16B, MM227, MM661, MM6K1, MME55, MMEE7, MMKE5, MMM07, MMM6B, MMMB1, MMMGB, MMMM7, MNM61, MNN3N, N00CD, N00KD, N03LN, N0A8D, N0AM7, N0D8D, N0KKD, N0L3N, N0LAD, N0NDD, N16L1, N3GIN, N3LAD, N3LIN, N3NNN, N61L1, N96M1, N9M61, NA0CD, NAK0D, NAKKD, NCA8D, NCM77, NDGIN, NDIIN, NDLLN, NF991, NGM07, NIIIN, NINNN, NKKDD, NLNAD, NN0LN, NN191, NN3NN, NN6L1, NN83D, NNAAD, NNDIN, NNGIN, NNL3N, NNLND, NNM61, NNNIN, 166G21, 16G621, 19MMM1, 1BBBMB, 1BBGM1, 1GCCC1, 1GCCM1, 1MMM1B, 200E27, 2E0027, 2HH0HH, 2HHC0H, 2KK0HH, 2M0E27, 2M22E7, 30NNNN, 3333M5, 333AID, 333I35, 33I555, 3F5FF5, 3I3MM5, 3I88GN, 3II8LN, 3IIII5, 400IJJ, 40J00J, 40JJ3J, 40JJJJ, 44403J, 444I0J, 44IJJJ, 44J0JJ, 44JJIJ, 48I8IN, 4I440J, 4II8IN, 4IJ0IJ, 4JIIIJ, 4JJ0JJ, 4JJJ0J, 50033D, 5003AD, 5008ID, 500D8D, 500G01, 500L11, 500LAD, 500MG1, 503LAD, 508ILD, 50DDLD, 50ILAD, 50M001, 516G61, 519MM1, 538NNN, 53NNNN, 55005D, 5508ID, 550D8D, 558ILD, 55F5I5, 56G661, 58333D, 58NNNN, 5999F5, 59AAF5, 5DNNNN, 5F55I5, 5FMMM1, 5G6661, 5K9AA5, 5KK9F5, 5KKK95, 5M0001, 5NDD8D, 5NDINN, 5NN33D, 5NNLAD, 5NNNAD, 5NNNDN, 608K0H, 61CCM1, 61G621, 661G21, 666621, 6666CB, 6666F1, 66K661, 6BCCC1, 6BKKC1, 6F6BBB, 6G6621, 6GCCC1, 6GMMM1, 6K6K61, 6M666B, 70A077, 70L991, 7722E7, 772E27, 7772E7, 777A27, 777L27, 77A777, 77EL27, 7A7077, 7A7777, 7E7227, 7L2E27, 7LEL27, 7LL2E7, 7LLE27, 7LLL27, 800GIN, 80NINN, 80NNNN, 8BBMGN, 8C888B, 8C88LB, 8MM0GN, 900001, 90043J, 959MM1, 96K661, 9999F1, 9999K1, 999AF5, 999FF5, 99EEE5, 99K991, 99MMM1, 9AAFF5, 9EIIIJ, 9F9991, 9F9MM1, 9FEAA5, 9G444J, 9K9991, 9M6661, A000CD, A000KD, A000M5, A0083D, A00I0D, A00M05, A022E7, A07E77, A0FF35, A0K3ID, A0K83D, A4AJJJ, A77777, AA0035, AA0355, AAA035, ADDD0D, ADDDDD, AF0035, AFFF35, AKK8ID, AM0M05, AM7777, B0F0MB, BBBBM1, BBLBMB, BFBBBB, BFM0MB, BFMMMB, BLBBMB, BLMBBB, C00071, C000E7, C007C1, C00G07, C07KKH, C0CC6H, C0CH6H, C0EEE7, C0HHHH, C777L7, C77L77, C7L777, C7LL07, C8088B, CAAK8D, CAKKAD, CC000H, CC0CHH, CD000H, CD0KKH, CE0007, CEE0E7, CELL77, CG0007, CGGL07, CH0CHH, CHCH0H, CHHH6H, CK0C0H, CKAK8D, CKKA8D, CL7707, D002FH, D0D0KD, D0DA77, D0DKKD, D0IIIN, D0K0DD, D0KDKD, D0KKDD, D0KKKH, DC0EE7, DCEEE7, DD0227, DD0D27, DD0DKD, DD0KKD, DD2E27, DDD0D7, DDD0LD, DDD227, DDDA77, DDDBCB, DDDCE7, DDDDFB, DDDMM7, DDEEE7, DDMBCB, DEEC07, DH000H, DHMEE7, DIIIGN, DK0KDD, DKMKKH, DMBBBB, DMEEE7, DMMM27, E00G27, E07727, E0C707, E0CE77, E0E027, E0EEG7, E0EGE7, E0EL27, E0GE27, E0L207, E0LE27, E0LL27, E2E2E7, E7L2E7, E900IJ, E9EEE5, EAAKK5, EC00E7, EC0G07, EC7007, ECEG07, EE0G07, EE0GE7, EE72E7, EE7L27, EECE07, EECEG7, EEEEE5, EEEK55, EEEKA5, EEEL27, EEGLL7, EELE27, EGLLL7, EKK595, EKKA55, EKKAK5, EKKKK5, F000E5, F0AA35, F0F035, F0FFFH, F0HKK5, F0KKE5, F16BB1, F16MM1, F1BBBB, F1MC6B, F666BB, F66BBB, F6GMM1, FB0BBB, FB1BBB, FBBB0B, FBMMG1, FC0FFH, FCFCCH, FEEE55, FEEEA5, FF03F5, FF0FFH, FF3F35, FFEE35, FFF2CH, FFFCCH, FFFFE5, FFI335, FFKFE5, FGLMMB, FK55I5, FKFE55, FLM8CB, FMC66B, FMMC6B, G0AA4J, G0CCC1, G0LE07, G666F1, GG0007, GG00G7, GG0L07, GGLLL7, GGLMM7, GI444J, GJJ33J, GLLE27, GLMMCB, GM0661, GMMM61, H00G07, H05555, H09FF5, H0C0E7, H0CE07, H0CEE7, H0E227, H0H007, H0H5F5, H0H995, H0HHE7, H0HHH7, H55505, H55II5, H5FII5, H99FF5, HEG007, HFFK55, HH0007, HH02M7, HH0C0H, HH7AEJ, HHC0E7, HHE227, HHH0C7, HHH0M7, HHH995, HHHC0H, HHHE27, HHHEAJ, HHHH07, HHHH8H, HHHHE7, HHHHI5, HHHHJH, HHHJ8H, HHHJCH, HHJ00H, HHK095, HHKKM5, HKK0F5, HKK5F5, HKKK55, HKKKK5, HKM555, HMEEE7, I00555, I05555, I0I94J, I333I5, I33555, I444IJ, I55055, I55505, I55555, IAAC8D, ID000D, IDD0LD, II9I4J, III4IJ, III505, IIIC8D, IIJIJJ, IJIIIJ, IM8LLN, IN00AD, INAACD, INCAAD, ININGN, J00CCH, J0IJJJ, J0J09J, J3333D, JIJIIJ, JJ68KH, JJIJIJ, JJJAJJ, JJJHGJ, JJJJAJ, JJJJGJ, JJJJIJ, K0008H, K00161, K001G1, K001L1, K002CH, K002HH, K00521, K00AKD, K00C0H, K00GF1, K00I0D, K00K95, K00M05, K01621, K05021, K0505D, K051G1, K059F5, K05K95, K0C0C1, K0L291, K0M005, K0M505, K1K661, K2CK0H, K33IAD, K3IIID, K5550D, K56121, K59AA5, K612K1, K61CC1, K66661, K6K611, K900C1, K99661, K9AFF5, K9C001, KAKI8D, KC00C1, KDK00D, KF9991, KI0IID, KK000D, KK01L1, KK0661, KK0I8D, KK0L11, KK0M2H, KK5661, KK59F5, KK61C1, KK9995, KK9EE5, KKA3ID, KKA83D, KKAI8D, KKC001, KKC0C1, KKC1C1, KKCCC1, KKD2HH, KKK595, KKK9A5, KKKK95, KKKKKH, KKKMCH, KKM505, KKMEE5, KKMKCH, KM0005, L222E7, L33AAD, L38I8N, LCCC11, LDFBCB, LEL2E7, LELE27, LELL27, LGMMM1, LLE2E7, LM2ME7, M000M5, M006MB, M00E27, M00MM1, M02227, M06M61, M06MM1, M0E227, M0EE27, M0KME5, M0M5GN, M0MM61, M0MMCB, M0MNNN, M0NNM1, M38LLN, M5K505, M77707, M7E227, M7E727, M8CHHH, MBMMCB, MBMMM1, MEEE77, MHH027, MHH505, MHHC6H, MHHH6H, MHHK05, MKK001, MM2ME7, MM7707, MM7E77, MMBMK1, MMM2E7, MMMC0B, MMMK61, N0003D, N0008D, N0030N, N030NN, N0C0AD, N0CKAD, N0DKDD, N0N3GN, N0NN3N, N333AD, N777A7, N77A77, NAACKD, NAAKDD, NACAKD, NACKAD, NC0AKD, NC0KAD, NCA0KD, NCKAKD, NDNNLN, NNNLAD, NNNNLD, 1BBBBBB, 1BBBBG1, 1M6MMMB, 1MMBBBB, 1MMMMK1, 2000227, 2000EE7, 20EEEE7, 2C0FFFH, 2E2EEE7, 2KKKHCH, 2MEE227, 2MEEE27, 333333D, 3333355, 3335555, 333FFF5, 333IIID, 388NNNN, 38INNNN, 3INNNNN, 4000IMN, 4000JJJ, 400IIIN, 444444J, 44JJJJJ, 488888N, 4IIJIIJ, 4JJJ33J, 50002M1, 5001G21, 5006621, 500LGM1, 555083D, 55555I5, 5616G21, 59MMMM1, 5K999A5, 61CCCC1, 66666K1, 6K0000H, 6K0080H, 70000A7, 70077A7, 70700A7, 7070A77, 77700A7, 77770A7, 7777227, 7777E27, 77L2227, 7LE22E7, 888888B, 888888N, 8888BBN, 8888IIN, 888B88B, 888I8IN, 88IINNN, 88NIINN, 88NNIIN, 8INNNNN, 90444IJ, 904I44J, 9666661, 9666FK1, 9666K61, 9966FK1, A00KK0D, A0K000D, AAAAA35, AAKKI8D, BB8888B, BBB0BLB, BBBB1BB, BBBBBB1, BBBBBGB, BBBBBLB, BBBLMBB, C0007KH, C000F11, C00FFFH, C00HH0H, C00K00H, C0C0HHH, C0CCHHH, C0CHH0H, C0CHHCH, C0FFFFH, C0H0H0H, C0KKC0H, CC0HH0H, CCCCC11, CCCCCC1, CCHHHHH, CDKKKKH, CEL7777, CGGG0G7, CGGGGG7, CH00HHH, CHGGGG7, CHHHH0H, CHHHHCH, CHHHHHH, CK0000H, CKDKKKH, D00DDKD, DD0DDD7, DDBBBLB, DDD2EE7, DDDBBLB, DDDDD27, DDDDDBB, DDDDDC7, DDDDDKD, DDDDDMB, DDDDEE7, DDDDKKD, DDDDLDB, DDDFBBB, DDDLFCB, DDDMEE7, DDM2227, DHHEEE7, DK000KD, DK00D0D, DK0D00D, DNN000N, E000CL7, E000EG7, E000GE7, E00C0G7, E00CE07, E00EE27, E0C00G7, E0C0EG7, E0CE007, E0EC0G7, E0EE207, E0G0007, E20EE27, E22EEE7, E2EE227, E2EEE27, E772227, E77LL27, E7L2227, E9IIIIJ, EAKKKA5, EC000G7, ECG00G7, EE00L27, EE0E0G7, EE20EE7, EEE0EG7, EEE22E7, EEEE727, EEEEE27, EEEEG07, EEEEGE7, EEEKKK5, EELLL27, EI0IIIJ, EKKKAA5, ELLLE27, F00FA35, F0333F5, F0F0FE5, F333335, FAAFF35, FCF0FCH, FEEEE35, FF03335, FF0FA35, FF0FE35, FF0FMCH, FFF0A35, FFF0F35, FFFAF35, FFFF5I5, FFFFM2H, FFFI3I5, FFFIII5, FFH5555, FH55555, FL1MMM1, G0000G7, G0000L7, GGGG007, GLE2227, GLLLLE7, GLLLLL7, H000007, H0000C7, H000HCH, H000HM7, H000M27, H000ME7, H00G227, H00HHM7, H02M227, H0C0HHH, H0CH00H, H0E0007, H0FFF35, H0FFFF5, H0H0ME7, H0HFII5, H0HHHCH, H0M0227, H555555, H5F5FF5, HC000G7, HC00H0H, HCCHHCH, HCHHHCH, HCHHHHH, HEEEE27, HFF5FF5, HFF5FI5, HFKKK05, HG00007, HH00E27, HH0G227, HHH2MM7, HHH55F5, HHH9FF5, HHHFFK5, HHHFK55, HHHH7EJ, HHHHCM7, HHHHHAJ, HHHHHF5, HHHHHHJ, HKK5505, I000055, I00A0ID, I0I4IIJ, I0IIIIJ, I88NIIN, III0055, III0I55, III444J, IINNLIN, J000IJJ, K0000DH, K0000KD, K00033D, K000A5D, K000K5D, K00555D, K009995, K00K00D, K00K8ID, K00KI8D, K00KIAD, K01GCC1, K05033D, K0999F5, K2KKKCH, K53333D, K956661, K999991, KCCC1C1, KFFFE55, KFFKKE5, KFKFKE5, KK009A5, KK00C11, KK01GC1, KK99001, KKIII05, KKK09F5, KKKE9E5, KKKEAK5, KKKKI05, KKKKKE5, KMMEEE5, L1BBBG1, LBMMMCB, LBMMMMB, LDEEE07, LEE22E7, LEE2E27, LEEE2E7, LLLLE27, M0000CB, M000C6B, M02EEE7, M0K0005, M0M0005, M2CHHHH, M2HHHHH, M6MMMM1, MC0000B, MCHHHHH, ME7E777, MEE7777, MEEE2E7, MG06661, MHHHCCH, MHHHH27, MHHHHH7, MHM0027, MM6666B, MM77777, MMC000B, MMM7727, MMNM777, N000NLN, N00333D, N003AAD, N0A00DD, N0NN33D, N0NNLLN, N30000N, N777777, NDNNNNN, NN0N0GN, NN0N30N, NNN300N, NNN333D, NNN3LLN, NNNDDDD, NNNNN3N, NNNNNND, 33333F35, 33FFFF35, 3555FFF5, 3FFFFF55, 3NNNNNLN, 40000I0J, 40I0IIIJ, 444440IJ, 4J0000IJ, 500006G1, 5D00DDDD, 5L1MMMM1, 5MMMMMG1, 5NNDDDDD, 5NNNNDDD, 5NNNNN8D, 6000080H, 777777A7, 77777A77, 7944444J, 800000IN, 996666K1, 999999I5, 9999FEA5, A00003ID, AAAAFF35, BBBGMMMB, C0000011, C000007H, C0CC0H0H, C666666B, CCCH0HHH, CCHH0HCH, CE777777, CEEEEE07, CHH0H00H, D00000GN, D000D0LD, D000IIGN, D0DDDDD7, DDD0E2E7, DDDDDDDB, DDDDDME7, DEEEELE7, DEEELEE7, DEELEEE7, DELEE0E7, E00000C7, E00000G7, E0000CG7, E000C0E7, E000G007, E00CG007, E00E0CG7, E0C00007, E0CGGGG7, E0GGGGG7, E20000E7, EAAKAAA5, EAKKAAA5, EE00E727, EE020007, EEEE2027, ELEE2227, F00003F5, F0000A35, F0003335, F0FFFA35, F1999991, F1999MM1, FAAAAF35, FBBBBBG1, FEAAAAA5, FF000A35, FF00FF35, FF0KEEE5, FFAAAF35, FFF555I5, FFFF33I5, FFFFF035, FFFFF3F5, FFFFFKI5, FFFFFMHH, FKFKEEE5, FKKFEEE5, FMMMMMCB, FMMMMMM1, G2000007, GGGGGMM7, GJJJJJ0J, GJJJJJ3J, H0000E27, H0000G27, H000C0G7, H000CEG7, H000CHH7, H000E0E7, H000EE27, H00CHHG7, H00EEE27, H00M0EE7, H05FF5F5, H0E00EE7, H55FF5F5, HCHH0H0H, HE000EE7, HFFIIII5, HGGG2227, HH00CEG7, HH00H0CH, HH0EEEE7, HH0FFFI5, HHEEEEE7, HHH000CH, HHH00EG7, HHHC00G7, HHHFFFF5, HHHHHKK5, HHHK5F55, HK5555F5, I4IIIIIJ, IA0000ID, II0005I5, III000I5, III055I5, III5NNNN, IIIII9IJ, IIIIIII5, IIINNNGN, JAJJJJJJ, JJAJJJJJ, K000005D, K00009A5, K0000M55, K000M555, K008IIID, K00D0K0D, K00III8D, K00K550D, K00LCC11, K0999951, K0D0000H, K0K00595, K0K9AAF5, K0KK0095, K0KK9FF5, K3333IID, KFKFEEE5, KFKKEEE5, KK00000H, KK0000M5, KK099991, KK55583D, KKKEEEA5, KKKKKKI5, LEEEE227, LLLEEE27, LLLL2E07, M000006B, M000M6CB, M0MMMMM1, M222EEE7, M777E777, M7E77777, ME222EE7, ME2EEEE7, MEE222E7, MM0NNNNN, MME77727, MMM6MMM1, N00003GN, N0000ADD, N000N0GN, N000NNND, N033333D, N0NN0NGN, NDDDDKDD, NN000N3N, NN00N03N, NN03000N, NNN003GN, NNNNDNLN, NNNNNADD, 199999MM1, 200FFFFFH, 222MEEEE7, 2FFFFFFCH, 30000000N, 30N00000N, 400000J3J, 500000M01, 5000166G1, 5000666G1, 50DDDDDDD, 8NN33333D, 999999991, C000000FH, C00000K0H, C000H00HH, C77777707, CH00H000H, CHH0000HH, D00000DKD, D00000DLD, D0000200H, D0000KK0D, D000KK00D, D00D0DDLD, D0D00DDLD, D0LEEEEE7, DDBBBBBBB, DDD000KDD, DDDDDDDE7, DK00000DD, DNNNNNNNN, E00000E27, E00007L27, E0000E727, E0E000C07, E20000027, EAAAAKAA5, EAKAAAAK5, EE0000C07, EEE000E27, EEEEEEGL7, F00FFFF35, F0FFFFF35, FF0000035, FF0FFFF35, FF5555FI5, FFFFFFA35, FFFFFFF35, FFFFFFFI5, FKKKKEEE5, FMMMMMMMB, GGGGG2227, GGGGGG207, GJJJJJJJJ, GLMMMMMMB, H000022M7, H000222M7, H000EEEE7, H0EEEEEE7, H0H0000CH, H0IIIIII5, HCH00000H, HE0EEEEE7, HFFFFFI35, HFFFFKKK5, HHHHHHG27, HHHHHHH55, HHHHHHHH7, HHHHHHM55, HHHKK5555, HHIIIII05, HIIIIII05, HKK5555I5, I000000AD, I000000ID, I000A000D, I00A0000D, IIIII0555, IIIIIII9J, K000000AD, K00000595, K000009F5, K0000550D, K099999A5, K0C00000H, K0I00000D, KK0000595, KK0000HCH, KKK000095, KKKFKFFE5, KKKIIIII5, M77777777, MEEEE2227, MMMMMMMM1, N0000000D, N0000003N, N000003NN, N00000N3N, N00000NGN, N00N000GN, N00NNNN8D, N0NNN00GN, N0NNNN3AD, N0NNNNNGN, N999999M1, NN0NNNNGN, NNN000NGN, NNNNNDD8D, NNNNNN0GN, 16MMMMMMMB, 1MMMMMMBCB, 3333333335, 33333333I5, 400000000N, 40IIIIIIJJ, 4IIIIIIIJJ, 4IIIIIIJIJ, 50000000M1, 70F9999991, 777E777727, 9999995MM1, ADD000000D, C00000088B, C000000CF1, C00000F0HH, CH00000H0H, D00KD0000D, D0D0DDDDLD, D0E2EEEEE7, D2EEEEEEE7, DBBBBBBBBB, DD000000KD, DD0000DDLD, DLE0EEEEE7, EEE0000727, EEEAAAAAA5, EEEEEE00G7, EEEEEEE0G7, F000000F35, F00FFKEEE5, F00KFFEEE5, F0M666666B, FCFFFFFFFH, FFFFFFF2HH, FFFKKKEEE5, GGGGGGG227, H00000C06H, H0000HHH6H, H555FFFFF5, H55FFFFF55, H5FFFFFF55, HF5FFFFFF5, HHHH0H0HCH, HHHHH0HHCH, HHHHHH0HCH, HHHHHHHHM5, HHHIIIIII5, IIDNNNNNLN, IIIIIJJIIJ, IIINNNNNLN, IINNNNNNGN, INNNNNNNLN, J0000000IJ, K0000II8ID, K099999995, K0I0000AID, K0K0009FF5, K9999999F5, KK00000095, KKFFFKEEE5, LLLLLLLME7, LLLMEEEEE7, LMEEEEEEE7, M000000005, MHHHHHHHH5, MK00000005, MMMMMMMBCB, NN000000GN, NN0000NNGN, NN99999991, 2HHHHHHHHHH, 38NNNNNNNNN, 3MNNNNNNNNN, 40IIIIIIIIJ, 4AJJJJJJJJJ, 4J000000J0J, 4JJJJJJJJJJ, 506666666G1, 5DDDDDDDDLD, 999999999F5, 99999999EA5, 99999999FE5, A0000000035, C0000000007, C00000000G7, C00000000KH, CEEEEEEEEL7, D0000000FMH, D000DDDDDLD, D0KD000000D, DDDD00000KD, DEEE0EEEEE7, DEL0EEEEEE7, DELEEEEEE07, E0000E20007, E7777777727, EE000000207, EEE20000007, FFFFFFFFMCH, FM66666666B, GGGGGGGG2M7, HFFFFFFFF55, HHHHHHHHH6H, HHHHHHHHHCH, HHHHHK55555, I9IIIIIIIIJ, IIIIIIII44J, IIIIIIIJJIJ, IINNNNNNNNN, JDDDDDDDDDD, JJIIIIIIIIJ, K00000I8IID, LLLLLLLLL27, 9999999EEAA5, AI000000000D, C77700000007, CH0HH000000H, D00D000000LD, DEEEEEEE0EE7, DN000000000N, EAKAAAAAAAA5, EKAAAAAAAAK5, F6666666666B, H000HHHHHH6H, H55FFFFFFFF5, HFFFFFFFFKK5, K00000000I8D, K999999999A5, 3555555555FF5, 5000000000001, 6G66666666661, 99999999999A5, C00000000000H, CFFFFFFFFFFCH, CHHH00000000H, D00000000K0KD, D0000000K00KD, D0D00000000LD, DEEEEEEEEEL07, E000E20000007, E00E200000007, EEE0000000027, GGGGGGGGGGGM7, GGGGGGGGGGM07, H00000000CHHH, H00HC0000000H, HFFFFFFFFFFK5, I0A000000000D, J000000000J9J, K000000000095, K000000000M2H, M0000000000M1, M0EEEEEEEEEE7, MHHHHHHHHHHHH, MMNNNNNNNNNNN, MNNNNNNNNNNNN, N000000DDDDDD, N000DDDDDDDDD, NNDDDDDDDDDDD, 22EEEEEEEEEEE7, 35FFFFFFFFFFF5, 400000000000JJ, 800000000000GN, DDDDDDDDDDD077, DDDDDDDDDDDDD7, E0000000000L27, EAAAAAAAAAAKA5, EEG00000000007, H0000000000C6H, I5500000000005, II0000000000I5, M0666666666661, M6MMMMMMMMMMMB, 4JJ00000000000J, 506666666666661, BGMMMMMMMMMMMCB, CFFFFFFFFFFFFFH, D0000000000KD0D, D0HEEEEEEEEEEE7, F0BBBBBBBBBBBBB, HGGGGGGGGGGGGG7, K0000000000000D, K00000000000MCH, M0M6MMMMMMMMMMB, 5DDDDDDDDDDDDDDD, C00000000000008B, D00000000000000H, DEEEEEEEEEEEE0L7, DEEEEEEEEEEEEEL7, EEE2EEEEEEEEEEE7, GM66666666666661, H5FFFFFFFFFFFFF5, IIIIIIIIIIIIIJJJ, BGMMMMMMMMMMMMMMB, DLEEEEEEEEEEEEEE7, H0000000000000CHH, H000000000C0000HH, I000000000000000D, IIIIIIIIIIIIIIIJJ, INNNNNNNNNNNNNNNN, J000000000000009J, M666666666666666B, N0000000000000LLN, N00DDDDDDDDDDDDDD, 355555555555555555, 60000000000000008H, 6M6666666666666661, C000000000000000F1, N0DDDDDDDDDDDDDDDD, 666666666666666666B, 800000000000000000N, AD000000000000000DD, DEEEEEEEEEEEEEEEEE7, I500000000000000005, 20000000000000000027, 4000000000000000003J, 400000000000000000IJ, 99999999999999999995, DD00DDDDDDDDDDDDDDLD, E2EEEEEEEEEEEEEEEEE7, N00000000000000000LN, 500000000066666666661, EE0000000000000000727, GGGGGGGGGGGGGGGGGGG07, H0000000000000000006H, 40000000000000IIIIIIIJ, AD0000000000000000000D, K0000000000000000000M5, CL777777777777777777777, D000000000000000000000N, D0000000000000000000IIN, HHHHHHHHHHHHHHHHHHHHHK5, NDDDDDDDDDDDDDDDDDDDDDD, 1MMMMMMMMMMMMMMMMMMMMMBB, D00DDDDDDDDDDDDDDDDDDDLD, FFFFFFFFFFFFFFFFFFFFFFFH, 4J0000000000000000000000J, 566666666666666666666666G1, EKKAAAAAAAAAAAAAAAAAAAAAA5, 6666666666666666666666666G1, AJJJJJJJJJJJJJJJJJJJJJJJJJJJ, H00000000000000000000000008H, N0000000000000000000000000GN, DD0000000000000000000000000LD, IIIIIIIIIIIIIIIIIIIIIIIIIIIIJ, G0666666666666666666666666666661, GGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGGG7, K000000000000000000000000000000000H, EAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA5, LLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLM7, M2EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE7, C000000000000000000000000000000000000000001, MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMCB, E00000000000000000000000000000000000000000000727, 777777777777777777777777777777777777777777777777727, EG000000000000000000000000000000000000000000000000000007, D000000000000000000000000000000000000000000000000000000000LD, 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===Base 36=== 11, 15, 17, 1B, 1H, 1N, 1P, 1V, 1Z, 21, 27, 2B, 2H, 2P, 2T, 2V, 2Z, 31, 35, 3J, 3N, 3T, 3V, 45, 47, 4D, 4J, 4N, 4T, 4Z, 51, 5B, 5D, 5H, 5J, 5V, 67, 6B, 6D, 6H, 6N, 6P, 6Z, 75, 7B, 7H, 7J, 7P, 7T, 7V, 85, 8J, 8N, 8P, 8T, 97, 9D, 9N, 9P, 9T, 9Z, A7, AD, AJ, AN, AT, B1, B5, BD, BN, BP, BZ, C1, C7, CB, CH, CP, CT, CV, CZ, DB, DJ, DN, DV, DZ, E5, EH, EJ, F1, F7, FH, FN, FT, FV, G1, GB, GH, GN, GP, GV, H1, H5, H7, HJ, HT, HV, HZ, I5, IB, ID, IP, IT, IZ, J7, JH, JP, JZ, K7, KD, KJ, KN, KV, L1, L5, LD, LH, LV, M5, MH, MJ, MT, MV, MZ, N1, NB, NP, NT, NV, NZ, OD, OH, OJ, ON, P7, PB, PJ, PT, Q1, Q5, QB, QH, QV, QZ, R5, RB, RJ, RP, S1, S5, SB, SD, SN, SP, SV, T5, T7, TH, TJ, TP, U7, UB, UD, UH, UN, UT, V1, V7, VD, VZ, W1, WB, WJ, WT, WZ, X5, XD, XP, XT, XZ, Y5, Y7, YD, YP, YZ, ZH, ZJ, ZN, ZT, ZV, 12D, 16J, 18D, 1CD, 1CJ, 1DD, 1GT, 1JD, 1JJ, 1LT, 1QJ, 1RD, 1RT, 1ST, 1XJ, 1YJ, 1YT, 22D, 22J, 22N, 23D, 255, 2A5, 2CD, 2ED, 2EN, 2F5, 2GD, 2GJ, 2IJ, 2JN, 2LN, 2MN, 2ND, 2O5, 2QD, 2QJ, 2U5, 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MYMU1, MYRK1, MYUU1, MYXR1, N000D, N0055, N006J, N007N, N00CN, N00IH, N00O7, N00S7, N029J, N02DD, N0307, N044H, N049H, N0505, N05M7, N05Q7, N06LJ, N070N, N0A55, N0CEN, N0CNN, N0CYN, N0DMD, N0DRD, N0F05, N0FF5, N0FMD, N0GED, N0HNN, N0I4H, N0IS7, N0IXJ, N0JLN, N0JNN, N0K4H, N0KIH, N0L6J, N0LI7, N0LLN, N0M0N, N0MQ7, N0NEN, N0O07, N0O77, N0Q0N, N0QEN, N0RRH, N0S07, N0SS7, N0U0J, N0ULJ, N0W77, N0WC5, N0WL7, N0XO7, N26JJ, N29XJ, N2CNN, N2CYN, N2MMD, N2N6J, N2N9J, N2NLJ, N2YYN, N30ED, N32RD, N33GD, N33MD, N3GGD, N3IW7, N3MR7, N3MX7, N3RGD, N3RI7, N3W2D, N3W77, N3X07, N3X77, N400H, N409H, N44YH, N494H, N49YH, N4AYH, N50K5, N5307, N5537, N5577, N57Q7, N5FK5, N5KK5, N5Q77, N5SQ7, N69LJ, N6L9J, N6U0J, N700D, N7077, N70EN, N70FD, N73RD, N7777, N777N, N7DMD, N7E2N, N7FED, N7M3D, N7MWD, N7SR7, N7WMD, N92LJ, N9XXJ, NA505, NA5F5, NAFK5, NAK05, NAKF5, NARYH, NAWW5, NC00N, NC0EN, NC0NN, NCE0N, NCN0N, NCNQN, NCW05, ND3GD, ND7DD, NDAW5, NDDGD, NDDRD, NDE3D, NDWM7, NDWW7, NE0GD, NED3D, NEDDD, NEL2N, NEMMD, NEMRD, NF0F5, NFEMD, NFKF5, NFWC5, NFWW5, NGDED, NGW2D, NH00H, NH2NN, NHH0H, NHMCN, NHMMD, NHNNN, NI00H, NI0EN, NI0MN, NI0NH, NI0NJ, NI0XJ, NI44H, NI7R7, NI7S7, NII77, NIIM7, NIIMN, NIIYN, NIM0N, NIN0H, NIN0J, NINJJ, NIW37, NIWM7, NIX77, NIXW7, NJ0EN, NJ0NN, NJ30D, NJ33D, NJFMD, NJIEN, NJJ2N, NJJIN, NJJJJ, NJJJN, NJL0J, NJLNN, NJN0N, NJSJJ, NK005, NK04H, NK3KH, NK5F5, NKA05, NKFK5, NKK55, NKKF5, NL09J, NL2IN, NLCNN, NLII7, NLIIN, NLJLN, NLL2J, NLLNJ, NLLNN, NLNCN, NLNIN, NLNLJ, NLNLN, NM00N, NM73D, NM7DD, NM7RD, NME0N, NMELN, NMLL7, NMLQ7, NMMRD, NMQEN, NMRD7, NMRR7, NMWWD, NN02J, NN0AH, NN0EN, NN30H, NN525, NN6UJ, NN92J, NNA55, NNASH, NNCLN, NNCMN, NNELN, NNF25, NNHHN, NNJ2J, NNJJJ, NNLNN, NNN55, NNN9J, NNNMN, NNNNJ, NNNSH, NNNXJ, NNQYN, NNS3H, NNU9J, NNW55, NNXJJ, NO077, NO0W7, NO777, NOI77, NOIM7, NOWL7, NQ00N, NQ0YN, NQ77N, NQ7EN, NQLLN, NQLNN, NQQQN, NQQYN, NRARH, NRAYH, NRDI7, NRHYH, NRQL7, NRRHH, NRW2D, NS0L7, NSA3H, NSHHH, NSIIJ, NSIJJ, NSS07, NSSJJ, NSSL7, NSSS7, NSULJ, NU00J, NUL6J, NUSIJ, NW0C5, NW3W7, NW5W5, NWI77, NWIM7, NWIO7, NWW55, NWWM7, NX0I7, NX0O7, NX3R7, NX9XJ, NXIM7, NXRL7, NXSSJ, NXWI7, NXWW7, NXXSJ, NY44H, NY94H, NYYIN, O0057, O005T, O0077, O0091, O009B, O00BT, O00EV, O00OV, O00TT, O00WP, O00WV, O040V, O0507, O056T, O065T, O08AV, O08E7, O0901, O09AB, O09I1, O09K1, O09WV, O0A4V, O0AI1, O0E9V, O0EAB, O0EAV, O0EFP, O0EKB, O0EOB, O0EU1, O0FMP, O0K6T, O0K91, O0MM1, O0O9B, O0OAB, O0OI1, O0OOB, O0OT1, O0OUV, O0OW7, O0OYT, O0T8V, O0TOT, O0TOV, O0TTV, O0TUV, O0TX1, O0U91, O0UE1, O0W87, O0WOV, O0YOT, O100T, O1EET, O4PAV, O4U0V, O5005, O50EP, O50I7, O50K5, O555P, O55FP, O55UP, O5EFP, O5I07, O5K05, O5KK5, O5OYT, O6LGT, O77E7, O7Q77, O80E7, O8AAZ, O8AFZ, O8AOV, O8FFZ, O8FKB, O8FZB, O8IM7, O8ZRZ, O8ZZB, O900V, O90WV, O96OV, O99LB, O9K01, O9OU1, O9R6V, O9U01, O9UI1, OA091, OA0O1, OA4AV, OA4IV, OA68V, OA8AZ, OA9I1, OAA4V, OAAFZ, OAAO1, OAAOV, OAAZZ, OAFAZ, OAFYB, OAI0V, OAI91, OAOAV, OAOE1, OAOU1, OAUE1, OAUI1, OAY01, OAY0B, OAYI1, OAZZZ, OB66T, OBE0T, OBEET, OE0BT, OE0FB, OE4AV, OE6ET, OE8AV, OE9EB, OEA8V, OEAA1, OEAFB, OEBET, OEBI7, OEEO7, OEFFP, OEFLP, OEKA1, OEKAB, OEKU1, OELMP, OEML7, OEO07, OEO0B, OEO4V, OEO6V, OEOAV, OEOO7, OEOOB, OEOVB, OEPU1, OERET, OERRV, OETY1, OEUK1, OEUU1, OEV9B, OEVFP, OEW8V, OEWW7, OF0FP, OF0OB, OFFFP, OFFMP, OFFZP, OFFZZ, OFMMP, OFMWP, OFOQP, OFPFZ, OFPUZ, OFZ0P, OFZFZ, OFZOB, OFZZZ, OG065, OG0YT, OI0EV, OI7Q7, OI9WV, OIIE7, OIQ77, OIQW7, OIR6V, OIYO1, OK8KB, OKAE1, OKAY1, OKE6T, OKEU1, OKFYB, OKIY1, OKYA1, OKYI1, OL08B, OL4EP, OL777, OL8E7, OLBI7, OLEL7, OLFOP, OLGYT, OLIW7, OLLSZ, OLLZZ, OLOUP, OLR6T, OLUMP, OMAU1, OMOU1, OMXM1, OO0AB, OO0I7, OO0T1, OO1ET, OO4AV, OO4MP, OO4QP, OO6KT, OO6LT, OO6OV, OO7O7, OOA0B, OOAIV, OOAOV, OOBGT, OOEMP, OOERT, OOET1, OOEU1, OOEY1, OOGTT, OOI4V, OOIRV, OOKET, OOLBT, OOMMP, OOMQ7, OOMU1, OOMWP, OOO0B, OOO6T, OOO77, OOOE7, OOOKT, OOOOV, OOORT, OOOW7, OOPY1, OOQ77, OOT01, OOT91, OOTAV, OOTOV, OOTUV, OOU91, OOUUV, OOWIV, OOWMP, OOWVP, OOY01, OOY0B, OOYKB, OOYU1, OOYYB, OP0EV, OP44V, OP8FZ, OPA91, OPAAZ, OPAOV, OPEAV, OPI4V, OPO4V, OPOIV, OPOUV, OPOY1, OPUAZ, OPUEV, OPWUV, OPYU1, OQ7W7, OQ8E7, OQLE7, OQQL7, OR8AZ, ORA8Z, ORKLT, ORS8Z, OS8RZ, OSR8Z, OSS8Z, OT09V, OT0K1, OT66T, OTT8V, OTUOV, OU091, OU0E1, OU0MP, OU4LP, OUAAZ, OUASZ, OUL4P, OUP8Z, OUP91, OUPAZ, OUPY1, OUU0P, OUUPZ, OUYM1, OUZ91, OV0FB, OV0LB, OV54P, OV5WP, OV9K5, OVALB, OVB6T, OVKFB, OVLEB, OVLFP, OVQWP, OW0EV, OW7O7, OW7Q7, OW7W7, OW90V, OWEEP, OWIWV, OWLW7, OWOM7, OWOVP, OWQ87, OWU0V, OWW9V, OWWL7, OWWOV, OXAA1, OXU01, OXUI1, OXXI1, OXXX1, OY00T, OY0E1, OY0RT, OY0U1, OY0YB, OY98B, OYE0T, OYFKB, OYKFB, OYLYB, OYYYB, OZ0OB, OZ0WP, OZ0YB, OZ9EB, OZ9K1, OZ9O1, OZALB, OZAYB, OZEAB, OZEOB, OZFEB, OZFFP, OZFFZ, OZFRZ, OZFZP, OZO0B, OZO91, OZOYB, OZPRZ, OZR8Z, OZUUP, OZZYB, P00W5, P02G5, P049V, P04YV, P0A0V, P0AW5, P0CC5, P0CU5, P0E9V, P0EWV, P0F25, P0FC5, P0G25, P0K3Z, P0KOZ, P0KRH, P0O55, P0OEV, P0PG5, P0PO5, P0PW5, P0R0H, P0ROZ, P0U25, P0UIV, P0UVP, P0V0P, P0VKP, P0W9V, P0YIV, P13WD, P22K5, P2D95, P2NC5, P33ZZ, P38DD, P3GGD, P3SAH, P409V, P444V, P44O1, P44Y1, P4AIV, P4AYV, P4E0V, P4OA1, P4OK1, P4OO1, P4U81, P4UEV, P4UR1, P55EP, P5GZ5, P6A41, P803H, P80AZ, P838D, P8AAZ, P8D8D, P8FKZ, P8K0Z, P8KKZ, P8KXH, P8OE1, P8UE1, P8X0H, P8X3H, P8XKH, P909V, P9941, P99EV, P99U1, P9E9V, P9K91, P9OE1, P9OK1, P9W9V, P9X9H, PA00H, PA441, PA5G5, PA8FZ, PA9O1, PAA8Z, PAI4V, PAI9V, PAMKP, PAOEV, PAR8H, PAUM1, PAVKP, PAZEP, PC0W5, PCAG5, PCAU5, PCCA5, PCWN5, PD0QP, PD0SH, PD33D, PD3AH, PD3MD, PDGC5, PDK3H, PDK8H, PDSAH, PDUAP, PE99V, PEA91, PEAEP, PEEA1, PEEE1, PEER1, PEI9V, PEK61, PEQQP, PERE1, PERK1, PEUY1, PEY0V, PF2K5, PF3CD, PF83D, PFC2D, PFCWD, PFDAP, PFEKP, PFMCD, PFMMD, PFO55, PFR2D, PFZEP, PG295, PG2WD, PG8CD, PGCD5, PGDC5, PGF5Z, PGG8D, PGKF5, PGKK5, PGO95, PGW8D, PH2MD, PHD8D, PK0O5, PK441, PK53Z, PK5GZ, PK83H, PK8KZ, PKAA1, PKAU1, PKKO5, PKP3Z, PKRXH, PKSIH, PMAA1, PMAAP, PMAU1, PMF3D, PMFCD, PMMAP, PMMMD, PMMMP, PMMPD, PMRR1, PMUO1, PN2D5, PN2W5, PNA3H, PNAF5, PNFK5, PNHFD, PNK55, PNKF5, PNSKH, PO09V, PO0EV, PO4AV, PO90V, POAY1, POI4V, POOE1, POUEV, POUY1, POWUV, POWWV, POYA1, POZAZ, POZFZ, PP0QP, PP2CN, PP2MD, PP33D, PP595, PP5W5, PP8CD, PPCNN, PPCRD, PPCU5, PPCW5, PPCWD, PPD05, PPDUP, PPECN, PPF8D, PPFC5, PPFW5, PPGC5, PPGF5, PPGFZ, PPKGZ, PPMCD, PPMCN, PPMEN, PPMND, PPNF5, PPNMN, PPNWD, PPP3Z, PPP95, PPPC5, PPPRZ, PPPZD, PPQKP, PPS0Z, PPW2N, PPWND, PPWWN, PPZC5, PPZSZ, PQQQP, PR1GD, PR8AH, PRA0H, PREK1, PRKO1, PRMM1, PRNFD, PRRM1, PRRRH, PRW2N, PS0ZZ, PS80H, PS80Z, PS8AH, PS8ZZ, PSIKH, PSR3Z, PU025, PU0IV, PU49V, PU4YV, PU8AZ, PU9EV, PUE91, PUE9V, PUI9V, PURK1, PUUM1, PUUR1, PUVAP, PUYIV, PUYO1, PUZMP, PV00P, PV2CN, PV2RN, PVAKP, PVFAP, PVFFP, PVFKP, PVK0P, PVKAP, PVNCN, PVNEN, PW005, PW0C5, PWCWD, PWF8D, PWNC5, PWO9V, PWOWV, PWWOV, PWWWN, PWWWV, PX0IH, PXAIH, PXIAH, PXIKH, PYKE1, PYKO1, PYRK1, PYRM1, PZ0G5, PZ1WD, PZ295, PZ2C5, PZ2G5, PZ481, PZ641, PZE61, PZEAP, PZG25, PZK61, PZKOZ, PZOFZ, PZOO1, PZR41, PZRZZ, PZS0Z, PZSUZ, PZUMP, Q00DT, Q00JD, Q00QT, Q00YN, Q020J, Q03RD, Q0CQN, Q0DRD, Q0FRD, Q0JTT, Q0Q3D, Q0QRN, Q0QXN, Q0XEN, Q0XIJ, Q0XLN, Q0XQN, Q0YYN, Q20RD, Q2D0D, Q32WD, Q3D3D, Q4EKP, Q4PQP, Q7777, Q77Q7, Q7ER7, Q7EW7, Q7LR7, Q7LS7, Q7LW7, Q7QQN, Q7QR7, Q7QYN, Q7RE7, Q7SS7, QAAPP, QAOOP, QAOWP, QAUAP, QAUWP, QAWEP, QAWLP, QCEQN, QCNYN, QCQEN, QD0GD, QD0RD, QD20D, QD33D, QD3DD, QDAWP, QDRET, QDWKP, QE7QN, QECLN, QEEL7, QEES7, QELCN, QEQCN, QEQTT, QERE7, QESR7, QF2WD, QFGUJ, QFR2D, QG00J, QG0IJ, QG7Q7, QG7R7, QGDDD, QGGGD, QGGQD, QGJ2J, QGMQ7, QGOO7, QGQQ7, QGRR7, QJ02J, QJ03D, QJ0SJ, QJ33D, QJ3WD, QJD0D, QJI0J, QK44P, QKPKP, QL7W7, QLES7, QLLL7, QLNNN, QLQ77, QLQL7, QLQM7, QLQPN, QLR87, QLSRT, QMLR7, QMQQ7, QN0YN, QN70N, QNCYN, QNLW7, QNNQN, QNNYN, QNQNN, QO777, QO7E7, QO8M7, QOUWP, QP3RD, QPDAP, QPEEP, QPFRD, QPGGD, QPPQP, QPR2D, QPRGD, QQ0RN, QQ0YN, QQ4LP, QQ4QP, QQAEP, QQCYN, QQELT, QQEPN, QQEQP, QQGDD, QQGGD, QQGO7, QQGQD, QQKKP, QQLAP, QQLPN, QQLRT, QQMR7, QQN77, QQNNN, QQNQN, QQPDD, QQPEN, QQPGD, QQPQP, QQPWD, QQQ0N, QQQ3D, QQQ87, QQQKP, QQQL7, QQQM7, QQQPD, QQQQN, QQQWD, QQRL7, QQRN7, QQRYN, QQSST, QQTDT, QQTQT, QQU4P, QQUAP, QQUUP, QQWPD, QQXEN, QQXQN, QQYXN, QR00N, QR0XN, QR2FD, QR32D, QREKT, QRN0N, QRNL7, QRQQD, QRRE7, QRXXN, QSES7, QSS2J, QSSE7, QSSS7, QSSSJ, QTDET, QTQ2D, QTQLT, QUAAP, QUAWP, QUPPP, QUUAP, QW7G7, QWG77, QWGM7, QWLEP, QWOEP, QWOLP, QX77N, QX7EN, QXELN, QXEQN, QXLLN, QXLQN, QXNQN, QXOE7, QXQXN, QXXNN, R0061, R008D, R00IH, R00NN, R00XN, R013D, R02YN, R030D, R04R1, R06E1, R06ET, R07M1, R08AH, R0A8H, R0AYH, R0D07, R0DDT, R0EK1, R0EYT, R0F3D, R0G8D, R0GDD, R0GGD, R0GQ7, R0H0N, R0KE1, R0LQ7, R0N0N, R0RYN, R0T2N, R0TLT, R0TNN, R0TYN, R0X0N, R0YST, R100T, R130D, R13WD, R166T, R1E0T, R1F3D, R1GFD, R1TGD, R200N, R28WD, R2W8D, R2WFD, R32DD, R330D, R332D, R3W8D, R44XV, R4AAV, R4AYV, R4MO1, R4OI1, R4RO1, R4U4V, R4X9V, R6001, R60K1, R66LT, R66YV, R6E01, R6EE1, R6MK1, R6SST, R6X01, R6X9V, R6XE1, R6XYV, R6Y6V, R7OO1, R7XE1, R80G7, R88FD, R8F8D, R8IE7, R8X8H, R9OAV, R9UVV, R9VOV, R9X6V, RA08H, RAA9V, RAS8Z, RAX0H, RD00D, RD8I7, RD8IH, RDA0H, RDDQT, RDQDT, RE061, RE6KT, RE6LT, REEK1, REEKT, RER01, RERK1, RERR1, RF0DD, RFD0D, RG28D, RGGFD, RGGWD, RGRM7, RGWQD, RH00H, RH0EN, RH8HH, RH9HH, RH9XH, RHH9H, RHHIN, RHHNN, RHIWN, RHNEN, RHWIN, RHWWN, RHX8H, RHXEN, RHY9H, RI00H, RI2IN, RIHEN, RIIWN, RIMRN, RK00T, RKEE1, RKELT, RKL0T, RKQLT, RLEI7, RLQM7, RLR07, RLRE7, RLSSZ, RLSYT, RLZE7, RLZI7, RLZLZ, RM0G7, RM0Q7, RMIWN, RMKO1, RMLQ7, RMOX1, RMR07, RMRR1, RN2YN, RN30D, RND3D, RNI2N, RNIMN, RNMEN, RNW2D, RO0TT, RO66T, RO6KT, ROAAV, ROOKT, ROOOT, ROU9V, ROXM1, ROZK1, RQK0T, RQLKT, RQLST, RQMR7, RQQLT, RQQQT, RQQWD, RQRE7, RQTQD, RR44V, RR4YV, RR601, RR66V, RR6WV, RR9HH, RR9UV, RRA9V, RRAYV, RRE01, RREE1, RRGM7, RRGR7, RRI2N, RRIIN, RRIN7, RRIXN, RRKE1, RRL07, RRLM7, RRM07, RRMG7, RRML7, RRMM1, RRMO1, RRMWN, RRNR7, RRO01, RROI1, RROK1, RRR0N, RRRE1, RRRG7, RRRIN, RRRL7, RRRM1, RRRO1, RRROV, RRRWV, RRU6V, RRV9H, RRVUV, RRVYH, RRWWV, RRXRN, RRY4V, RRY9H, RRYHH, RRYIN, RRYRN, RS8FZ, RSSSZ, RSSYT, RSYST, RT3ZZ, 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====Additional known quasi-minimal primes (not necessarily the next)==== P₈₁₉₉₃SZ ==Unsolved families== Families for which not even a probable prime is known nor can be ruled out as only contain composites (only count the numbers > base (''b'')) by covering congruence, algebraic factorization, or combine of them. {|class=wikitable |base (''b'')||unsolved family (base-''b'' form)||unsolved family (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||current search limit of length||factorization of numbers in this family |- |13||9{5}||(113×13^''n''−5)/12||116000||[http://factordb.com/index.php?query=%28113*13%5En-5%29%2F12&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |13||A{3}A||(41×13^''n''+1+27)/4||111000||[http://factordb.com/index.php?query=%2841*13%5E%28n%2B1%29%2B27%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |16||{3}AF||(16^''n''+2+619)/5||98000||[http://factordb.com/index.php?query=%2816%5E%28n%2B2%29%2B619%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1{7}||(23×17^''n''−7)/16||32074||[http://factordb.com/index.php?query=%2823*17%5En-7%29%2F16&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||1F{0}7||32×17^''n''+1+7||32074||[http://factordb.com/index.php?query=32*17%5E%28n%2B1%29%2B7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||4{7}A||(71×17^''n''+1+41)/16||32074||[http://factordb.com/index.php?query=%2871*17%5E%28n%2B1%29%2B41%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||51{0}D||86×17^''n''+1+13||32074||[http://factordb.com/index.php?query=86*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||70F{0}D||2038×17^''n''+1+13||32074||[http://factordb.com/index.php?query=2038*17%5E%28n%2B1%29%2B13&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||8{B}9||(139×17^''n''+1−43)/16||32074||[http://factordb.com/index.php?query=%28139*17%5E%28n%2B1%29-43%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||9{5}9||(149×17^''n''+1+59)/16||32074||[http://factordb.com/index.php?query=%28149*17%5E%28n%2B1%29%2B59%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||95{F}||(2543×17^''n''−15)/16||32074||[http://factordb.com/index.php?query=%282543*17%5En-15%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||A{D}F||(173×17^''n''+1+19)/16||32074||[http://factordb.com/index.php?query=%28173*17%5E%28n%2B1%29%2B19%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}B3||11×17^''n''+2+190||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B190&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||B{0}DB||11×17^''n''+2+232||32074||[http://factordb.com/index.php?query=11*17%5E%28n%2B2%29%2B232&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2BE||(11×17^''n''+3−41579)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B3%29-41579%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}2E||(11×17^''n''+2−2411)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29-2411%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}E9||(11×17^''n''+2+773)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B773%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |17||{B}EE||(11×17^''n''+2+853)/16||32074||[http://factordb.com/index.php?query=%2811*17%5E%28n%2B2%29%2B853%29%2F16&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- 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|21||B9{0}E5||240×21^''n''+2+299||20000||[http://factordb.com/index.php?query=240*21%5E%28n%2B2%29%2B299&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B{D}B||(233×21^''n''+1−53)/20||20000||[http://factordb.com/index.php?query=%28233*21%5E%28n%2B1%29-53%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||B{H}6H||(237×21^''n''+2−4637)/20||20000||[http://factordb.com/index.php?query=%28237*21%5E%28n%2B2%29-4637%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||DH{D}||(5813×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%285813*21%5En-13%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||F{9}D||(309×21^''n''+1+71)/20||20000||[http://factordb.com/index.php?query=%28309*21%5E%28n%2B1%29%2B71%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{F}35||(3×21^''n''+2−1051)/4||20000||[http://factordb.com/index.php?query=%283*21%5E%28n%2B2%29-1051%29%2F4&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||G{0}FK||16×21^''n''+2+335||506722||[http://factordb.com/index.php?query=16*21%5E%28n%2B2%29%2B335&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{0}7771||17×21^''n''+4+68062||20000||[http://factordb.com/index.php?query=17*21%5E%28n%2B4%29%2B68062&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||H{D}||(353×21^''n''−13)/20||20000||[http://factordb.com/index.php?query=%28353*21%5En-13%29%2F20&use=n&n=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |21||{J}6J||(19×21^''n''+2−5479)/20||20000||[http://factordb.com/index.php?query=%2819*21%5E%28n%2B2%29-5479%29%2F20&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||7{K}Z||(53×36^''n''+1+101)/7||20000||[http://factordb.com/index.php?query=%2853*36%5E%28n%2B1%29%2B101%29%2F7&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||B{0}EUV||11×36^''n''+3+19255||20000||[http://factordb.com/index.php?query=11*36%5E%28n%2B3%29%2B19255&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||HM{0}N||634×36^''n''+1+23||20000||[http://factordb.com/index.php?query=634*36%5E%28n%2B1%29%2B23&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||N{0}YYN||23×36^''n''+3+45311||20000||[http://factordb.com/index.php?query=23*36%5E%28n%2B3%29%2B45311&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||O{L}Z||(123×36^''n''+1+67)/5||100000||[http://factordb.com/index.php?query=%28123*36%5E%28n%2B1%29%2B67%29%2F5&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |- |36||S{0}8H||28×36^''n''+2+305||20000||[http://factordb.com/index.php?query=28*36%5E%28n%2B2%29%2B305&use=n&n=0&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1&sent=Show] |} For base 17, the smallest prime in family {B}2BE may or may not be quasi-minimal prime, since another unsolved family is {B}2E For base 19, the smallest prime in family {2}7A may or may not be quasi-minimal prime, since another unsolved family is {2}7, and the smallest prime in family 333{5} may or may not be quasi-minimal prime, since another unsolved family is 3{5}, and the smallest prime in family 5{H}05 may or may not be quasi-minimal prime, since another unsolved family is 5{H}5, and the smallest prime in family FHHH0{H} may or may not be quasi-minimal prime, since another unsolved family is FH0{H} For base 21, the smallest prime in families {9}0D and F{9}D may or may not be quasi-minimal primes, since another unsolved family is {9}D, and the smallest prime in family DH{D} may or may not be quasi-minimal prime, since another unsolved family is H{D} ==Primality certificates for the proven primes > 10²⁹⁹== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 See also: [[w:Primality certificate|Primality certificate]] and [[w:Elliptic curve primality|Elliptic curve primality]] {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b''||quasi-minimal prime (base-''b'' form)||quasi-minimal prime (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this prime||primality certificate of this prime |- |9||149||76₃₂₉2||(31×9³³⁰−19)/4||[http://factordb.com/index.php?id=1100000002359003642]||[http://factordb.com/cert.php?id=1100000002359003642] |- |9||150||27₆₈₆07||(23×9⁶⁸⁸−511)/8||[http://factordb.com/index.php?id=1100000002495467486]||[http://factordb.com/cert.php?id=1100000002495467486] |- |9||151||30₁₁₅₈11||3×9¹¹⁶⁰+10||[http://factordb.com/index.php?id=1100000002376318423]||[http://factordb.com/cert.php?id=1100000002376318423] |- |11||1065||A₇₁₃58||11⁷¹⁵−58||[http://factordb.com/index.php?id=1100000003576826487]||[http://factordb.com/cert.php?id=1100000003576826487] |- |11||1066||7₇₅₉44||(7×11⁷⁶¹−367)/10||[http://factordb.com/index.php?id=1100000002505568840]||[http://factordb.com/cert.php?id=1100000002505568840] |- |11||1067||557₁₀₁₁||(607×11¹⁰¹¹−7)/10||[http://factordb.com/index.php?id=1100000002361376522]||[http://factordb.com/cert.php?id=1100000002361376522] |- |13||3165||50₂₇₀44||5×13²⁷²+56||[http://factordb.com/index.php?id=1100000002632397005]||[http://factordb.com/cert.php?id=1100000002632397005] |- |13||3166||9₂₇₁095||(3×13²⁷⁴−6103)/4||[http://factordb.com/index.php?id=1100000003590431654]||[http://factordb.com/cert.php?id=1100000003590431654] |- |13||3167||10₂₈₆7771||13²⁹⁰+16654||[http://factordb.com/index.php?id=1100000003590431633]||[http://factordb.com/cert.php?id=1100000003590431633] |- |13||3168||9₃₀₈1||(3×13³⁰⁹−35)/4||[http://factordb.com/index.php?id=1100000000840126705]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=308&c0=-&EN= 13³⁰⁸−1] |- |13||3169||B₃₄₁C4||(11×13³⁴³+61)/12||[http://factordb.com/index.php?id=1100000003590431618]||[http://factordb.com/cert.php?id=1100000003590431618] |- |13||3170||8B₃₄₃||(107×13³⁴³−11)/12||[http://factordb.com/index.php?id=1100000002321018736]||[http://factordb.com/cert.php?id=1100000002321018736] |- |13||3171||710₃₇₁111||92×13³⁷⁴+183||[http://factordb.com/index.php?id=1100000003590431609]||[http://factordb.com/cert.php?id=1100000003590431609] |- |13||3172||75₃₇₅7||(89×13³⁷⁶+19)/12||[http://factordb.com/index.php?id=1100000003590431596]||[http://factordb.com/cert.php?id=1100000003590431596] |- |13||3173||9B0₃₉₁9||128×13³⁹²+9||[http://factordb.com/index.php?id=1100000002632396790]||[http://factordb.com/cert.php?id=1100000002632396790] |- |13||3174||7B0B₃₉₇||(15923×13³⁹⁷−11)/12||[http://factordb.com/index.php?id=1100000003590431574]||[http://factordb.com/cert.php?id=1100000003590431574] |- |13||3175||10₄₁₄93||13⁴¹⁶+120||[http://factordb.com/index.php?id=1100000002523249240]||[http://factordb.com/cert.php?id=1100000002523249240] |- |13||3176||81010₄₁₅1||17746×13⁴¹⁶+1||[http://factordb.com/index.php?id=1100000003590431555]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3177||8110₄₃₅1||1366×13⁴³⁶+1||[http://factordb.com/index.php?id=1100000002373259109]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3178||B7₄₈₆||(139×13⁴⁸⁶−7)/12||[http://factordb.com/index.php?id=1100000002321015892]||[http://factordb.com/cert.php?id=1100000002321015892] |- |13||3179||B₅₆₃C||(11×13⁵⁶⁴+1)/12||[http://factordb.com/index.php?id=1100000000000217927]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=564&c0=-&EN= 13⁵⁶⁴−1] |- |13||3180||1B₅₇₆||(23×13⁵⁷⁶−11)/12||[http://factordb.com/index.php?id=1100000002321021456]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], factor ''N''−1 is equivalent to factor [http://myfactorcollection.mooo.com:8090/cgi-bin/showSingleEntry?Base=13&Exp=576&c0=-&EN= 13⁵⁷⁶−1] |- |13||3181||80₆₉₃87||8×13⁶⁹⁵+111||[http://factordb.com/index.php?id=1100000002615636527]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000002615636532], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000002615636532] |- |13||3182||CC5₇₁₃||(2021×13⁷¹³−5)/12||[http://factordb.com/index.php?id=1100000002615627353]||[http://factordb.com/cert.php?id=1100000002615627353] |- |13||3183||B₈₃₄74||(11×13⁸³⁶−719)/12||[http://factordb.com/index.php?id=1100000003590430871]||[http://factordb.com/cert.php?id=1100000003590430871] |- |13||3184||9₉₆₈B||(3×13⁹⁶⁹+5)/4||[http://factordb.com/index.php?id=1100000000258566244]||[http://factordb.com/cert.php?id=1100000000258566244] |- |13||3185||10₁₂₉₅181||13¹²⁹⁸+274||[http://factordb.com/index.php?id=1100000002615445013]||[http://factordb.com/cert.php?id=1100000002615445013] |- |13||3186||9₁₃₆₂5||(3×13¹³⁶³−19)/4||[http://factordb.com/index.php?id=1100000002321017776]||[http://factordb.com/cert.php?id=1100000002321017776] |- |13||3187||7₁₅₀₄1||(7×13¹⁵⁰⁵−79)/12||[http://factordb.com/index.php?id=1100000002320890755]||[http://factordb.com/cert.php?id=1100000002320890755] |- |13||3188||930₁₅₅₁1||120×13¹⁵⁵²+1||[http://factordb.com/index.php?id=1100000000765961452]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3189||720₂₂₉₇2||93×13²²⁹⁸+2||[http://factordb.com/index.php?id=1100000002632396910]||[http://factordb.com/cert.php?id=1100000002632396910] |- |13||3190||1770₂₇₀₃17||267×13²⁷⁰⁵+20||[http://factordb.com/index.php?id=1100000003590430825]||[http://factordb.com/cert.php?id=1100000003590430825] |- |13||3191||390₆₂₆₆1||48×13⁶²⁶⁷+1||[http://factordb.com/index.php?id=1100000000765961441]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |13||3192||B0₆₅₄₀BBA||11×13⁶⁵⁴³+2012||[http://factordb.com/index.php?id=1100000002616382906]||[http://factordb.com/cert.php?id=1100000002616382906] |- |13||3193||C₁₀₆₃₁92||13¹⁰⁶³³−50||[http://factordb.com/index.php?id=1100000003590493750]||[http://factordb.com/cert.php?id=1100000003590493750] |- |14||649||34D₇₀₈||47×14⁷⁰⁸−1||[http://factordb.com/index.php?id=1100000001540144903]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |14||650||4D₁₉₆₉₈||5×14¹⁹⁶⁹⁸−1||[http://factordb.com/index.php?id=1100000000884560233]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2328||880₂₄₆7||136×16²⁴⁷+7||[http://factordb.com/index.php?id=1100000002468140199]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, and this prime factor is < 10²⁹⁹ |- |16||2329||D4₂₆₃D||(199×16²⁶⁴+131)/15||[http://factordb.com/index.php?id=1100000002468170238]||[http://factordb.com/cert.php?id=1100000002468170238] |- |16||2330||E0₂₆₁4DD||14×16²⁶⁴+1245||[http://factordb.com/index.php?id=1100000003588388352]||[http://factordb.com/cert.php?id=1100000003588388352] |- |16||2331||8C0₂₉₀ED||140×16²⁹²+237||[http://factordb.com/index.php?id=1100000003588388307]||[http://factordb.com/cert.php?id=1100000003588388307] |- |16||2332||DA₃₀₅5||(41×16³⁰⁶−17)/3||[http://factordb.com/index.php?id=1100000003588388284]||[http://factordb.com/cert.php?id=1100000003588388284] |- |16||2333||CE80₄₂₂D||3304×16⁴²³+13||[http://factordb.com/index.php?id=1100000003588388257]||[http://factordb.com/cert.php?id=1100000003588388257] |- |16||2334||5F₅₄₄6F||6×16⁵⁴⁶−145||[http://factordb.com/index.php?id=1100000002604723967]||[http://factordb.com/cert.php?id=1100000002604723967] |- |16||2335||88F₅₄₅||137×16⁵⁴⁵−1||[http://factordb.com/index.php?id=1100000000413679658]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |16||2336||BE0₇₉₂BB||190×16⁷⁹⁴+187||[http://factordb.com/index.php?id=1100000003588387938]||[http://factordb.com/cert.php?id=1100000003588387938] |- |16||2337||D9₁₀₅₂||(68×16¹⁰⁵²−3)/5||[http://factordb.com/index.php?id=1100000002321036020]||[http://factordb.com/cert.php?id=1100000002321036020] |- |16||2338||FAF₁₀₆₂45||251×16¹⁰⁶⁴−187||[http://factordb.com/index.php?id=1100000003588387610]||[http://factordb.com/cert.php?id=1100000003588387610] |- |16||2339||F8₁₅₁₇F||(233×16¹⁵¹⁸+97)/15||[http://factordb.com/index.php?id=1100000000633744824]||[http://factordb.com/cert.php?id=1100000000633744824] |- |16||2340||20₁₇₁₃321||2×16¹⁷¹⁶+801||[http://factordb.com/index.php?id=1100000003588386735]||[http://factordb.com/cert.php?id=1100000003588386735] |- |16||2341||300F₁₉₆₀AF||769×16¹⁹⁶²−81||[http://factordb.com/index.php?id=1100000003588368750]||[http://factordb.com/cert.php?id=1100000003588368750] |- |16||2342||90₃₅₄₂91||9×16³⁵⁴⁴+145||[http://factordb.com/index.php?id=1100000000633424191]||[http://factordb.com/cert.php?id=1100000000633424191] |- |16||2343||5BC₃₇₀₀D||(459×16³⁷⁰¹+1)/5||[http://factordb.com/index.php?id=1100000000993764322]||[http://factordb.com/cert.php?id=1100000000993764322] |- |16||2344||D0B₁₇₈₀₄||(3131×16¹⁷⁸⁰⁴−11)/15||[http://factordb.com/index.php?id=1100000003589278511]||[http://factordb.com/cert.php?id=1100000003589278511] |- |18||547||80₂₉₈B||8×18²⁹⁹+11||[http://factordb.com/index.php?id=1100000002355574745]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has sum-of-two-cubes algebraic factorization, 6×18⁹⁹+1 is an algebraic factor of ''N''+1, factordb entry of 6×18⁹⁹+1 is [http://factordb.com/index.php?id=1100000000900149167] |- |18||548||H₇₆₆FH||18⁷⁶⁸−37||[http://factordb.com/index.php?id=1100000003590430490]||[http://factordb.com/cert.php?id=1100000003590430490] |- |18||549||C0₆₂₆₈C5||12×18⁶²⁷⁰+221||[http://factordb.com/index.php?id=1100000003590442437]||[http://factordb.com/cert.php?id=1100000003590442437] |- |20||3301||H₂₄₇A0H||(17×20²⁵⁰−59677)/19||[http://factordb.com/index.php?id=1100000003590502619]||[http://factordb.com/cert.php?id=1100000003590502619] |- |20||3302||7₂₄₉A7||(7×20²⁵¹+1133)/19||[http://factordb.com/index.php?id=1100000003590502602]||[http://factordb.com/cert.php?id=1100000003590502602] |- |20||3303||J7₂₇₀||(368×20²⁷⁰−7)/19||[http://factordb.com/index.php?id=1100000002325395462]||[http://factordb.com/cert.php?id=1100000002325395462] |- |20||3304||J₃₃₀CCC7||20³³⁴−58953||[http://factordb.com/index.php?id=1100000003590502572]||[http://factordb.com/cert.php?id=1100000003590502572] |- |20||3305||40₃₈₇404B||4×20³⁹¹+32091||[http://factordb.com/index.php?id=1100000003590502563]||[http://factordb.com/cert.php?id=1100000003590502563] |- |20||3306||EC0₄₂₉7||292×20⁴³⁰+7||[http://factordb.com/index.php?id=1100000002633348702]||[http://factordb.com/cert.php?id=1100000002633348702] |- |20||3307||G₄₄₇99||(16×20⁴⁴⁹−2809)/19||[http://factordb.com/index.php?id=1100000000840126753]||[http://factordb.com/cert.php?id=1100000000840126753] |- |20||3308||3A₅₂₇3||(67×20⁵²⁸−143)/19||[http://factordb.com/index.php?id=1100000003590502531]||[http://factordb.com/cert.php?id=1100000003590502531] |- |20||3309||E₅₆₆C7||(14×20⁵⁶⁸−907)/19||[http://factordb.com/index.php?id=1100000003590502516]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003810023889], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003810023889] |- |20||3310||JCJ₆₂₉||393×20⁶²⁹−1||[http://factordb.com/index.php?id=1100000001559454258]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |20||3311||J₆₅₅05J||20⁶⁵⁸−7881||[http://factordb.com/index.php?id=1100000003590502490]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 has a large prime factor, factordb entry of this prime factor is [http://factordb.com/index.php?id=1100000003591067052], and primality certificate of this prime factor is [http://factordb.com/cert.php?id=1100000003591067052] |- |20||3312||50₁₁₆₃AJ||5×20¹¹⁶⁵+219||[http://factordb.com/index.php?id=1100000003590502412]||[http://factordb.com/cert.php?id=1100000003590502412] |- |20||3313||CD₂₄₄₉||(241×20²⁴⁴⁹−13)/19||[http://factordb.com/index.php?id=1100000002325393915]||[http://factordb.com/cert.php?id=1100000002325393915] |- |20||3314||G0₆₂₆₉D||16×20⁶²⁷⁰+13||[http://factordb.com/index.php?id=1100000003590539457]||[http://factordb.com/cert.php?id=1100000003590539457] |- |22||7984||I7G0₂₅₄H||8882×22²⁵⁵+17||[http://factordb.com/index.php?id=1100000003591372788]||[http://factordb.com/cert.php?id=1100000003591372788] |- |22||7985||D0₂₅₅5EEF||13×22²⁵⁹+60339||[http://factordb.com/index.php?id=1100000003591371932]||[http://factordb.com/cert.php?id=1100000003591371932] |- |22||7986||IK₃₂₂F||(398×22³²³−125)/21||[http://factordb.com/index.php?id=1100000000840384145]||[http://factordb.com/cert.php?id=1100000000840384145] |- |22||7987||C0₃₄₀G9||12×22³⁴²+361||[http://factordb.com/index.php?id=1100000000840384159]||[http://factordb.com/cert.php?id=1100000000840384159] |- |22||7988||77E₃₄₈K7||(485×22³⁵⁰+373)/3||[http://factordb.com/index.php?id=1100000003591369779]||[http://factordb.com/cert.php?id=1100000003591369779] |- |22||7989||J₃₇₉KJ||(19×22³⁸¹+443)/21||[http://factordb.com/index.php?id=1100000003591369027]||[http://factordb.com/cert.php?id=1100000003591369027] |- |22||7990||J₃₈₈EJ||(19×22³⁹⁰−2329)/21||[http://factordb.com/index.php?id=1100000003591367729]||[http://factordb.com/cert.php?id=1100000003591367729] |- |22||7991||DJ₄₀₀||(292×22⁴⁰⁰−19)/21||[http://factordb.com/index.php?id=1100000002325880110]||[http://factordb.com/cert.php?id=1100000002325880110] |- |22||7992||E₄₀₄K7||(2×22⁴⁰⁶+373)/3||[http://factordb.com/index.php?id=1100000003591366298]||[http://factordb.com/cert.php?id=1100000003591366298] |- |22||7993||66F₄₅₃B3||(971×22⁴⁵⁵−705)/7||[http://factordb.com/index.php?id=1100000003591365809]||[http://factordb.com/cert.php?id=1100000003591365809] |- |22||7994||L0₄₅₄B63||21×22⁴⁵⁷+5459||[http://factordb.com/index.php?id=1100000003591365331]||[http://factordb.com/cert.php?id=1100000003591365331] |- |22||7995||L₄₈₃G3||22⁴⁸⁵−129||[http://factordb.com/index.php?id=1100000003591364730]||[http://factordb.com/cert.php?id=1100000003591364730] |- |22||7996||E60₄₉₆L||314×22⁴⁹⁷+21||[http://factordb.com/index.php?id=1100000000632703239]||[http://factordb.com/cert.php?id=1100000000632703239] |- |22||7997||I₆₂₆AF||(6×22⁶²⁸−1259)/7||[http://factordb.com/index.php?id=1100000000632724334]||[http://factordb.com/cert.php?id=1100000000632724334] |- |22||7998||K0₇₆₀EC1||20×22⁷⁶³+7041||[http://factordb.com/index.php?id=1100000000632724415]||[http://factordb.com/cert.php?id=1100000000632724415] |- |22||7999||J0₇₆₇IGGJ||19×22⁷⁷¹+199779||[http://factordb.com/index.php?id=1100000003591362567]||[http://factordb.com/cert.php?id=1100000003591362567] |- |22||8000||7₉₅₉K7||(22⁹⁶¹+857)/3||[http://factordb.com/index.php?id=1100000003591361817]||[http://factordb.com/cert.php?id=1100000003591361817] |- |22||8001||L₂₃₈₅KE7||22²³⁸⁸−653||[http://factordb.com/index.php?id=1100000003591360774]||[http://factordb.com/cert.php?id=1100000003591360774] |- |22||8002||7₃₈₁₅2L||(22³⁸¹⁷−289)/3||[http://factordb.com/index.php?id=1100000003591359839]||[http://factordb.com/cert.php?id=1100000003591359839] |- |24||3400||I0₂₄₁I5||18×24²⁴³+437||[http://factordb.com/index.php?id=1100000002633360037]||[http://factordb.com/cert.php?id=1100000002633360037] |- |24||3401||D0₂₅₉KKD||13×24²⁶²+12013||[http://factordb.com/index.php?id=1100000003593270725]||[http://factordb.com/cert.php?id=1100000003593270725] |- |24||3402||C7₂₉₈||(283×24²⁹⁸−7)/23||[http://factordb.com/index.php?id=1100000002326181235]||[http://factordb.com/cert.php?id=1100000002326181235] |- |24||3403||20₃₁₃7||2×24³¹⁴+7||[http://factordb.com/index.php?id=1100000002355610241]||[http://factordb.com/cert.php?id=1100000002355610241] |- |24||3404||BC0₃₃₁B||276×24³³²+11||[http://factordb.com/index.php?id=1100000002633359842]||[http://factordb.com/cert.php?id=1100000002633359842] |- |24||3405||N₂₆₄₄LLN||24²⁶⁴⁷−1201||[http://factordb.com/index.php?id=1100000003593270089]||[http://factordb.com/cert.php?id=1100000003593270089] |- |24||3406||D₂₆₉₈LD||(13×24²⁷⁰⁰+4403)/23||[http://factordb.com/index.php?id=1100000003593269876]||[http://factordb.com/cert.php?id=1100000003593269876] |- |24||3407||A0₂₉₅₁8ID||10×24²⁹⁵⁴+5053||[http://factordb.com/index.php?id=1100000003593269654]||[http://factordb.com/cert.php?id=1100000003593269654] |- |24||3408||88N₅₉₅₁||201×24⁵⁹⁵¹−1||[http://factordb.com/index.php?id=1100000003593275880]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |- |24||3409||N00N₈₁₂₉LN||13249×24⁸¹³¹−49||[http://factordb.com/index.php?id=1100000003593391606]||[http://factordb.com/cert.php?id=1100000003593391606] |- |30||2613||AN₂₀₆||(313×30²⁰⁶−23)/29||[http://factordb.com/index.php?id=1100000002327651073]||[http://factordb.com/cert.php?id=1100000002327651073] |- |30||2614||M₂₄₁QB||(22×30²⁴³+3139)/29||[http://factordb.com/index.php?id=1100000003593408295]||[http://factordb.com/cert.php?id=1100000003593408295] |- |30||2615||M0₅₄₇SS7||22×30⁵⁵⁰+26047||[http://factordb.com/index.php?id=1100000003593407988]||[http://factordb.com/cert.php?id=1100000003593407988] |- |30||2616||C0₁₀₂₂1||12×30¹⁰²³+1||[http://factordb.com/index.php?id=1100000000785448736]||proven prime by [https://primes.utm.edu/prove/prove3_1.html ''N''−1 primality test], ''N''−1 is trivially 100% factored |- |30||2617||5₄₈₈₂J||(5×30⁴⁸⁸³+401)/29||[http://factordb.com/index.php?id=1100000002327649423]||[http://factordb.com/cert.php?id=1100000002327649423] |- |30||2619||OT₃₄₂₀₅||25×30³⁴²⁰⁵−1||[http://factordb.com/index.php?id=1100000000800812865]||proven prime by [https://primes.utm.edu/prove/prove3_2.html ''N''+1 primality test], ''N''+1 is trivially 100% factored |} ==Unproven PRPs== Only list bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 20, 22, 24, 30 {|class=wikitable |base (''b'')||index of this quasi-minimal prime in base ''b'' (assuming the primality of all PRP in base ''b'')||unproven PRP (base-''b'' form)||unproven PRP (algebraic ((''a''×''b''^''n''+''c'')/''d'') form)||factordb entry of this PRP |- |11||1068||57₆₂₆₆₈||(57×11⁶²⁶⁶⁸−7)/10||[http://factordb.com/index.php?id=1100000003573679860] |- |13||3194||C5₂₃₇₅₅C||(149×13²³⁷⁵⁶+79)/12||[http://factordb.com/index.php?id=1100000003590647776] |- |13||3195||80₃₂₀₁₇111||8×13³²⁰²⁰+183||[http://factordb.com/index.php?id=1100000000490878060] |- |16||2345||DB₃₂₂₃₄||(206×16³²²³⁴−11)/15||[http://factordb.com/index.php?id=1100000002383583629] |- |16||2346||4₇₂₇₈₅DD||(4×16⁷²⁷⁸⁷+2291)/15||[http://factordb.com/index.php?id=1100000003615909841] |- |22||8003||BK₂₂₀₀₁5||(251×22²²⁰⁰²−335)/21||[http://factordb.com/index.php?id=1100000003594696838] |- |30||2618||I0₂₄₆₀₈D||18×30²⁴⁶⁰⁹+13||[http://factordb.com/index.php?id=1100000003593967511] |} All these PRPs pass the [[w:Miller–Rabin primality test|Miller–Rabin primality test]] to bases 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59 and 61, and pass the [[w:Lucas pseudoprime#Strong Lucas pseudoprimes|strong Lucas primality test]] with parameters (''P'', ''Q'') defined by Selfridge's Method ''A'', and [[w:Trial division|trial factored]] to 10¹⁶. (Thus, they pass the [[w:Baillie–PSW primality test|Baillie–PSW primality test]]) ==Proof== ===Base 2=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. ===Base 3=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (2,1), (2,2) * Case (1,1): ** Since 12, 21, 111 are primes, we only need to consider the family 1{0}1 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 1{0}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 12 are primes, we only need to consider the family 2{0,2}2 (since any digits 1 between them will produce smaller primes) *** All numbers of the form 2{0,2}2 are divisible by 2, thus cannot be prime. ===Base 4=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (2,1), (2,3), (3,1), (3,3) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 11, 31, 221 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 13, 23 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. ===Base 5=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4) * Case (1,1): ** Since 12, 21, 111, 131 are primes, we only need to consider the family 1{0,4}1 (since any digits 1, 2, 3 between them will produce smaller primes) *** All numbers of the form 1{0,4}1 are divisible by 2, thus cannot be prime. * Case (1,2): ** 12 is prime, and thus the only minimal prime in this family. * Case (1,3): ** Since 12, 23, 43, 133 are primes, we only need to consider the family 1{0,1}3 (since any digits 2, 3, 4 between them will produce smaller primes) *** Since 111 is prime, we only need to consider the families 1{0}3 and 1{0}1{0}3 (since any digit combo 11 between (1,3) will produce smaller primes) **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** For the 1{0}1{0}3 family, since 10103 is prime, we only need to consider the families 1{0}13 and 11{0}3 (since any digit combo 010 between (1,3) will produce smaller primes) ***** The smallest prime of the form 1{0}13 is 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013, which can be written as 1(0^93)13 and equal the prime 5^95+8 ([http://factordb.com/index.php?id=1100000000034686071 factordb]) ***** All numbers of the form 11{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** Since 12, 34, 104 are primes, we only need to consider the family 1{1,4}4 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 111, 414 are primes, we only need to consider the families 1{4}4 and 11{4}4 (since any digit combo 11 or 41 between them will produce smaller primes) **** The smallest prime of the form 1{4}4 is 14444. **** All numbers of the form 11{4}4 are divisible by 2, thus cannot be prime. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,2): ** Since 21, 23, 12, 32 are primes, we only need to consider the family 2{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 21, 23, 34 are primes, we only need to consider the family 2{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 2{0,2,4}4 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 34, 21 are primes, we only need to consider the family 3{0,1,3}1 (since any digits 2, 4 between them will produce smaller primes) *** Since 313, 111, 131, 3101 are primes, we only need to consider the families 3{0,3}1 and 3{0,3}11 (since any digit combo 10, 11, 13 between (3,1) will produce smaller primes) **** For the 3{0,3}1 family, we can separate this family to four families: ***** For the 30{0,3}01 family, we have the prime 30301, and the remain case is the family 30{0}01. ****** All numbers of the form 30{0}01 are divisible by 2, thus cannot be prime. ***** For the 30{0,3}31 family, note that there must be an even number of 3's between (30,31), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (30,31) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 30{0}31, and this prime is 300031. ***** For the 33{0,3}01 family, note that there must be an even number of 3's between (33,01), or the result number will be divisible by 2 and cannot be prime. ****** Since 33331 is prime, any digit combo 33 between (33,01) will produce smaller primes. ******* Thus, the only possible prime is the smallest prime in the family 33{0}01, and this prime is 33001. ***** For the 33{0,3}31 family, we have the prime 33331, and the remain case is the family 33{0}31. ****** All numbers of the form 33{0}31 are divisible by 2, thus cannot be prime. **** All numbers of the form 3{0,3}11 are divisible by 3, thus cannot be prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 34, 23, 43, 313 are primes, we only need to consider the family 3{0,3}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** All numbers of the form 3{0,3}3 are divisible by 3, thus cannot be prime. * Case (3,4): ** 34 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 43, 21, 401 are primes, we only need to consider the family 4{1,4}1 (since any digits 0, 2, 3 between them will produce smaller primes) *** Since 414, 111 are primes, we only need to consider the families 4{4}1 and 4{4}11 (since any digit combo 14 or 11 between them will produce smaller primes) **** The smallest prime of the form 4{4}1 is 44441. **** All numbers of the form 4{4}11 are divisible by 2, thus cannot be prime. * Case (4,2): ** Since 43, 12, 32 are primes, we only need to consider the family 4{0,2,4}2 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 43, 34, 414 are primes, we only need to consider the family 4{0,2,4}4 (since any digits 1, 3 between them will produce smaller primes) *** All numbers of the form 4{0,2,4}4 are divisible by 2, thus cannot be prime. ===Base 6=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (2,1), (2,5), (3,1), (3,5), (4,1), (4,5), (5,1), (5,5) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 11, 21, 31, 51 are primes, we only need to consider the family 4{0,4}1 (since any digits 1, 2, 3, 5 between them will produce smaller primes) *** Since 4401 and 4441 are primes, we only need to consider the families 4{0}1 and 4{0}41 (since any digits combo 40 and 44 between them will produce smaller primes) **** All numbers of the form 4{0}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 4{0}41 is 40041 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 15, 25, 35, 45 are primes, we only need to consider the family 5{0,5}5 (since any digits 1, 2, 3, 4 between them will produce smaller primes) *** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. ===Base 7=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,1), (2,2), (2,3), (2,4), (2,5), (2,6), (3,1), (3,2), (3,3), (3,4), (3,5), (3,6), (4,1), (4,2), (4,3), (4,4), (4,5), (4,6), (5,1), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6) * Case (1,1): ** Since 14, 16, 41, 61, 131 are primes, we only need to consider the family 1{0,1,2,5}1 (since any digits 3, 4, 6 between them will produce smaller primes) *** Since the digit sum of primes must be odd (otherwise the number will be divisible by 2, thus cannot be prime), there is an odd total number of 1 and 5 in the {} **** If there are >=3 number of 1 and 5 in the {}: ***** If there is 111 in the {}, then we have the prime 11111 ***** If there is 115 in the {}, then the prime 115 is a subsequence ***** If there is 151 in the {}, then the prime 115 is a subsequence ***** If there is 155 in the {}, then the prime 155 is a subsequence ***** If there is 511 in the {}, then the current number is 15111, which has digit sum = 12, but digit sum divisible by 3 will cause the number divisible by 3 and cannot be prime, and we cannot add more 1 or 5 to this number (to avoid 11111, 155, 515, 551 as subsequence), thus we must add at least one 2 to this number, but then the number has both 2 and 5, and will have either 25 or 52 as subsequence, thus cannot be minimal prime ***** If there is 515 in the {}, then the prime 515 is a subsequence ***** If there is 551 in the {}, then the prime 551 is a subsequence ***** If there is 555 in the {}, then the prime 551 is a subsequence **** Thus there is only one 1 (and no 5) or only one 5 (and no 1) in the {}, i.e. we only need to consider the families 1{0,2}1{0,2}1 and 1{0,2}5{0,2}1 ***** For the 1{0,2}1{0,2}1 family, since 1211 is prime, we only need to consider the family 1{0}1{0,2}1 ****** Since all numbers of the form 1{0}1{0}1 are divisible by 3 and cannot be prime, we only need to consider the family 1{0}1{0}2{0}1 ******* Since 11201 is prime, we only need to consider the family 1{0}1{0}21 ******** The smallest prime of the form 11{0}21 is 1100021 ******** All numbers of the form 101{0}21 are divisible by 5, thus cannot be prime ******** The smallest prime of the form 1001{0}21 is 100121 ********* Since this prime has no 0 between 1{0}1 and 21, we do not need to consider more families ***** For the 1{0,2}5{0,2}1 family, since 25 and 52 are primes, we only need to consider the family 1{0}5{0}1 ****** Since 1051 is prime, we only need to consider the family 15{0}1 ******* The smallest prime of the form 15{0}1 is 150001 * Case (1,2): ** Since 14, 16, 32, 52 are primes, we only need to consider the family 1{0,1,2}2 (since any digits 3, 4, 5, 6 between them will produce smaller primes) *** Since 1112 and 1222 are primes, there is at most one 1 and at most one 2 in {} **** If there are one 1 and one 2 in {}, then the digit sum is 6, and the number will be divisible by 6 and cannot be prime. **** If there is one 1 but no 2 in {}, then the digit sum is 4, and the number will be divisible by 2 and cannot be prime. **** If there is no 1 but one 2 in {}, then the form is 1{0}2{0}2 ***** Since 1022 and 1202 are primes, we only need to consider the number 122 ****** 122 is not prime. **** If there is no 1 and no 2 in {}, then the digit sum is 3, and the number will be divisible by 3 and cannot be prime. * Case (1,3): ** Since 14, 16, 23, 43, 113, 133 are primes, we only need to consider the family 1{0,5}3 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** Since 155 is prime, we only need to consider the family 1{0}3 and 1{0}5{0}3 **** All numbers of the form 1{0}3 are divisible by 2, thus cannot be prime. **** All numbers of the form 1{0}5{0}3 are divisible by 3, thus cannot be prime. * Case (1,4): ** 14 is prime, and thus the only minimal prime in this family. * Case (1,5): ** Since 14, 16, 25, 65, 115, 155 are primes, we only need to consider the family 1{0,3}5 (since any digits 1, 2, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 1{0,3}5 are divisible by 3, thus cannot be prime. * Case (1,6): ** 16 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 25, 41, 61, 221 are primes, we only need to consider the family 2{0,1}1 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 2111 is prime, we only need to consider the families 2{0}1 and 2{0}1{0}1 **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** All numbers of the form 2{0}1{0}1 are divisible by 2, thus cannot be prime. * Case (2,2): ** Since 23, 25, 32, 52, 212 are primes, we only need to consider the family 2{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,4): ** Since 23, 25, 14 are primes, we only need to consider the family 2{0,2,4,6}4 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}4 are divisible by 2, thus cannot be prime. * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,6): ** Since 23, 25, 16, 56 are primes, we only need to consider the family 2{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 2{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (3,1): ** Since 32, 41, 61 are primes, we only need to consider the family 3{0,1,3,5}1 (since any digits 2, 4, 6 between them will produce smaller primes) *** Since 551 is prime, we only need to consider the family 3{0,1,3}1 and 3{0,1,3}5{0,1,3}1 (since any digits combo 55 between (3,1) will produce smaller primes) **** For the 3{0,1,3}1 family, since 3031 and 131 are primes, we only need to consider the families 3{0,1}1 and 3{3}3{0,1}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes, thus for the digits between (3,1), all 3's must be before all 0's and 1's, and thus we can let the red 3 in 3{3}3{0,1}1 be the rightmost 3 between (3,1), all digits before this 3 must be 3's, and all digits after this 3 must be either 0's or 1's) ***** For the 3{0,1}1 family: ****** If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ****** If there are no 1's between (3,1), then the form will be 3{0}1 ******* All numbers of the form 3{0}1 are divisible by 2, thus cannot be prime. ****** If there are no 0's between (3,1), then the form will be 3{1}1 ******* The smallest prime of the form 3{1}1 is 31111 ****** If there are exactly 1 0's between (3,1), then there must be <3 1's between (3,1), or 31111 will be a subsequence. ******* If there are 2 1's between (3,1), then the digit sum is 6, thus the number is divisible by 6 and cannot be prime. ******* If there are 1 1's between (3,1), then the number can only be either 3101 or 3011 ******** Neither 3101 nor 3011 is prime. ******* If there are no 1's between (3,1), then the number must be 301 ******** 301 is not prime. ***** For the 3{3}3{0,1}1 family: ****** If there are at least one 3 between (3,3{0,1}1) and at least one 1 between (3{3}3,1), then 33311 will be a subsequence. ****** If there are no 3 between (3,3{0,1}1), then the form will be 33{0,1}1 ******* If there are at least 3 1's between (33,1), then 31111 will be a subsequence. ******* If there are exactly 2 1's between (33,1), then the digit sum is 12, thus the number is divisible by 3 and cannot be prime. ******* If there are exactly 1 1's between (33,1), then the digit sum is 11, thus the number is divisible by 2 and cannot be prime. ******* If there are no 1's between (33,1), then the form will be 33{0}1 ******** The smallest prime of the form 33{0}1 is 33001 ****** If there are no 1 between (3{3}3,1), then the form will be 3{3}3{0}1 ******* If there are at least 2 0's between (3{3}3,1), then 33001 will be a subsequence. ******* If there are exactly 1 0's between (3{3}3,1), then the form is 3{3}301 ******** The smallest prime of the form 3{3}301 is 33333301 ******* If there are no 0's between (3{3}3,1), then the form is 3{3}31 ******** The smallest prime of the form 3{3}31 is 33333333333333331 **** For the 3{0,1,3}5{0,1,3}1 family, since 335 is prime, we only need to consider the family 3{0,1}5{0,1,3}1 ***** Numbers containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1,3}3{0,1,3}1 ******* Since 3031 and 131 are primes, we only need to consider the family 35{3}3{0,1,3}1 (since any digits combo 03, 13 between (3,1) will produce smaller primes) ******** Since 533 is prime, we only need to consider the family 353{0,1}1 (since any digits combo 33 between (35,1) will produce smaller primes) ********* Since 5011 is prime, we only need to consider the family 353{1}{0}1 (since any digits combo 01 between (353,1) will produce smaller primes) ********** If there are at least 3 1's between (353,{0}1), then 31111 will be a subsequence. ********** If there are exactly 2 1's between (353,{0}1), then the digit sum is 20, thus the number is divisible by 2 and cannot be prime. ********** If there are exactly 1 1's between (353,{0}1), then the form is 3531{0}1 *********** The smallest prime of the form 3531{0}1 is 3531001, but it is not minimal prime since 31001 is prime. ********** If there are no 1's between (353,{0}1), then the digit sum is 15, thus the number is divisible by 6 and cannot be prime. ***** Numbers not containing 3 between (3{0,1}5,1): ****** The form is 3{0,1}5{0,1}1 ******* If there are >=2 0's and >=1 1's between (3,1), then at least one of 30011, 30101, 31001 will be a subsequence. ******* If there are no 1's between (3,1), then the form will be 3{0}5{0}1 ******** All numbers of the form 3{0}5{0}1 are divisible by 3, thus cannot be prime. ******* If there are no 0's between (3,1), then the form will be 3{1}5{1}1 ******** If there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 31151, 31511, 35111 ********* None of 31151, 31511, 35111 are primes. ******** If there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 351 ********* 351 is not prime. ******* If there are exactly 1 0's between (3,1), then the form will be 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1 ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are >=3 1's between (3,1), then 31111 will be a subsequence. ******** If there are exactly 2 1's between (3,1), then the number can only be 311051, 310151, 310511, 301151, 301511, 305111, 311501, 315101, 315011, 351101, 351011, 350111 ********* Of these numbers, 311051, 301151, 311501, 351101, 350111 are primes. ********** However, 311051, 301151, 311501 have 115 as subsequence, and 350111 has 5011 as subsequence, thus only 351101 is minimal prime. ******** No matter 3{1}0{1}5{1}1 or 3{1}5{1}0{1}1, if there are exactly 1 1's between (3,1), then the digit sum is 13, thus the number is divisible by 2 and cannot be prime. ******** If there are no 1's between (3,1), then the number is 3051 for 3{1}0{1}5{1}1 or 3501 for 3{1}5{1}0{1}1 ********* Neither 3051 nor 3501 is prime. * Case (3,2): ** 32 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 32, 23, 43, 313 are primes, we only need to consider the family 3{0,3,5,6}3 (since any digits 1, 2, 4 between them will produce smaller primes) *** If there are >=2 5's in {}, then 553 will be a subsequence. *** If there are no 5's in {}, then the family will be 3{0,3,6}3 **** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. *** If there are exactly 1 5's in {}, then the family will be 3{0,3,6}5{0,3,6}3 **** Since 335, 65, 3503, 533, 56 are primes, we only need to consider the family 3{0}53 (since any digit 3, 6 between (3,5{0,3,6}3) and any digit 0, 3, 6 between (3{0,3,6}5,3) will produce smaller primes) ***** The smallest prime of the form 3{0}53 is 300053 * Case (3,4): ** Since 32, 14, 304, 344, 364 are primes, we only need to consider the family 3{3,5}4 (since any digits 0, 1, 2, 4, 6 between them will produce smaller primes) *** Since 3334 and 335 are primes, we only need to consider the family 3{5}4 and 3{5}34 (since any digits combo 33, 35 between them will produce smaller primes) **** The smallest prime of the form 3{5}4 is 35555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555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with 9234 5's, which can be written as 3(5^9234)4 and equal the prime (23*7^9235-11)/6 ([http://factordb.com/index.php?id=1100000002766595757 factordb]) ([http://factordb.com/cert.php?id=1100000002766595757 primality certificate]) (not minimal prime, since 35555 and 5554 are primes) **** The smallest prime of the form 3{5}34 is 355555555555555555555555555555555555555555555555555555555555555534 (not minimal prime, since 35555, 553, and 5554 are primes) * Case (3,5): ** Since 32, 25, 65, 335 are primes, we only need to consider the family 3{0,1,4,5}5 (since any digits 2, 3, 6 between them will produce smaller primes) *** If there are at least one 1's and at least one 5's in {}, then either 155 or 515 will be a subsequence. *** If there are at least one 1's and at least one 4's in {}, then either 14 or 41 will be a subsequence. *** If there are at least two 1's in {}, then 115 will be a subsequence. *** If there are exactly one 1's and no 4's or 5's in {}, then the family will be 3{0}1{0}5 **** All numbers of the form 3{0}1{0}5 are divisible by 3, thus cannot be prime. *** If there is no 1's in {}, then the family will be 3{0,4,5}5 **** If there are at least to 4's in {}, then 344 and 445 will be subsequences. **** If there is no 4's in {}, then the family will be 3{0,5}5 ***** Since 3055 and 3505 are primes, we only need to consider the families 3{0}5 and 3{5}5 ****** All numbers of the form 3{0}5 are divisible by 2, thus cannot be prime. ****** The smallest prime of the form 3{5}5 is 35555 **** If there is exactly one 4's in {}, then the family will be 3{0,5}4{0,5}5 ***** Since 304, 3545 are primes, we only need to consider the families 34{0,5}5 (since any digits 0 or 5 between (3,4{0,5}5) will produce small primes) ****** All numbers of the form 34{0,5}5 are divisible by 5, thus cannot be prime. * Case (3,6): ** Since 32, 16, 56, 346 are primes, we only need to consider the family 3{0,3,6}6 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}6 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,2): ** Since 41, 43, 32, 52 are primes, we only need to consider the family 4{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,4): ** Since 41, 43, 14 are primes, we only need to consider the family 4{0,2,4,5,6}4 (since any digits 1, 3 between them will produce smaller primes) *** If there is no 5's in {}, then the family will be 4{0,2,4,6}4 **** All numbers of the form 4{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there is at least one 5's in {}, then there cannot be 2 in {} (since if so, then either 25 or 52 will be a subsequence) and there cannot be 6 in {} (since if so, then either 65 or 56 will be a subsequence), thus the family is 4{0,4,5}5{0,4,5}4 **** Since 445, 4504, 544 are primes, we only need to consider the family 4{0,5}5{5}4 (since any digit 4 between (4,5{0,4,5}4) and any digit 0, 4 between (4{0,4,5}5,4) will produce smaller primes) ***** If there are at least two 0's between (4,5{0,4,5}4), then 40054 will be a subsequence. ***** If there is no 0's between (4,5{0,4,5}4), then the family will be 4{5}5{5}4, which is equivalent to 4{5}4 ****** The smallest prime of the form 4{5}4 is 45555555555555554 (not minimal prime, since 4555 and 5554 are primes) ***** If there is exactly one 0's between (4,5{0,4,5}4), then the family will be 4{5}0{5}5{5}4 ****** Since 4504 is prime, we only need to consider the family 40{5}5{5}4 (since any digit 5 between (4,0{5}5{5}4) will produce small primes), which is equivalent to 40{5}4 ******* The smallest prime of the form 40{5}4 is 405555555555555554 (not minimal prime, since 4555 and 5554 are primes) * Case (4,5): ** Since 41, 43, 25, 65, 445 are primes, we only need to consider the family 4{0,5}5 (since any digits 1, 2, 3, 4, 6 between them will produce smaller primes) *** If there are at least two 5's in {}, then 4555 will be a subsequence. *** If there is exactly one 5's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there is no 5's in {}, then the family will be 4{0}5 **** All numbers of the form 4{0}5 are divisible by 3, thus cannot be prime. * Case (4,6): ** Since 41, 43, 16, 56 are primes, we only need to consider the family 4{0,2,4,6}6 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 4{0,2,4,6}6 are divisible by 2, thus cannot be prime. * Case (5,1): ** Since 52, 56, 41, 61, 551 are primes, we only need to consider the family 5{0,1,3}1 (since any digits 2, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 3's in {}, then 533 will be a subsequence. *** If there is no 3's in {}, then the family will be 5{0,1}1 **** Since 5011 is prime, we only need to consider the family 5{1}{0}1 ***** Since 11111 is prime, we only need to consider the families 5{0}1, 51{0}1, 511{0}1, 5111{0}1 (since any digits combo 1111 between (5,1) will produce small primes) ****** All numbers of the form 5{0}1 are divisible by 6, thus cannot be prime. ****** The smallest prime of the form 51{0}1 is 5100000001 ****** All numbers of the form 511{0}1 are divisible by 2, thus cannot be prime. ****** All numbers of the form 5111{0}1 are divisible by 3, thus cannot be prime. *** If there is exactly one 3's in {}, then the family will be 5{0,1}3{0,1}1 **** If there is at least one 1's between (5,3{0,1}1), then 131 will be a subsequence. ***** Thus we only need to consider the family 5{0}3{0,1}1 ****** If there are no 1's between (5{0}3,1), then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. ****** If there are exactly one 1's between (5{0}3,1), then the digit sum is 13, and the number will be divisible by 2 and cannot be prime. ****** If there are exactly three 1's between (5{0}3,1), then the digit sum is 15, and the number will be divisible by 6 and cannot be prime. ****** If there are at least four 1's between (5{0}3,1), then 11111 will be a subsequence. ****** If there are exactly two 1's between (5{0}3,1), then the family will be 5{0}3{0}1{0}1{0}1 ******* Since 5011 is prime, we only need to consider the family 5311{0}1 (since any digit 0 between (5,1{0}1) will produce small primes, this includes the leftmost three {} in 5{0}3{0}1{0}1{0}1, and thus only the rightmost {} can contain 0) ******** The smallest prime of the form 5311{0}1 is 531101 * Case (5,2): ** 52 is prime, and thus the only minimal prime in this family. * Case (5,3): ** Since 52, 56, 23, 43, 533, 553 are primes, we only need to consider the family 5{0,1}3 (since any digits 2, 3, 4, 5, 6 between them will produce smaller primes) *** If there are at least two 1's in {}, then 113 will be a subsequence. *** If there is exactly one 1's in {}, then the digit sum is 12, and the number will be divisible by 3 and cannot be prime. *** If there is no 1's in {}, then the digit sum is 11, and the number will be divisible by 2 and cannot be prime. * Case (5,4): ** Since 52, 56, 14, 544 are primes, we only need to consider the family 5{0,3,5}4 (since any digits 1, 2, 4, 6 between them will produce smaller primes) *** If there are no 5's in {}, then the family will be 5{0,3}4 **** All numbers of the form 5{0,3}4 are divisible by 3, thus cannot be prime. *** If there are at least one 5's and at least one 3's in {}, then either 535 or 553 will be a subsequence. *** If there are exactly one 5's and no 3's in {}, then the digit sum is 20, and the number will be divisible by 2 and cannot be prime. *** If there are at least two 5's in {}, then 5554 will be a subsequence. * Case (5,5): ** Since 52, 56, 25, 65, 515, 535 are primes, we only need to consider the family 5{0,4,5}5 (since any digits 1, 2, 3, 6 between them will produce smaller primes) *** If there are no 4's in {}, then the family will be 5{0,5}5 **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. *** If there are no 5's in {}, then the family will be 5{0,4}5 **** All numbers of the form 5{0,4}5 are divisible by 2, thus cannot be prime. *** If there are at least one 4's and at least one 5's in {}, then either 5455 or 5545 will be a subsequence. * Case (5,6): ** 56 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,2): ** Since 61, 65, 32, 52 are primes, we only need to consider the family 6{0,2,4,6}2 (since any digits 1, 3, 5 between them will produce smaller primes) *** All numbers of the form 6{0,2,4,6}2 are divisible by 2, thus cannot be prime. * Case (6,3): ** Since 61, 65, 23, 43 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,4): ** Since 61, 65, 14 are primes, we only need to consider the family 6{0,2,3,4,6}4 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are exactly two 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 **** All numbers of the form 6{0,2,4,6}3{0,2,4,6}3{0,2,4,6}4 are divisible by 2, thus cannot be prime. *** If there are at least three 3's in {}, then 3334 will be a subsequence. *** If there is exactly one 3's in {}, then the family will be 6{0,2,4,6}3{0,2,4,6}4 **** If there is 0 between (6,3{0,2,4,6}4), then 6034 will be a subsequence. **** If there is 2 between (6,3{0,2,4,6}4), then 23 will be a subsequence. **** If there is 4 between (6,3{0,2,4,6}4), then 43 will be a subsequence. **** If there is 6 between (6,3{0,2,4,6}4), then 6634 will be a subsequence. **** If there is 0 between (6{0,2,4,6}3,4), then 304 will be a subsequence. **** If there is 2 between (6{0,2,4,6}3,4), then 32 will be a subsequence. **** If there is 4 between (6{0,2,4,6}3,4), then 344 will be a subsequence. **** If there is 6 between (6{0,2,4,6}3,4), then 364 will be a subsequence. **** Thus the number can only be 634 ***** 634 is not prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,6): ** Since 61, 65, 16, 56 are primes, we only need to consider the family 6{0,2,3,4,6}6 (since any digits 1, 5 between them will produce smaller primes) *** If there is no 3's in {}, then the family will be 6{0,2,4,6}6 **** All numbers of the form 6{0,2,4,6}6 are divisible by 2, thus cannot be prime. *** If there is no 2's and no 4's in {}, then the family will be 6{0,3,6}6 **** All numbers of the form 6{0,3,6}6 are divisible by 3, thus cannot be prime. *** If there is at least one 3's and at least one 2's in {}, then either 32 or 23 will be a subsequence. *** If there is at least one 3's and at least one 4's in {}, then either 346 or 43 will be a subsequence. ===Base 8=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,5), (1,7), (2,1), (2,3), (2,5), (2,7), (3,1), (3,3), (3,5), (3,7), (4,1), (4,3), (4,5), (4,7), (5,1), (5,3), (5,5), (5,7), (6,1), (6,3), (6,5), (6,7), (7,1), (7,3), (7,5), (7,7) * Case (1,1): ** Since 13, 15, 21, 51, 111, 141, 161 are primes, we only need to consider the family 1{0,7}1 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** Since 107, 177, 701 are primes, we only need to consider the number 171 and the family 1{0}1 (since any digits combo 07, 70, 77 between them will produce smaller primes) **** 171 is not prime. **** All numbers of the form 1{0}1 factored as 10^n+1 = (2^n+1) * (4^n-2^n+1) (n≥1) (and since if n≥1, 2^n+1 ≥ 2^1+1 = 3 > 1, 4^n-2^n+1 ≥ 4^1-2^1+1 = 3 > 1, this factorization is nontrivial), thus cannot be prime. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** Since 13, 15, 27, 37, 57, 107, 117, 147, 177 are primes, we only need to consider the family 1{6}7 (since any digits 0, 1, 2, 3, 4, 5, 7 between them will produce smaller primes) *** The smallest prime of the form 1{6}7 is 16667 (not minimal prime, since 667 is prime) * Case (2,1): ** 21 is prime, and thus the only minimal prime in this family. * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,5): ** Since 21, 23, 27, 15, 35, 45, 65, 75, 225, 255 are primes, we only need to consider the family 2{0}5 (since any digits 1, 2, 3, 4, 5, 6, 7 between them will produce smaller primes) *** All numbers of the form 2{0}5 are divisible by 7, thus cannot be prime. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (3,1): ** Since 35, 37, 21, 51, 301, 361 are primes, we only need to consider the family 3{1,3,4}1 (since any digits 0, 2, 5, 6, 7 between them will produce smaller primes) *** Since 13, 343, 111, 131, 141, 431, 3331, 3411 are primes, we only need to consider the families 3{3}11, 33{1,4}1, 3{3,4}4{4}1 (since any digits combo 11, 13, 14, 33, 41, 43 between them will produce smaller primes) **** All numbers of the form 3{3}11 are divisible by 3, thus cannot be prime. **** For the 33{1,4}1 family, since 111 and 141 are primes, we only need to consider the families 33{4}1 and 33{4}11 (since any digits combo 11, 14 between them will produce smaller primes) ***** The smallest prime of the form 33{4}1 is 3344441 ***** All numbers of the form 33{4}11 are divisible by 301, thus cannot be prime. **** For the 3{3,4}4{4}1 family, since 3331 and 3344441 are primes, we only need to consider the families 3{4}1, 3{4}31, 3{4}341, 3{4}3441, 3{4}34441 (since any digits combo 33 or 34444 between (3,1) will produce smaller primes) ***** All numbers of the form 3{4}1 are divisible by 31, thus cannot be prime. ***** Since 4443 is prime, we only need to consider the numbers 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 (since any digit combo 444 between (3,3{4}1) will produce smaller primes) ****** None of 3431, 34431, 34341, 344341, 343441, 3443441, 3434441, 34434441 are primes. * Case (3,3): ** Since 35, 37, 13, 23, 53, 73, 343 are primes, we only need to consider the family 3{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 3{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 21, 51, 401, 431, 471 are primes, we only need to consider the family 4{1,4,6}1 (since any digits 0, 2, 3, 5, 7 between them will produce smaller primes) *** Since 111, 141, 161, 661, 4611 are primes, we only need to consider the families 4{4}11, 4{4,6}4{1,4,6}1, 4{4}6{4}1 (since any digits combo 11, 14, 16, 61, 66 between them will produce smaller primes) **** The smallest prime of the form 4{4}11 is 44444444444444411 (not minimal prime, since 444444441 is prime) **** For the 4{4,6}4{1,4,6}1 family, we can separate this family to 4{4,6}41, 4{4,6}411, 4{4,6}461 ***** For the 4{4,6}41 family, since 661 and 6441 are primes, we only need to consider the families 4{4}41 and 4{4}641 (since any digits combo 64 or 66 between (4,41) will produce smaller primes) ****** The smallest prime of the form 4{4}41 is 444444441 ****** The smallest prime of the form 4{4}641 is 444641 ***** For the 4{4,6}411 family, since 661 and 6441 are primes, we only need to consider the families 4{4}411 and 4{4}6411 (since any digits combo 64 or 66 between (4,411) will produce smaller primes) ****** The smallest prime of the form 4{4}411 is 444444441 ****** The smallest prime of the form 4{4}6411 is 4444444444444446411 (not minimal prime, since 444444441 and 444641 are primes) ***** For the 4{4,6}461 family, since 661 is prime, we only need to consider the family 4{4}461 ****** The smallest prime of the form 4{4}461 is 4444444461 (not minimal prime, since 444444441 is prime) **** For the 4{4}6{4}1 family, since 6441 is prime, we only need to consider the families 4{4}61 and 4{4}641 (since any digits combo 44 between (4{4}6,1) will produce smaller primes) ***** The smallest prime of the form 4{4}61 is 4444444461 (not minimal prime, since 444444441 is prime) ***** The smallest prime of the form 4{4}641 is 444641 * Case (4,3): ** Since 45, 13, 23, 53, 73, 433, 463 are primes, we only need to consider the family 4{0,4}3 (since any digits 1, 2, 3, 5, 6, 7 between them will produce smaller primes) *** Since 4043 and 4443 are primes, we only need to consider the families 4{0}3 and 44{0}3 (since any digits combo 04, 44 between them will produce smaller primes) **** All numbers of the form 4{0}3 are divisible by 7, thus cannot be prime. **** All numbers of the form 44{0}3 are divisible by 3, thus cannot be prime. * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 27, 37, 57, 407, 417, 467 are primes, we only need to consider the family 4{4,7}7 (since any digits 0, 1, 2, 3, 5, 6 between them will produce smaller primes) *** Since 747 is prime, we only need to consider the families 4{4}7, 4{4}77, 4{7}7, 44{7}7 (since any digits combo 74 between (4,7) will produce smaller primes) **** The smallest prime of the form 4{4}7 is 44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444447, with 220 4's, which can be written as (4^220)7 and equal the prime (4*8^221+17)/7 ([http://factordb.com/index.php?id=1100000000416605822 factordb]) **** The smallest prime of the form 4{4}77 is 4444477 **** The smallest prime of the form 4{7}7 is 47777 **** The smallest prime of the form 44{7}7 is 4477777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777, with 851 7's, which can be written as 44(7^851) and equal the prime 37*8^851-1 ([http://factordb.com/index.php?id=1100000000413677646 factordb]) (not minimal prime, since 47777 is prime) * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 53, 57, 15, 35, 45, 65, 75 are primes, we only need to consider the family 5{0,2,5}5 (since any digits 1, 3, 4, 6, 7 between them will produce smaller primes) *** Since 225, 255, 5205 are primes, we only need to consider the families 5{0,5}5 and 5{0,5}25 (since any digits combo 20, 22, 25 between them will produce smaller primes) **** All numbers of the form 5{0,5}5 are divisible by 5, thus cannot be prime. **** For the 5{0,5}25 family, since 500025 and 505525 are primes, we only need to consider the number 500525 the families 5{5}25, 5{5}025, 5{5}0025, 5{5}0525, 5{5}00525, 5{5}05025 (since any digits combo 000, 055 between (5,25) will produce smaller primes) ***** 500525 is not prime. ***** The smallest prime of the form 5{5}25 is 555555555555525 ***** The smallest prime of the form 5{5}025 is 55555025 ***** The smallest prime of the form 5{5}0025 is 5555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555555550025, with 184 5's, which can be written as (5^183)0025 and equal the prime (5*8^187-20333)/7 ([http://factordb.com/index.php?id=1100000002350205912 factordb]) (not minimal prime, since 55555025 and 555555555555525 are primes) ***** The smallest prime of the form 5{5}0525 is 5550525 ***** The smallest prime of the form 5{5}00525 is 5500525 ***** The smallest prime of the form 5{5}05025 is 5555555555555555555555505025 (not minimal prime, since 5550525, 55555025, and 555555555555525 are primes) * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (6,1): ** Since 65, 21, 51, 631, 661 are primes, we only need to consider the family 6{0,1,4,7}1 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Numbers containing 4: (note that the number cannot contain two or more 4's, or 6441 will be a subsequence) **** The form is 6{0,1,7}4{0,1,7}1 ***** Since 141, 401, 471 are primes, we only need to consider the family 6{0,7}4{1}1 ****** Since 111 is prime, we only need to consider the families 6{0,7}41 and 6{0,7}411 ******* For the 6{0,7}41 family, since 60741 is prime, we only need to consider the family 6{7}{0}41 ******** Since 6777 is prime, we only need to consider the families 6{0}41, 67{0}41, 677{0}41 ********* All numbers of the form 6{0}41 are divisible by 3, thus cannot be prime. ********* All numbers of the form 67{0}41 are divisible by 13, thus cannot be prime. ********* All numbers of the form 677{0}41 are divisible by 3, thus cannot be prime. ******* For the 6{0,7}411 family, since 60411 is prime, we only need to consider the family 6{7}411 ******** The smallest prime of the form 6{7}411 is 67777411 (not minimal prime, since 6777 is prime) *** Numbers not containing 4: **** The form is 6{0,1,7}1 ***** Since 111 is prime, we only need to consider the families 6{0,7}1 and 6{0,7}1{0,7}1 ****** All numbers of the form 6{0,7}1 are divisible by 7, thus cannot be prime. ****** For the 6{0,7}1{0,7}1 family, since 711 and 6101 are primes, we only need to consider the family 6{0}1{7}1 ******* Since 60171 is prime, we only need to consider the families 6{0}11 and 61{7}1 ******** All numbers of the form 6{0}11 are divisible by 3, thus cannot be prime. ******** The smallest prime of the form 61{7}1 is 617771 (not minimal prime, since 6777 is prime) * Case (6,3): ** Since 65, 13, 23, 53, 73, 643 are primes, we only need to consider the family 6{0,3,6}3 (since any digits 1, 2, 4, 5, 7 between them will produce smaller primes) *** All numbers of the form 6{0,3,6}3 are divisible by 3, thus cannot be prime. * Case (6,5): ** 65 is prime, and thus the only minimal prime in this family. * Case (6,7): ** Since 65, 27, 37, 57, 667 are primes, we only need to consider the family 6{0,1,4,7}7 (since any digits 2, 3, 5, 6 between them will produce smaller primes) *** Since 107, 117, 147, 177, 407, 417, 717, 747, 6007, 6477, 6707, 6777 are primes, there cannot be digits combo 00, 10, 11, 14, 17, 40, 41, 47, 70, 71, 74, 77 between them **** If there is 1 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}17 ***** All numbers of the form 6{0}17 are divisible by 3, thus cannot be prime. **** If there is 7 between them, then there cannot be 1, 4, 7 before it and cannot be 0, 1, 4, 7 after it, thus the form will be 6{0}77 ***** All numbers of the form 6{0}77 are divisible by 3, thus cannot be prime. **** If there is neither 1 nor 7 between them, then the form is 6{0,4}7 ***** Since 6007, 407 at primes, we only need to consider the families 6{4}7 and 60{4}7 (since any digits combo 00, 40 between them will produce smaller primes) ****** All numbers of the form 6{4}7 are divisible by 3, 5, or 15, thus cannot be prime. ****** All numbers of the form 60{4}7 are divisible by 21, thus cannot be prime. * Case (7,1): ** Since 73, 75, 21, 51, 701, 711 are primes, we only need to consider the family 7{4,6,7}1 (since any digits 0, 1, 2, 3, 5 between them will produce smaller primes) *** Since 747, 767, 471, 661, 7461, 7641 are primes, we only need to consider the families 7{4,7}4{4}1, 7{7}61, 7{7}7{4,6,7}1 (since any digits combo 46, 47, 64, 66, 67 between them will produce smaller primes) **** For the 7{4,7}4{4}1 family, since 747, 471 are primes, we only need to consider the family 7{7}{4}1 (since any digits combo 47 between (7,4{4}1) will produce smaller primes) ***** The smallest prime of the form 7{7}1 is 7777777777771 ***** The smallest prime of the form 7{7}41 is 777777777777777777777777777777777777777777777777777777777777777777777777777777741, with 79 7's, which can be written as (7^79)41 and equal the prime 8^81-31 ([http://factordb.com/index.php?id=1100000000294462449 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777441, with 84 7's, which can be written as (7^84)441 and equal the prime 8^87-223 ([http://factordb.com/index.php?id=1100000000294462776 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}4441 is 777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777774441, with 233 7's, which can be written as (7^233)4441 and equal the prime 8^237-1759 ([http://factordb.com/index.php?id=1100000002352073382 factordb]) (not minimal prime, since 7777777777771 is prime) ***** The smallest prime of the form 7{7}44441 is 7777777777777777777777777777777777777777777777777777777744441, with 56 7's, which can be written as (7^56)44441 and equal the prime 8^61-14047 ([http://factordb.com/index.php?id=1100000002350250002 factordb]) (not minimal prime, since 7777777777771 is prime) ***** All numbers of the form 7{7}444441 are divisible by 7, thus cannot be prime. ***** The smallest prime of the form 7{7}4444441 is 77774444441 ****** Since this prime has just 4 7's, we only need to consider the families with <=3 7's ******* The smallest prime of the form 7{4}1 is 744444441 ******* All numbers of the form 77{4}1 are divisible by 5, thus cannot be prime. ******* The smallest prime of the form 777{4}1 is 777444444444441 (not minimal prime, since 444444441 and 744444441 are primes) * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 73, 75, 27, 37, 57, 717, 747, 767 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. ===Base 10=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,3), (1,7), (1,9), (2,1), (2,3), (2,7), (2,9), (3,1), (3,3), (3,7), (3,9), (4,1), (4,3), (4,7), (4,9), (5,1), (5,3), (5,7), (5,9), (6,1), (6,3), (6,7), (6,9), (7,1), (7,3), (7,7), (7,9), (8,1), (8,3), (8,7), (8,9), (9,1), (9,3), (9,7), (9,9) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,3): ** 13 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,9): ** 19 is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 23, 29, 11, 31, 41, 61, 71, 251, 281 are primes, we only need to consider the family 2{0,2}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 2221 and 20201 are primes, we only need to consider the families 2{0}1, 2{0}21, 22{0}1 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 2{0}1 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 2{0}21 is 20021 **** The smallest prime of the form 22{0}1 is 22000001 * Case (2,3): ** 23 is prime, and thus the only minimal prime in this family. * Case (2,7): ** Since 23, 29, 17, 37, 47, 67, 97, 227, 257, 277 are primes, we only need to consider the family 2{0,8}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 9 between them will produce smaller primes) *** Since 887 and 2087 are primes, we only need to consider the families 2{0}7 and 28{0}7 (since any digit combo 08 or 88 between them will produce smaller primes) **** All numbers of the form 2{0}7 are divisible by 3, thus cannot be prime. **** All numbers of the form 28{0}7 are divisible by 7, thus cannot be prime. * Case (2,9): ** 29 is prime, and thus the only minimal prime in this family. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,3): ** Since 31, 37, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 3{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,9): ** Since 31, 37, 19, 29, 59, 79, 89, 349 are primes, we only need to consider the family 3{0,3,6,9}9 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 3{0,3,6,9}9 are divisible by 3, thus cannot be prime. * Case (4,1): ** 41 is prime, and thus the only minimal prime in this family. * Case (4,3): ** 43 is prime, and thus the only minimal prime in this family. * Case (4,7): ** 47 is prime, and thus the only minimal prime in this family. * Case (4,9): ** Since 41, 43, 47, 19, 29, 59, 79, 89, 409, 449, 499 are primes, we only need to consider the family 4{6}9 (since any digits 0, 1, 2, 3, 4, 5, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 4{6}9 are divisible by 7, thus cannot be prime. * Case (5,1): ** Since 53, 59, 11, 31, 41, 61, 71, 521 are primes, we only need to consider the family 5{0,5,8}1 (since any digits 1, 2, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 881 is prime, we only need to consider the families 5{0,5}1 and 5{0,5}8{0,5}1 (since any digit combo 88 between them will produce smaller primes) **** For the 5{0,5}1 family, since 5051 and 5501 are primes, we only need to consider the families 5{0}1 and 5{5}1 (since any digit combo 05 or 50 between them will produce smaller primes) ***** All numbers of the form 5{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 5{5}1 is 555555555551 **** For the 5{0,5}8{0,5}1 family, since 5081, 5581, 5801, 5851 are primes, we only need to consider the number 581 ***** 581 is not prime. * Case (5,3): ** 53 is prime, and thus the only minimal prime in this family. * Case (5,7): ** Since 53, 59, 17, 37, 47, 67, 97, 557, 577, 587 are primes, we only need to consider the family 5{0,2}7 (since any digits 1, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 227 and 50207 are primes, we only need to consider the families 5{0}7, 5{0}27, 52{0}7 (since any digits combo 22 or 020 between them will produce smaller primes) **** All numbers of the form 5{0}7 are divisible by 3, thus cannot be prime. **** The smallest prime of the form 5{0}27 is 5000000000000000000000000000027 **** The smallest prime of the form 52{0}7 is 5200007 * Case (5,9): ** 59 is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,3): ** Since 61, 67, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 6{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 6{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,9): ** Since 61, 67, 19, 29, 59, 79, 89 are primes, we only need to consider the family 6{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 6{0,3,6,9}9 and 6{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 6{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 6{0,3,6,9}4{0,3,6,9}9 family, since 409, 43, 6469, 499 are primes, we only need to consider the family 6{0,3,6,9}49 ***** Since 349, 6949 are primes, we only need to consider the family 6{0,6}49 ****** Since 60649 is prime, we only need to consider the family 6{6}{0}49 (since any digits combo 06 between {6,49} will produce smaller primes) ******* The smallest prime of the form 6{6}49 is 666649 ******** Since this prime has just 4 6's, we only need to consider the families with <=3 6's ********* The smallest prime of the form 6{0}49 is 60000049 ********* The smallest prime of the form 66{0}49 is 66000049 ********* The smallest prime of the form 666{0}49 is 66600049 * Case (7,1): ** 71 is prime, and thus the only minimal prime in this family. * Case (7,3): ** 73 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 71, 73, 79, 17, 37, 47, 67, 97, 727, 757, 787 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9 between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,9): ** 79 is prime, and thus the only minimal prime in this family. * Case (8,1): ** Since 83, 89, 11, 31, 41, 61, 71, 821, 881 are primes, we only need to consider the family 8{0,5}1 (since any digits 1, 2, 3, 4, 6, 7, 8, 9 between them will produce smaller primes) *** Since 8501 is prime, we only need to consider the family 8{0}{5}1 (since any digits combo 50 between them will produce smaller primes) **** Since 80051 is prime, we only need to consider the families 8{0}1, 8{5}1, 80{5}1 (since any digits combo 005 between them will produce smaller primes) ***** All numbers of the form 8{0}1 are divisible by 3, thus cannot be prime. ***** The smallest prime of the form 8{5}1 is 8555555555555555555551 (not minimal prime, since 555555555551 is prime) ***** The smallest prime of the form 80{5}1 is 80555551 * Case (8,3): ** 83 is prime, and thus the only minimal prime in this family. * Case (8,7): ** Since 83, 89, 17, 37, 47, 67, 97, 827, 857, 877, 887 are primes, we only need to consider the family 8{0}7 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9 between them will produce smaller primes) *** All numbers of the form 8{0}7 are divisible by 3, thus cannot be prime. * Case (8,9): ** 89 is prime, and thus the only minimal prime in this family. * Case (9,1): ** Since 97, 11, 31, 41, 61, 71, 991 are primes, we only need to consider the family 9{0,2,5,8}1 (since any digits 1, 3, 4, 6, 7, 9 between them will produce smaller primes) *** Since 251, 281, 521, 821, 881, 9001, 9221, 9551, 9851 are primes, we only need to consider the families 9{2,5,8}0{2,5,8}1, 9{0}2{0}1, 9{0}5{0,8}1, 9{0,5}8{0}1 (since any digits combo 00, 22, 25, 28, 52, 55, 82, 85, 88 between them will produce smaller primes) **** For the 9{2,5,8}0{2,5,8}1 family, since any digits combo 22, 25, 28, 52, 55, 82, 85, 88 between (9,1) will produce smaller primes, we only need to consider the numbers 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 901, 9021, 9051, 9081, 9201, 9501, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0}2{0}1 family, since 9001 is prime, we only need to consider the numbers 921, 9201, 9021 ***** None of 921, 9201, 9021 are primes. **** For the 9{0}5{0,8}1 family, since 9001 and 881 are primes, we only need to consider the numbers 951, 9051, 9501, 9581, 90581, 95081, 95801 ***** 95801 is the only prime among 951, 9051, 9501, 9581, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. **** For the 9{0,5}8{0}1 family, since 9001 and 5581 are primes, we only need to consider the numbers 981, 9081, 9581, 9801, 90581, 95081, 95801 ***** 95801 is the only prime among 981, 9081, 9581, 9801, 90581, 95081, 95801, but it is not minimal prime since 5801 is prime. * Case (9,3): ** Since 97, 13, 23, 43, 53, 73, 83 are primes, we only need to consider the family 9{0,3,6,9}3 (since any digits 1, 2, 4, 5, 7, 8 between them will produce smaller primes) *** All numbers of the form 9{0,3,6,9}3 are divisible by 3, thus cannot be prime. * Case (9,7): ** 97 is prime, and thus the only minimal prime in this family. * Case (9,9): ** Since 97, 19, 29, 59, 79, 89 are primes, we only need to consider the family 9{0,3,4,6,9}9 (since any digits 1, 2, 5, 7, 8 between them will produce smaller primes) *** Since 449 is prime, we only need to consider the families 9{0,3,6,9}9 and 9{0,3,6,9}4{0,3,6,9}9 (since any digit combo 44 between them will produce smaller primes) **** All numbers of the form 9{0,3,6,9}9 are divisible by 3, thus cannot be prime. **** For the 9{0,3,6,9}4{0,3,6,9}9 family, since 9049, 349, 9649, 9949 are primes, we only need to consider the family 94{0,3,6,9}9 ***** Since 409, 43, 499 are primes, we only need to consider the family 94{6}9 (since any digits 0, 3, 9 between (94,9) will produce smaller primes) ****** The smallest prime of the form 94{6}9 is 946669 ===Base 12=== The possible (first digit,last digit) combo for a quasi-minimal prime with ≥3 digits are: (1,1), (1,5), (1,7), (1,B), (2,1), (2,5), (2,7), (2,B), (3,1), (3,5), (3,7), (3,B), (4,1), (4,5), (4,7), (4,B), (5,1), (5,5), (5,7), (5,B), (6,1), (6,5), (6,7), (6,B), (7,1), (7,5), (7,7), (7,B), (8,1), (8,5), (8,7), (8,B), (9,1), (9,5), (9,7), (9,B), (A,1), (A,5), (A,7), (A,B), (B,1), (B,5), (B,7), (B,B) * Case (1,1): ** 11 is prime, and thus the only minimal prime in this family. * Case (1,5): ** 15 is prime, and thus the only minimal prime in this family. * Case (1,7): ** 17 is prime, and thus the only minimal prime in this family. * Case (1,B): ** 1B is prime, and thus the only minimal prime in this family. * Case (2,1): ** Since 25, 27, 11, 31, 51, 61, 81, 91, 221, 241, 2A1, 2B1 are primes, we only need to consider the family 2{0}1 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B between them will produce smaller primes) *** The smallest prime of the form 2{0}1 is 2001 * Case (2,5): ** 25 is prime, and thus the only minimal prime in this family. * Case (2,7): ** 27 is prime, and thus the only minimal prime in this family. * Case (2,B): ** Since 25, 27, 1B, 3B, 4B, 5B, 6B, 8B, AB, 2BB are primes, we only need to consider the family 2{0,2,9}B (since any digits 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 90B, 200B, 202B, 222B, 229B, 292B, 299B are primes, we only need to consider the numbers 20B, 22B, 29B, 209B, 220B (since any digits combo 00, 02, 22, 29, 90, 92, 99 between them will produce smaller primes) **** None of 20B, 22B, 29B, 209B, 220B are primes. * Case (3,1): ** 31 is prime, and thus the only minimal prime in this family. * Case (3,5): ** 35 is prime, and thus the only minimal prime in this family. * Case (3,7): ** 37 is prime, and thus the only minimal prime in this family. * Case (3,B): ** 3B is prime, and thus the only minimal prime in this family. * Case (4,1): ** Since 45, 4B, 11, 31, 51, 61, 81, 91, 401, 421, 471 are primes, we only need to consider the family 4{4,A}1 (since any digit 0, 1, 2, 3, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since A41 and 4441 are primes, we only need to consider the families 4{A}1 and 44{A}1 (since any digit combo 44, A4 between them will produce smaller primes) **** All numbers of the form 4{A}1 are divisible by 5, thus cannot be prime. **** The smallest prime of the form 44{A}1 is 44AAA1 * Case (4,5): ** 45 is prime, and thus the only minimal prime in this family. * Case (4,7): ** Since 45, 4B, 17, 27, 37, 57, 67, 87, A7, B7, 447, 497 are primes, we only need to consider the family 4{0,7}7 (since any digit 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** Since 4707 and 4777 are primes, we only need to consider the families 4{0}7 and 4{0}77 (since any digit combo 70, 77 between them will produce smaller primes) **** All numbers of the form 4{0}7 are divisible by B, thus cannot be prime. **** The smallest prime of the form 4{0}77 is 400000000000000000000000000000000000000077 * Case (4,B): ** 4B is prime, and thus the only minimal prime in this family. * Case (5,1): ** 51 is prime, and thus the only minimal prime in this family. * Case (5,5): ** Since 51, 57, 5B, 15, 25, 35, 45, 75, 85, 95, B5, 565 are primes, we only need to consider the family 5{0,5,A}5 (since any digits 1, 2, 3, 4, 6, 7, 8, 9, B between them will produce smaller primes) *** All numbers of the form 5{0,5,A}5 are divisible by 5, thus cannot be prime. * Case (5,7): ** 57 is prime, and thus the only minimal prime in this family. * Case (5,B): ** 5B is prime, and thus the only minimal prime in this family. * Case (6,1): ** 61 is prime, and thus the only minimal prime in this family. * Case (6,5): ** Since 61, 67, 6B, 15, 25, 35, 45, 75, 85, 95, B5, 655, 665 are primes, we only need to consider the family 6{0,A}5 (since any digits 1, 2, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 6A05 and 6AA5 are primes, we only need to consider the families 6{0}5 and 6{0}A5 (since any digit combo A0, AA between them will produce smaller primes) **** All numbers of the form 6{0}5 are divisible by B, thus cannot be prime. **** The smallest prime of the form 6{0}A5 is 600A5 * Case (6,7): ** 67 is prime, and thus the only minimal prime in this family. * Case (6,B): ** 6B is prime, and thus the only minimal prime in this family. * Case (7,1): ** Since 75, 11, 31, 51, 61, 81, 91, 701, 721, 771, 7A1 are primes, we only need to consider the family 7{4,B}1 (since any digits 0, 1, 2, 3, 5, 6, 7, 8, 9, A between them will produce smaller primes) *** Since 7BB, 7441 and 7B41 are primes, we only need to consider the numbers 741, 7B1, 74B1 **** None of 741, 7B1, 74B1 are primes. * Case (7,5): ** 75 is prime, and thus the only minimal prime in this family. * Case (7,7): ** Since 75, 17, 27, 37, 57, 67, 87, A7, B7, 747, 797 are primes, we only need to consider the family 7{0,7}7 (since any digits 1, 2, 3, 4, 5, 6, 8, 9, A, B between them will produce smaller primes) *** All numbers of the form 7{0,7}7 are divisible by 7, thus cannot be prime. * Case (7,B): ** Since 75, 1B, 3B, 4B, 5B, 6B, 8B, AB, 70B, 77B, 7BB are primes, we only need to consider the family 7{2,9}B (since any digits 0, 1, 3, 4, 5, 6, 7, 8, A, B between them will produce smaller primes) *** Since 222B, 729B is prime, we only need to consider the families 7{9}B, 7{9}2B, 7{9}22B (since any digits combo 222, 29 between them will produce smaller primes) **** The smallest prime of the form 7{9}B is 7999B **** The smallest prime of the form 7{9}2B is 79992B (not minimal prime, since 992B and 7999B are primes) **** The smallest prime of the form 7{9}22B is 79922B (not minimal prime, since 992B is prime) * Case (8,1): ** 81 is prime, and thus the only minimal prime in this family. * Case (8,5): ** 85 is prime, and thus the only minimal prime in this family. * Case (8,7): ** 87 is prime, and thus the only minimal prime in this family. * Case (8,B): ** 8B is prime, and thus the only minimal prime in this family. * Case (9,1): ** 91 is prime, and thus the only minimal prime in this family. * Case (9,5): ** 95 is prime, and thus the only minimal prime in this family. * Case (9,7): ** Since 91, 95, 17, 27, 37, 57, 67, 87, A7, B7, 907 are primes, we only need to consider the family 9{4,7,9}7 (since any digit 0, 1, 2, 3, 5, 6, 8, A, B between them will produce smaller primes) *** Since 447, 497, 747, 797, 9777, 9947, 9997 are primes, we only need to consider the numbers 947, 977, 997, 9477, 9977 (since any digits combo 44, 49, 74, 77, 79, 94, 99 between them will produce smaller primes) **** None of 947, 977, 997, 9477, 9977 are primes. * Case (9,B): ** Since 91, 95, 1B, 3B, 4B, 5B, 6B, 8B, AB, 90B, 9BB are primes, we only need to consider the family 9{2,7,9}B (since any digit 0, 1, 3, 4, 5, 6, 8, A, B between them will produce smaller primes) *** Since 27, 77B, 929B, 992B, 997B are primes, we only need to consider the families 9{2,7}2{2}B, 97{2,9}B, 9{7,9}9{9}B (since any digits combo 27, 29, 77, 92, 97 between them will produce smaller primes) **** For the 9{2,7}2{2}B family, since 27 and 77B are primes, we only need to consider the families 9{2}2{2}B and 97{2}2{2}B (since any digits combo 27, 77 between (9,2{2}B) will produce smaller primes) ***** The smallest prime of the form 9{2}2{2}B is 9222B (not minimal prime, since 222B is prime) ***** The smallest prime of the form 97{2}2{2}B is 9722222222222B (not minimal prime, since 222B is prime) **** For the 97{2,9}B family, since 729B and 929B are primes, we only need to consider the family 97{9}{2}B (since any digits combo 29 between (97,B) will produce smaller primes) ***** Since 222B is prime, we only need to consider the families 97{9}B, 97{9}2B, 97{9}22B (since any digit combo 222 between (97,B) will produce smaller primes) ****** All numbers of the form 97{9}B are divisible by 11, thus cannot be prime. ****** The smallest prime of the form 97{9}2B is 979999992B (not minimal prime, since 9999B is prime) ****** All numbers of the form 97{9}22B are divisible by 11, thus cannot be prime. **** For the 9{7,9}9{9}B family, since 77B and 9999B are primes, we only need to consider the numbers 99B, 999B, 979B, 9799B, 9979B ***** None of 99B, 999B, 979B, 9799B, 9979B are primes. * Case (A,1): ** Since A7, AB, 11, 31, 51, 61, 81, 91, A41 are primes, we only need to consider the family A{0,2,A}1 (since any digits 1, 3, 4, 5, 6, 7, 8, 9, B between them will produce smaller primes) *** Since 221, 2A1, A0A1, A201 are primes, we only need to consider the families A{A}{0}1 and A{A}{0}21 (since any digits combo 0A, 20, 22, 2A between them will produce smaller primes) **** For the A{A}{0}1 family: ***** All numbers of the form A{0}1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form AA{0}1 is AA000001 ***** The smallest prime of the form AAA{0}1 is AAA0001 ***** The smallest prime of the form AAAA{0}1 is AAAA1 ****** Since this prime has no 0's, we do not need to consider the families {A}1, {A}01, {A}001, etc. **** All numbers of the form A{A}{0}21 are divisible by 5, thus cannot be prime. * Case (A,5): ** Since A7, AB, 15, 25, 35, 45, 75, 85, 95, B5 are primes, we only need to consider the family A{0,5,6,A}5 (since any digits 1, 2, 3, 4, 7, 8, 9, B between them will produce smaller primes) *** Since 565, 655, 665, A605, A6A5, AA65 are primes, we only need to consider the families A{0,5,A}5 and A{0}65 (since any digits combo 56, 60, 65, 66, 6A, A6 between them will produce smaller primes) **** All numbers of the form A{0,5,A}5 are divisible by 5, thus cannot be prime. **** The smallest prime of the form A{0}65 is A00065 * Case (A,7): ** A7 is prime, and thus the only minimal prime in this family. * Case (A,B): ** AB is prime, and thus the only minimal prime in this family. * Case (B,1): ** Since B5, B7, 11, 31, 51, 61, 81, 91, B21 are primes, we only need to consider the family B{0,4,A,B}1 (since any digits 1, 2, 3, 5, 6, 7, 8, 9 between them will produce smaller primes) *** Since 4B, AB, 401, A41, B001, B0B1, BB01, BB41 are primes, we only need to consider the families B{A}0{4,A}1, B{0,4}4{4,A}1, B{0,4,A,B}A{0,A}1, B{B}B{A,B}1 (since any digits combo 00, 0B, 40, 4B, A4, AB, B0, B4 between them will produce smaller primes) **** For the B{A}0{4,A}1 family, since A41 is prime, we only need consider the families B0{4}{A}1 and B{A}0{A}1 ***** For the B0{4}{A}1 family, since B04A1 is prime, we only need to consider the families B0{4}1 and B0{A}1 ****** The smallest prime of the form B0{4}1 is B04441 (not minimal prime, since 4441 is prime) ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) ***** For the B{A}0{A}1 family, since A0A1 is prime, we only need to consider the families B{A}01 and B0{A}1 ****** The smallest prime of the form B{A}01 is BAA01 ****** The smallest prime of the form B0{A}1 is B0AAAAA1 (not minimal prime, since AAAA1 is prime) **** For the B{0,4}4{4,A}1 family, since 4441 is prime, we only need to consider the families B{0}4{4,A}1 and B{0,4}4{A}1 ***** For the B{0}4{4,A}1 family, since B001 is prime, we only need to consider the families B4{4,A}1 and B04{4,A}1 ****** For the B4{4,A}1 family, since A41 is prime, we only need to consider the family B4{4}{A}1 ******* Since 4441 and BAAA1 are primes, we only need to consider the numbers B41, B441, B4A1, B44A1, B4AA1, B44AA1 ******** None of B41, B441, B4A1, B44A1, B4AA1, B44AA1 are primes. ****** For the B04{4,A}1 family, since B04A1 is prime, we only need to consider the family B04{4}1 ******* The smallest prime of the form B04{4}1 is B04441 (not minimal prime, since 4441 is prime) ***** For the B{0,4}4{A}1 family, since 401, 4441, B001 are primes, we only need to consider the families B4{A}1, B04{A}1, B44{A}1, B044{A}1 (since any digits combo 00, 40, 44 between (B,4{A}1) will produce smaller primes) ****** The smallest prime of the form B4{A}1 is B4AAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B04{A}1 is B04A1 ****** The smallest prime of the form B44{A}1 is B44AAAAAAA1 (not minimal prime, since BAAA1 is prime) ****** The smallest prime of the form B044{A}1 is B044A1 (not minimal prime, since B04A1 is prime) **** For the B{0,4,A,B}A{0,A}1 family, since all numbers in this family with 0 between (B,1) are in the B{A}0{4,A}1 family, and all numbers in this family with 4 between (B,1) are in the B{0,4}4{4,A}1 family, we only need to consider the family B{A,B}A{A}1 ***** Since BAAA1 is prime, we only need to consider the families B{A,B}A1 and B{A,B}AA1 ****** For the B{A,B}A1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}A1 and B{B}AA1 ******* All numbers of the form B{B}A1 are divisible by B, thus cannot be prime. ******* The smallest prime of the form B{B}AA1 is BBBAA1 ****** For the B{A,B}AA1 family, since BAAA1 is prime, we only need to consider the families B{B}AA1 ******* The smallest prime of the form B{B}AA1 is BBBAA1 **** For the B{B}B{A,B}1 family, since AB and BAAA1 are primes, we only need to consider the families B{B}B1, B{B}BA1, B{B}BAA1 (since any digits combo AB or AAA between (B{B}B,1) will produce smaller primes) ***** The smallest prime of the form B{B}B1 is BBBB1 ***** All numbers of the form B{B}BA1 are divisible by B, thus cannot be prime. ***** The smallest prime of the form B{B}BAA1 is BBBAA1 * Case (B,5): ** B5 is prime, and thus the only minimal prime in this family. * Case (B,7): ** B7 is prime, and thus the only minimal prime in this family. * Case (B,B): ** Since B5, B7, 1B, 3B, 4B, 5B, 6B, 8B, AB, B2B are primes, we only need to consider the family B{0,9,B}B (since any digits 1, 2, 3, 4, 5, 6, 7, 8, A between them will produce smaller primes) *** Since 90B and 9BB are primes, we only need to consider the families B{0,B}{9}B **** Since 9999B is prime, we only need to consider the families B{0,B}B, B{0,B}9B, B{0,B}99B, B{0,B}999B ***** All numbers of the form B{0,B}B are divisible by B, thus cannot be prime. ***** For the B{0,B}9B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}9B and B{B}9B (since any digits combo 0B, B0 between (B,9B) will produce smaller primes) ******* The smallest prime of the form B{0}9B is B0000000000000000000000000009B ******* All numbers of the from B{B}9B is either divisible by 11 (if totally number of B's is even) or factored as 10^(2*n)-21 = (10^n-5) * (10^n+5) (if totally number of B's is odd number 2*n-1 (n≥1)) (and since if n≥1, 10^n-5 ≥ 10^1-5 = 7 > 1, 10^n+5 ≥ 10^1+5 = 15 > 1, this factorization is nontrivial), thus cannot be prime. ***** For the B{0,B}99B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}99B and B{B}99B (since any digits combo 0B, B0 between (B,99B) will produce smaller primes) ******* The smallest prime of the form B{0}99B is B00099B ******* The smallest prime of the form B{B}99B is BBBBBB99B ***** For the B{0,B}999B family: ****** Since B0B9B and BB09B are primes, we only need to consider the families B{0}999B and B{B}999B (since any digits combo 0B, B0 between (B,999B) will produce smaller primes) ******* The smallest prime of the form B{0}999B is B0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000999B, with 1765 0's, which can be written as B(0^1765)999B and equal the prime 11*12^1769+16967 ([http://factordb.com/index.php?id=1100000002378273165 factordb]) ([http://factordb.com/cert.php?id=1100000002378273165 primality certificate]) (not minimal prime, since B00099B and B0000000000000000000000000009B are primes) ******* The smallest prime of the form B{B}999B is BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB999B, with 245 B's, which can be written as (B^244)999B and equal the prime 12^248-3769 ([http://factordb.com/index.php?id=1100000002378270237 factordb]) (not minimal prime, since BBBBBB99B is prime) == Examples of families which can be ruled out as contain no primes > ''b'' == It is not known if this problem is solvable: Problem: Given strings ''x'', ''y'', ''z'', and a base ''b'', does there exist a prime number whose base-''b'' expansion is of the form ''x''{''y''}''z''? It will be necessary for our algorithm to determine if families of the form ''x''{''y''}''z'' contain a prime > ''b'' or not. We use two different heuristic strategies to show that such families contain no primes > ''b''. In the first strategy, we mimic the well-known technique of “covering congruences”, by finding some finite set ''S'' of primes ''p'' such that every number in a given family is divisible by some element of ''S''. In the second strategy, we attempt to find an algebraic factorization, such as difference-of-squares factorization, difference-of-cubes factorization, and Aurifeuillian factorization for numbers of the form ''x''⁴+4''y''⁴. Examples of first strategy: (we can show that the corresponding numbers are > all elements in ''S'', if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥1 for 5{1} in base 9 and 2{5} in base 11 and {4}D in base 16 and {8}F in base 16, ''n''≥0 for other examples), thus these factorizations are nontrivial) * In base 10, all numbers of the form 4{6}9 are divisible by 7 * In base 6, all numbers of the form 4{0}1 are divisible by 5 * In base 15, all numbers of the form 9{6}8 are divisible by 11 * In base 9, all numbers of the form 5{1} are divisible by some element of {2, 5} * In base 11, all numbers of the form 2{5} are divisible by some element of {2, 3} * In base 14, all numbers of the form B{0}1 are divisible by some element of {3, 5} * In base 8, all numbers of the form 6{4}7 are divisible by some element of {3, 5, 13} * In base 13, all numbers of the form 3{0}95 are divisible by some element of {5, 7, 17} * In base 16, all numbers of the form {4}D are divisible by some element of {3, 7, 13} * In base 16, all numbers of the form {8}F are divisible by some element of {3, 7, 13} Examples of second strategy: (we can show that both factors are > 1, if ''n'' makes corresponding numbers > ''b'' (i.e. ''n''≥2 for {1} in base 9, ''n''≥0 for 1{0}1 in base 8 and B{4}1 in base 16, ''n''≥1 for other examples), thus these factorizations are nontrivial) * In base 9, all numbers of the form {1} factored as difference of squares * In base 8, all numbers of the form 1{0}1 factored as sum of cubes * In base 9, all numbers of the form 3{8} factored as difference of squares * In base 16, all numbers of the form 8{F} factored as difference of squares * In base 16, all numbers of the form {F}7 factored as difference of squares * In base 9, all numbers of the form 3{1} factored as difference of squares * In base 16, all numbers of the form {4}1 factored as difference of squares * In base 16, all numbers of the form 1{5} factored as difference of squares * In base 16, all numbers of the from {C}D factored as ''x''⁴+4''y''⁴ * In base 16, all numbers of the form B{4}1 factored as difference of squares Examples of combine of the two strategies: (we can show that for the part of the first strategy, the corresponding numbers are > all elements in S, and for the part of the second strategy, both factors are > 1, if n makes corresponding numbers > b, thus these factorizations are nontrivial) * In base 14, numbers of the form 8{D} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 12, numbers of the form {B}9B are divisible by 13 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 14, numbers of the form {D}5 are divisible by 5 if ''n'' is even and factored as difference of squares if ''n'' is odd * In base 17, numbers of the form 1{9} are divisible by 2 if ''n'' is odd and factored as difference of squares if ''n'' is even * In base 19, numbers of the form 1{6} are divisible by 5 if ''n'' is odd and factored as difference of squares if ''n'' is even == Bases 2≤''b''≤1024 such that these families can be ruled out as contain no primes > ''b'' == (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-powers factorization === 1{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 1{0}3 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === 1{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 1{0}5 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 5: Finite covering set {5} === 1{0}6 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === 1{0}7 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 7: Finite covering set {7} === 1{0}z === (none) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === (none) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === * ''b'' == 1 mod 3: Finite covering set {3} === {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {1} === * ''b'' = ''m''^''r'' with ''r''>1: Difference-of-''r''th-powers factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 4 (length 2), 8 (length 3), 16 (length 2), 27 (length 3), 36 (length 2), 100 (length 2), 128 (length 7), 196 (length 2), 256 (length 2), 400 (length 2), 512 (length 3), 576 (length 2), 676 (length 2)) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' == 0 mod 2: Finite covering set {2} === 1{2} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 1{3} === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' such that ''b'' and 3(''b''+2) are both squares: Difference-of-squares factorization (such bases are 25, 361) * ''b'' == 1 mod 2 such that 3(''b''+2) is square: Combine of finite covering set {2} (when length is even) and difference-of-squares factorization (when length is odd) (such bases are 25, 73, 145, 241, 361, 505, 673, 865) === 1{4} === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' such that ''b'' and 4(''b''+3) are both squares: Difference-of-squares factorization === 1{z} === (none) === 2{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} === 2{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} === 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}) === * ''b'' such that ''b'' and 2''b''−1 are both squares: Difference-of-squares factorization (such bases are 25, 841) === {2}1 === * ''b'' such that ''b'' and 2(''b''+1) are both squares: Difference-of-squares factorization (such bases are 49) === 2{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 3{0}2 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 3{0}4 === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 7: Finite covering set {7} === {3}1 === * ''b'' such that ''b'' and 3(2''b''+1) are both squares: Difference-of-squares factorization (such bases are 121) === 3{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 4{0}1 === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ === 4{0}3 === * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 7: Finite covering set {7} === {4}1 === * ''b'' such that ''b'' and 4(3''b''+1) are both squares: Difference-of-squares factorization (such bases are 16, 225) === 4{z} === * ''b'' == 1 mod 2: Finite covering set {2} === 5{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 5{z} === * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} * ''b'' = 6''m''² with ''m'' == 2 or 3 mod 5: Combine of finite covering set {5} (when length is odd) and difference-of-squares factorization (when length is even) (such bases are 24, 54, 294, 384, 864, 1014) === 6{0}1 === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 34 mod 35: Finite covering set {5, 7} === 6{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} === 7{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} === 7{z} === * ''b'' == 1 mod 7: Finite covering set {7} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 83, 307 mod 455: Finite covering set {5, 7, 13} (such bases are 83, 307, 538, 762, 993) * ''b'' = ''m''³: Difference-of-cubes factorization === 8{0}1 === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 20 mod 21: Finite covering set {3, 7} * ''b'' == 47, 83 mod 195: Finite covering set {3, 5, 13} (such bases are 47, 83, 242, 278, 437, 473, 632, 668, 827, 863, 1022) * ''b'' = 467: Finite covering set {3, 5, 7, 19, 37} * ''b'' = 722: Finite covering set {3, 5, 13, 73, 109} * ''b'' = ''m''³: Sum-of-cubes factorization * ''b'' = 128: Cannot have primes since 7''n''+3 cannot be power of 2 === 8{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === 9{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} === 9{z} === * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{0}1 === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 32 mod 33: Finite covering set {3, 11} === A{z} === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{0}1 === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} === B{z} === * ''b'' == 1 mod 11: Finite covering set {11} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 307: Finite covering set {5, 11, 29} * ''b'' = 901: Finite covering set {7, 11, 13, 19} === C{0}1 === * ''b'' == 1 mod 13: Finite covering set {13} * ''b'' == 142 mod 143: Finite covering set {11, 13} * ''b'' = 296, 901: Finite covering set {7, 11, 13, 19} * ''b'' = 562, 828, 900: Finite covering set {7, 13, 19} * ''b'' = 563: Finite covering set {5, 7, 13, 19, 29} * ''b'' = 597: Finite covering set {5, 13, 29} === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === (none) === {y}z === (none) === z{0}1 === (none) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === * ''b'' = ''m''^''r'' with odd ''r''>1: Sum-of-''r''th-power factorization (some bases still have primes, since for the corresponding length this factorization is trivial, but they only have this prime, they are 128 (length 7), 216 (length 3), 343 (length 3), 729 (length 3)) * ''b'' = 4''m''⁴: Aurifeuillian factorization of ''x''⁴+4''y''⁴ (base 4 still have primes, since for the corresponding length this factorization is trivial, but it only have this prime, at length 2) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}t === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 0 mod 7: Finite covering set {7} === {z}u === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} * ''b'' == 1 mod 5: Finite covering set {5} * ''b'' == 34 mod 35: Finite covering set {5, 7} === {z}v === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 5: Finite covering set {5} === {z}w === * ''b'' == 0 mod 2: Finite covering set {2} * ''b'' == 1 mod 3: Finite covering set {3} * ''b'' == 14 mod 15: Finite covering set {3, 5} * ''b'' = ''m''²: Difference-of-squares factorization * ''b'' == 4 mod 5: Combine of finite covering set {5} (when length is even) and difference-of-squares factorization (when length is odd) === {z}x === * ''b'' == 1 mod 2: Finite covering set {2} * ''b'' == 0 mod 3: Finite covering set {3} === {z}y === * ''b'' == 0 mod 2: Finite covering set {2} == Large known (probable) primes (length ≥10000) in these families (for bases 2≤''b''≤1024) == Format: base (length) (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) === 1{0}1 === (none) === 1{0}2 === (none) === 1{0}3 === (none) === 1{0}4 === 53 (13403) 113 (10647) === 1{0}z === 113 (20089) 123 (64371) === 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === (none) === 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) === 208 (26682) 607 (11032) 828 (19659) === 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1) === 201 (31276) 222 (52727) 227 (36323) 327 (135983) 425 (11231) 710 (24112) 717 (37508) 719 (13420) === {1} === 152 (270217) 184 (16703) 200 (17807) 311 (36497) 326 (26713) 331 (25033) 371 (15527) 485 (99523) 629 (32233) 649 (43987) 670 (18617) 684 (22573) 691 (62903) 693 (41189) 731 (15427) 752 (32833) 872 (10093) 932 (20431) === {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}) === (none) === 1{z} === 107 (21911) 170 (166429) 278 (43909) 303 (40175) 383 (20957) 515 (58467) 522 (62289) 578 (129469) 590 (15527) 647 (21577) 662 (16591) 698 (127559) 704 (62035) 845 (39407) 938 (40423) 969 (24097) 989 (26869) === 2{0}1 === 101 (192276) 206 (46206) 218 (333926) 236 (161230) 257 (12184) 305 (16808) 467 (126776) 578 (44166) 626 (174204) 695 (94626) 752 (26164) 788 (72918) 869 (49150) 887 (27772) 899 (15732) 932 (13644) === 2{z} === 432 (16003) === 3{0}1 === 912 (132174) === 3{z} === 72 (1119850) 212 (34414) 218 (23050) 270 (89662) 303 (198358) 312 (51566) 422 (21738) 480 (93610) 513 (38032) 527 (46074) 566 (23874) 650 (498102) 686 (16584) 758 (15574) 783 (12508) 800 (33838) 921 (98668) 947 (10056) === 4{0}1 === 107 (32587) 227 (13347) 257 (160423) 355 (10990) 410 (144079) 440 (56087) 452 (14155) 482 (30691) 542 (15983) 579 (67776) 608 (20707) 635 (11723) 650 (96223) 679 (69450) 737 (269303) 740 (58043) 789 (149140) 797 (468703) 920 (103687) 934 (101404) 962 (84235) === 4{z} === 14 (19699) 68 (13575) 254 (15451) 800 (20509) === 5{0}1 === 326 (400786) 350 (20392) 554 (10630) 662 (13390) 926 (40036) === 5{z} === 258 (212135) 272 (148427) 299 (64898) 307 (26263) 354 (25566) 433 (283919) 635 (36163) 678 (40859) 692 (45447) 719 (20552) 768 (70214) 857 (23083) 867 (61411) 972 (36703) === 6{0}1 === 108 (16318) 129 (16797) 409 (369833) 522 (52604) 587 (24120) 643 (164916) 762 (11152) 789 (27297) 986 (21634) === 6{z} === 68 (25396) 332 (15222) 338 (42868) 362 (146342) 488 (33164) 566 (164828) 980 (50878) 986 (12506) 1016 (23336) === 7{0}1 === 398 (17473) 1004 (54849) === 7{z} === 97 (192336) 170 (15423) 194 (38361) 202 (155772) 282 (21413) 283 (164769) 332 (13205) 412 (29792) 560 (19905) 639 (10668) 655 (53009) 811 (31784) 814 (17366) 866 (108591) 908 (61797) 962 (31841) 992 (10605) 997 (15815) === 8{0}1 === 23 (119216) 53 (227184) 158 (123476) 254 (67716) 320 (52004) 410 (279992) 425 (94662) 513 (19076) 518 (11768) 596 (148446) 641 (87702) 684 (23387) 695 (39626) 785 (900326) 788 (11408) 893 (86772) 908 (243440) 920 (107822) 962 (47222) 998 (81240) 1013 (43872) === 8{z} === 138 (35686) 412 (12154) 788 (11326) 990 (23032) === 9{0}1 === 248 (39511) 592 (96870) === 9{z} === 431 (43574) 446 (152028) 458 (126262) 599 (11776) 846 (12781) === A{0}1 === 173 (264235) 198 (47665) 311 (314807) 341 (106009) 449 (18507) 492 (42843) 605 (12395) 708 (17563) 710 (31039) 743 (285479) 744 (137056) 786 (68169) 800 (15105) 802 (149320) 879 (25004) 929 (13065) 977 (125873) 986 (48279) 1004 (10645) === A{z} === 368 (10867) 488 (10231) 534 (80328) 662 (13307) 978 (14066) === B{0}1 === 710 (15272) 740 (33520) 878 (227482) === B{z} === 153 (21660) 186 (112718) 439 (18752) 593 (16064) 602 (36518) 707 (10573) 717 (67707) === C{0}1 === 68 (656922) 219 (29231) 230 (94751) 312 (21163) 334 (83334) 353 (20262) 359 (61295) 457 (10024) 481 (45941) 501 (20140) 593 (42779) 600 (11242) 604 (17371) 641 (26422) 700 (91953) 887 (13961) 919 (45359) 923 (64365) 992 (10300) === {#}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3) === (none) === {#}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) === (none) === #{z} (for even bases b, # = b/2−1) === (none) === y{z} === 38 (136212) 83 (21496) 113 (286644) 188 (13508) 401 (103670) 417 (21003) 458 (46900) 494 (21580) 518 (129372) 527 (65822) 602 (17644) 608 (36228) 638 (74528) 663 (47557) 723 (24536) 758 (50564) 833 (12220) 904 (13430) 938 (50008) 950 (16248) === z{0}1 === 202 (46774) 251 (102979) 272 (16681) 297 (14314) 298 (60671) 326 (64757) 347 (69661) 363 (142877) 452 (71941) 543 (10042) 564 (38065) 634 (84823) 788 (13541) 869 (12289) 890 (37377) 953 (60995) 1004 (29685) === {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family) === 53 (21942) 124 (16426) 175 (31626) 188 (22036) 316 (48538) 365 (25578) 373 (24006) 434 (10090) 530 (11086) 545 (12346) 560 (15072) 596 (12762) 701 (12576) 706 (10656) 821 (13536) 833 (17116) 966 (14820) 983 (11272) === {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y) === (none) === {z}1 === (none) === {z}y === 317 (13896) == Bases 2≤''b''≤1024 which have these families as unsolved families == Unsolved families are families which are neither primes (>''b'') found nor can be ruled out as contain no primes > ''b'' (using A−Z to represent digit values 10 to 35, z−a to represent digit values ''b''−1 to ''b''−26 (e.g. "z" means 1 in base 2, 2 in base 3, 3 in base 4, ..., 8 in base 9, 9 in base 10, A in base 11, B in base 12, ..., Y in base 35, Z in base 36, ...), only consider bases which these families are interpretable, e.g. digit "7" is only interpretable for bases ≥8, and digit "u" (means ''b''−6) is only interpretable for bases ≥7) 1{0}1: 38, 50, 62, 68, 86, 92, 98, 104, 122, 144, 168, 182, 186, 200, 202, 212, 214, 218, 244, 246, 252, 258, 286, 294, 298, 302, 304, 308, 322, 324, 338, 344, 354, 356, 362, 368, 380, 390, 394, 398, 402, 404, 410, 416, 422, 424, 446, 450, 454, 458, 468, 480, 482, 484, 500, 514, 518, 524, 528, 530, 534, 538, 552, 558, 564, 572, 574, 578, 580, 590, 602, 604, 608, 620, 622, 626, 632, 638, 648, 650, 662, 666, 668, 670, 678, 684, 692, 694, 698, 706, 712, 720, 722, 724, 734, 744, 746, 752, 754, 762, 766, 770, 792, 794, 802, 806, 812, 814, 818, 836, 840, 842, 844, 848, 854, 868, 870, 872, 878, 888, 896, 902, 904, 908, 922, 924, 926, 932, 938, 942, 944, 948, 954, 958, 964, 968, 974, 978, 980, 988, 994, 998, 1002, 1006, 1014, 1016 (length limit: ≥8388608) 1{0}2: 167, 257, 323, 353, 383, 527, 557, 563, 623, 635, 647, 677, 713, 719, 803, 815, 947, 971, 1013 (length limit: 2000) 1{0}3: 646, 718, 998 (length limit: 2000) 1{0}4: 139, 227, 263, 315, 335, 365, 485, 515, 647, 653, 683, 773, 789, 797, 815, 857, 875, 893, 939, 995, 1007 (length limit: 2000) 1{0}5 1{0}6 1{0}7 1{0}8 1{0}9 1{0}A 1{0}B 1{0}C 1{0}D 1{0}E 1{0}F 1{0}G 1{0}z: 173, 179, 257, 277, 302, 333, 362, 392, 422, 452, 467, 488, 512, 527, 545, 570, 575, 614, 622, 650, 677, 680, 704, 707, 734, 740, 827, 830, 851, 872, 886, 887, 902, 904, 908, 929, 932, 942, 947, 949, 962, 973, 1022 (length limit: 2000) 1{0}11 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 198, 213, 318, 327, 353, 375, 513, 591, 647, 732, 734, 738, 759, 948, 951, 957, 1013, 1014 (length limit: 2000) 10{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}): 575 (length limit: 247000) 11{0}1 (not quasi-minimal prime if there is smaller prime of the form 1{0}1): 813, 863, 962, 1017 (length limit: ≥100000) {1}0z (not quasi-minimal prime if there is smaller prime of the form {1} or {1}z): 137, 161, 167, 217, 229, 232, 253, 261, 317, 325, 337, 347, 355, 375, 403, 411, 421, 427, 457, 479, 483, 505, 507, 537, 547, 577, 597, 599, 601, 613, 627, 631, 632, 641, 643, 649, 657, 679, 688, 697, 707, 711, 729, 733, 737, 742, 762, 773, 787, 793, 797, 817, 819, 841, 843, 853, 859, 861, 874, 877, 895, 899, 907, 913, 916, 917, 927, 957, 959, 997, 1003, 1009, 1015, 1017 (length limit: 2000) {1}: 185, 269, 281, 380, 384, 385, 394, 452, 465, 511, 574, 601, 631, 632, 636, 711, 713, 759, 771, 795, 861, 866, 881, 938, 948, 951, 956, 963, 1005, 1015 (length limit: ≥100000) 11{z} (not quasi-minimal prime if there is smaller prime of the form 1{z}) {1}2 (not quasi-minimal prime if there is smaller prime of the form {1}): 31, 61, 91, 93, 143, 247, 253, 293, 313, 329, 371, 383, 391, 393, 403, 415, 435, 443, 451, 491, 493, 513, 523, 527, 537, 541, 553, 565, 581, 587, 601, 613, 615, 623, 627, 635, 663, 729, 735, 757, 763, 775, 783, 823, 843, 865, 873, 877, 883, 897, 931, 941, 943, 955, 983, 1013, 1015, 1021, 1023 (length limit: 2000) {1}z 1{2}: 265, 355, 379, 391, 481, 649, 661, 709, 745, 811, 877, 977 (length limit: 2000) 1{3}: 107, 133, 179, 281, 305, 365, 473, 485, 487, 491, 535, 541, 601, 617, 665, 737, 775, 787, 802, 827, 905, 911, 928, 953, 955, 995 1{4}: 83, 143, 185, 239, 269, 293, 299, 305, 319, 325, 373, 383, 395, 431, 471, 503, 551, 577, 581, 593, 605, 617, 631, 659, 743, 761, 773, 781, 803, 821, 857, 869, 897, 911, 917, 923, 935, 983, 1019 (length limit: 2000) 1{z}: 581, 992, 1019 (length limit: ≥100000) 2{0}1: 365, 383, 461, 512, 542, 647, 773, 801, 836, 878, 908, 914, 917, 947, 1004 (length limit: ≥100000) 2{0}3: 79, 149, 179, 254, 359, 394, 424, 434, 449, 488, 499, 532, 554, 578, 664, 683, 694, 749, 794, 839, 908, 944, 982 (length limit: 2000) 2{1} (not quasi-minimal prime if there is smaller prime of the form {1}): 109, 117, 137, 147, 157, 175, 177, 201, 227, 235, 256, 269, 271, 297, 310, 331, 335, 397, 417, 427, 430, 437, 442, 451, 465, 467, 481, 502, 517, 547, 557, 567, 572, 577, 591, 597, 607, 627, 649, 654, 655, 667, 679, 687, 691, 697, 715, 727, 739, 759, 766, 782, 787, 796, 797, 808, 817, 821, 829, 841, 852, 877, 881, 899, 903, 907, 937, 947, 955, 1007, 1011, 1021 (length limit: 2000) {2}1: 106, 238, 262, 295, 364, 382, 391, 397, 421, 458, 463, 478, 517, 523, 556, 601, 647, 687, 754, 790, 793, 832, 872, 898, 962, 1002, 1021 (length limit: 2000) 2{z}: 588, 972 (length limit: ≥100000) 3{0}1: 718 (length limit: ≥100000) 3{0}2: 223, 283, 359, 489, 515, 529, 579, 619, 669, 879, 915, 997 (length limit: 2000) 3{0}4: 167, 391, 447, 487, 529, 653, 657, 797, 853, 913, 937 (length limit: 2000) {3}1: 79, 101, 189, 215, 217, 235, 243, 253, 255, 265, 313, 338, 341, 378, 379, 401, 402, 413, 489, 498, 499, 508, 525, 535, 589, 591, 599, 611, 621, 635, 667, 668, 681, 691, 711, 717, 719, 721, 737, 785, 804, 805, 813, 831, 835, 837, 849, 873, 911, 915, 929, 933, 941, 948, 959, 999, 1013, 1019 (length limit: 2000) 3{z}: 275, 438, 647, 653, 812, 927, 968 (length limit: ≥100000) 4{0}1: 32, 53, 155, 174, 204, 212, 230, 332, 334, 335, 395, 467, 512, 593, 767, 803, 848, 875, 1024 (length limit: ≥100000) 4{0}3: 83, 88, 97, 167, 188, 268, 289, 293, 412, 419, 425, 433, 503, 517, 529, 548, 613, 620, 622, 650, 668, 692, 706, 727, 763, 818, 902, 913, 937, 947, 958 (length limit: 2000) {4}1: 46, 77, 103, 107, 119, 152, 198, 203, 211, 217, 229, 257, 263, 291, 296, 305, 332, 371, 374, 407, 413, 416, 440, 445, 446, 464, 467, 500, 542, 545, 548, 557, 566, 586, 587, 605, 611, 614, 632, 638, 641, 653, 659, 698, 701, 731, 733, 736, 755, 786, 812, 820, 821, 827, 830, 887, 896, 899, 901, 922, 923, 935, 941, 953, 977, 983, 991, 1004 (length limit: 2000) 4{z}: 338, 998 (length limit: ≥100000) 5{0}1: 308, 512, 824 (length limit: ≥100000) 5{z}: 234, 412, 549, 553, 573, 619, 750, 878, 894, 954 (length limit: ≥100000) 6{0}1: 212, 509, 579, 625, 774, 794, 993, 999 (length limit: ≥100000) 6{z}: 308, 392, 398, 518, 548, 638, 662, 878 (length limit: ≥100000) 7{0}1: (none) 7{z}: 321, 328, 374, 432, 665, 697, 710, 721, 727, 728, 752, 800, 815, 836, 867, 957, 958, 972 (length limit: ≥100000) 8{0}1: 86, 140, 182, 263, 353, 368, 389, 395, 422, 426, 428, 434, 443, 488, 497, 558, 572, 575, 593, 606, 698, 710, 746, 758, 770, 773, 824, 828, 866, 911, 930, 953, 957, 983, 993, 1014 (length limit: ≥100000) 8{z}: 378, 438, 536, 566, 570, 592, 636, 688, 718, 830, 852, 926, 1010 (length limit: ≥100000) 9{0}1: 724, 884 (length limit: ≥100000) 9{z}: 80, 233, 530, 551, 611, 899, 912, 980 (length limit: ≥100000) A{0}1: 185, 338, 417, 432, 614, 668, 773, 863, 935, 1000 (length limit: ≥100000) A{z}: 214, 422, 444, 452, 458, 542, 638, 668, 804, 872, 950, 962 (length limit: ≥100000) B{0}1: 560, 770, 968 (length limit: ≥100000) B{z}: 263, 615, 912, 978 (length limit: ≥100000) C{0}1: 163, 207, 354, 362, 368, 480, 620, 692, 697, 736, 753, 792, 978, 998, 1019, 1022 (length limit: ≥100000) C{z} D{0}1 D{z} E{0}1 E{z} F{0}1 F{z} G{0}1 {&#35;}$ (for bases ''b'' == 1 mod 3, # = (''b''−1)/3, $ = (''b''+2)/3): 808, 829, 859, 1006 (length limit: 2000) {&#35;}$ (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2): 31, 37, 55, 63, 67, 77, 83, 89, 91, 93, 97, 99, 107, 109, 117, 123, 127, 133, 135, 137, 143, 147, 149, 151, 155, 161, 177, 179, 183, 189, 193, 197, 207, 211, 213, 215, 217, 223, 225, 227, 233, 235, 241, 247, 249, 255, 257, 263, 265, 269, 273, 277, 281, 283, 285, 287, 291, 293, 297, 303, 307, 311, 319, 327, 347, 351, 355, 357, 359, 361, 367, 369, 377, 381, 383, 385, 387, 389, 393, 397, 401, 407, 411, 413, 417, 421, 423, 437, 439, 443, 447, 457, 465, 467, 469, 473, 475, 481, 483, 489, 493, 495, 497, 509, 511, 515, 533, 541, 547, 549, 555, 563, 591, 593, 597, 601, 603, 611, 615, 619, 621, 625, 627, 629, 633, 635, 637, 645, 647, 651, 653, 655, 659, 663, 667, 671, 673, 675, 679, 683, 687, 691, 693, 697, 707, 709, 717, 731, 733, 735, 737, 741, 743, 749, 753, 755, 757, 759, 765, 767, 771, 773, 775, 777, 783, 785, 787, 793, 797, 801, 807, 809, 813, 817, 823, 825, 849, 851, 853, 865, 867, 873, 877, 887, 889, 893, 897, 899, 903, 907, 911, 915, 923, 927, 933, 937, 939, 941, 943, 945, 947, 953, 957, 961, 967, 975, 977, 983, 987, 993, 999, 1003, 1005, 1009, 1017 (length limit: ≥262143) &#35;{z} (for even bases ''b'', # = ''b''/2−1): 108, 278, 296, 338, 386, 494, 626, 920 (length limit: 2000) ${&#35;} (for odd bases ''b'', # = (''b''−1)/2, $ = (''b''+1)/2) x{z} y{z}: 128, 233, 268, 383, 478, 488, 533, 554, 665, 698, 779, 863, 878, 932, 941, 1010 (length limit: ≥200000) z{0}1: 123, 342, 362, 422, 438, 479, 487, 512, 542, 602, 757, 767, 817, 830, 872, 893, 932, 992, 997, 1005, 1007 (length limit: ≥100000) {y}z: 143, 173, 176, 213, 235, 248, 253, 279, 327, 343, 353, 358, 373, 383, 401, 413, 416, 427, 439, 448, 453, 463, 481, 513, 522, 527, 535, 547, 559, 565, 583, 591, 598, 603, 621, 623, 653, 659, 663, 679, 691, 698, 711, 743, 745, 757, 768, 785, 793, 796, 801, 808, 811, 821, 835, 845, 847, 853, 856, 883, 898, 903, 927, 955, 961, 971, 973, 993, 1005, 1013, 1019, 1021 (length limit: 2000) {z0}z1 (almost cannot be quasi-minimal prime, since this is not simple family): 97, 103, 113, 186, 187, 220, 304, 306, 309, 335, 414, 416, 428, 433, 445, 459, 486, 498, 539, 550, 557, 587, 592, 597, 598, 617, 624, 637, 659, 665, 671, 677, 696, 717, 726, 730, 740, 754, 766, 790, 851, 873, 890, 914, 923, 929, 943, 944, 965, 984, 985, 996, 1004, 1005 (length limit: ≥17326) zy{z} (not quasi-minimal prime if there is smaller prime of the form y{z}) {z}yz (not quasi-minimal prime if there is smaller prime of the form {z}y): 215, 353, 517, 743, 852, 899, 913 (length limit: 2000) {z}01 (not quasi-minimal prime if there is smaller prime of the form {z}1) {z}1: 93, 113, 152, 158, 188, 217, 218, 226, 227, 228, 233, 240, 275, 278, 293, 312, 338, 350, 353, 383, 404, 438, 464, 471, 500, 533, 576, 614, 641, 653, 704, 723, 728, 730, 758, 779, 788, 791, 830, 878, 881, 899, 908, 918, 929, 944, 953, 965, 968, 978, 983, 986, 1013 (length limit: 2000) {z}k {z}l {z}m {z}n {z}o {z}p {z}q {z}r {z}s {z}t {z}u {z}v {z}w: 207, 221, 293, 375, 387, 533, 633, 647, 653, 687, 701, 747, 761, 785, 863, 897, 905, 965, 1017 (length limit: 2000) {z}x: (none) {z}y: 305, 353, 397, 485, 487, 535, 539, 597, 641, 679, 731, 739, 755 (length limit: 2000) == List of lengths for quasi-minimal primes in some simple families == [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list of lengths for quasi-minimal primes in some simple families for bases 2≤''b''≤1024] NB: this family is not interpretable in this base (e.g. family 7{0}1 and 7{z} in bases <=7, family {z}x in bases <=3) (including the case which this family has either leading zeros (leading zeros do not count) or ending zeros (numbers ending in zero cannot be prime > base) in this base) RC: this family can be proven to only contain composite numbers (only count numbers > base) unknown: this family has no primes or PRPs found, nor can this family be proven to only contain composite numbers (only count numbers > base) Background color: red for title (bases or families), green for length > 10000, orange for 2500 < length ≤ 10000, white for length ≤ 2500, cyan for "RC", pink for "NB", yellow for "unknown". Search limit for lengths: ≥8388608 for 1{0}1, ≥200000 for y{z}, ≥100000 for ''d''{0}1 (''d'' = one of digits in {2, 3, 4, 5, 6, 7, 8, 9, A, B, C}) and ''d''{z} (''d'' = one of digits in {1, 2, 3, 4, 5, 6, 7, 8, 9, A, B}) and z{0}1 and {1}, ≥5000 for 1{0}2, {z}y, 1{0}z, {z}1, {y}z, ≥2500 for other families. == References == * [https://mersenneforum.org/showthread.php?t=24972 mersenneforum thread of this problem] * [https://docs.google.com/document/d/e/2PACX-1vQct6Hx-IkJd5-iIuDuOKkKdw2teGmmHW-P75MPaxqBXB37u0odFBml5rx0PoLa0odTyuW67N_vn96J/pub Minimal elements for the base ''b'' representations of the primes which are > ''b'' for bases ''b''≤16] * [https://primes.utm.edu/glossary/xpage/MinimalPrime.html article “minimal prime” in The Prime Glossary] * [https://en.wikipedia.org/wiki/Minimal_prime_(recreational_mathematics article “minimal prime” in Wikipedia] * [https://www.primepuzzles.net/puzzles/puzz_178.htm the puzzle of minimal primes (when the restriction of prime>base is not required) in The Prime Puzzles & Problems Connection] * [https://www.primepuzzles.net/problems/prob_083.htm the problem of minimal primes in The Prime Puzzles & Problems Connection] * [https://github.com/xayahrainie4793/non-single-digit-primes my data for these M(Lb) sets for 2 ≤ b ≤ 16] * [http://www.cs.uwaterloo.ca/~shallit/Papers/minimal5.pdf Shallit’s proof of base 10 minimal primes, when the restriction of prime>base is not required] * [https://scholar.colorado.edu/downloads/hh63sw661 proofs of minimal primes in bases b≤10, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/reports/mepn.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://cs.uwaterloo.ca/~cbright/talks/minimal-slides.pdf the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://doi.org/10.1080/10586458.2015.1064048 the article for this minimal prime problem in bases b≤30, when the restriction of prime>base is not required] * [https://github.com/curtisbright/mepn-data data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 30, when the restriction of prime>base is not required, search limits of lengths: 1000000 for b=17, 707000 for b=19, 506000 for b=21, 292000 for b=25, 486000 for b=26, 543000 for b=28, 233000 for b=29] * [https://github.com/RaymondDevillers/primes data for these M(Lb) sets and unsolved families for 2 ≤ b ≤ 50, when the restriction of prime>base is not required, search limits of lengths: 10000 for all b] * [http://www.bitman.name/math/article/730 article for minimal primes, when the restriction of prime>base is not required] * [http://www.bitman.name/math/table/497 data for minimal primes in bases 2 ≤ b ≤ 16, when the restriction of prime>base is not required] * [http://www.prothsearch.com/sierp.html the Sierpinski problem] * [http://www.prothsearch.com/rieselprob.html the Riesel problem] * [https://oeis.org/A076336/a076336c.html the dual Sierpinski problem] * [http://www.noprimeleftbehind.net/crus/Sierp-conjectures.htm generalized Sierpinski conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/Riesel-conjectures.htm generalized Riesel conjectures in bases b≤1030, some primes found in these conjectures are minimal primes in base b, especially, all primes for k&lt;b (if exist for a (k,b) combo) are always minimal primes in the base b) (also some examples for simple families contain no primes &gt; b] * [http://www.noprimeleftbehind.net/crus/tab/CRUS_tab.htm list for the status of the generalized Sierpinski conjectures and the generalized Riesel conjectures in bases b≤1030] * [https://www.utm.edu/staff/caldwell/preprints/2to100.pdf article for generalized Sierpinski conjectures in bases b≤100] * [http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/i61/i61.pdf article for the mixed (original+dual) Sierpinski problem] * [http://www.fermatquotient.com/PrimSerien/GenRepu.txt generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤160, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://web.archive.org/web/20021111141203/http://www.users.globalnet.co.uk/~aads/primes.html generalized repunit primes (primes of the form (bn−1)/(b−1)) in bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://jeppesn.dk/generalized-fermat.html generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1000, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.noprimeleftbehind.net/crus/GFN-primes.htm generalized Fermat primes (primes of the form b2^n+1) in even bases b≤1030, the smallest such prime for base b (if exists) is always minimal prime in base b] * [http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt list of generalized half Fermat primes (primes of the form (b2^n+1)/2) sorted by n, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://harvey563.tripod.com/wills.txt primes of the form (b−1)*bn−1 for bases b≤2049, the smallest such prime for base b (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MM_least the smallest primes of the form (b−1)*bn−1 for bases b≤2049, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Williams_prime_MP_least the smallest primes of the form (b−1)*bn+1 for bases b≤1024, these primes (if exists) is always minimal prime in base b] * [https://www.rieselprime.de/ziki/Riesel_prime_small_bases_least_n the smallest primes of the form k*bn−1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://www.rieselprime.de/ziki/Proth_prime_small_bases_least_n the smallest primes of the form k*bn+1 for k≤12 and bases b≤1024, these primes (if exists) is always minimal prime in base b if b>k] * [https://docs.google.com/spreadsheets/d/e/2PACX-1vTKkSNKGVQkUINlp1B3cXe90FWPwiegdA07EE7-U7sqXntKAEQrynoI1sbFvvKriieda3LfkqRwmKME/pubhtml list for the smallest primes in given simple family in bases b≤1024] * [https://www.rose-hulman.edu/~rickert/Compositeseq/ a problem related to this project] * [http://www.worldofnumbers.com/Appending%201s%20to%20n.txt a problem related to this project] * [https://stdkmd.net/nrr/prime/primecount.txt near- and quasi- repdigit (probable) primes sorted by count] * [https://stdkmd.net/nrr/prime/primedifficulty.txt near- and quasi- repdigit (probable) primes sorted by difficulty] * [http://www.prothsearch.com/fermat.html factoring status of Fermat numbers] * [http://www.rieselprime.de/dl/CRUS_pack.zip srsieve, sr1sieve, sr2sieve, pfgw, and llr softwares] * [https://www.bc-team.org/app.php/dlext/?cat=3 srsieve, sr1sieve, sr2sieve, sr5sieve software] * [https://sourceforge.net/projects/openpfgw/ pfgw software] * [http://jpenne.free.fr/index2.html llr software] * [http://www.ellipsa.eu/public/primo/primo.html PRIMO software] * [https://primes.utm.edu/prove/index.html website for primality proving] * [https://primes.utm.edu/curios/page.php?number_id=22380 the largest base 10 minimal prime in Prime Curios!] * [https://oeis.org/A071062 OEIS sequence for base 10 minimal primes, when the restriction of prime>base is not required] * [https://oeis.org/A326609 OEIS sequence for the largest base b minimal prime, when the restriction of prime>base is not required] * [https://primes.utm.edu/primes/lists/all.txt top proven primes] * [http://www.primenumbers.net/prptop/prptop.php top PRPs] * [http://factordb.com online factor database, including many primes which are minimal primes in a small base] 31nix5id4aqs8q56s6kz4snyqtmnqy2 0 273015 2416340 2269745 2022-08-18T22:59:36Z Sophivorus 85690 Remove myself from the board, but feel free to contact me or specific technical needs wikitext text/x-wiki <noinclude>{{WikiJ top menu}} __NOTOC__ :''See also [[meta:WikiJournal_User_Group#Administrative_board]]'' The administrative board of WikiJournal consists of users who participate in common interests of the journals. Members of the administrative board also direct the creation and/or acceptance of additional journals in this group. It has a more administrative role such as financials, while the editorial boards of individual journals are more of subject-specific nature such as acceptance of articles. Discussions should take place on this [[Talk:WikiJournal/WikiJournal User Group|Discuss page]] or at the [https://lists.wikimedia.org/mailman/listinfo/wikijournal-en public mailing list]. Important discussions or decisions should be shared with individual journal boards as well. Membership in the administrative is ''not'' a requirement to contribute to WikiJournal or any of the included journals, even as a journal official such as a board member. For more information, see [[WikiJournal User Group/Bylaws|Bylaws of WikiJournal]]. {{wjh_h2|Administrative board members}} </noinclude>{{div col|colwidth=35em}} {{editor info | Q = Q38382414 |'''Chair''' of the administrative board since June 2018. Biochemistry researcher in Australia. Editor-in-chief of ''[[Wikiversity:WikiJournal of Science|Wiki.J.Science]]''. Also editor for ''[[v:WikiJournal of Medicine|Wiki.J.Med]]'' and [http://topicpageswiki.plos.org ''PLOS'' Topics Pages]. }} {{editor info | Q = Q38381954 | ''[[Wikipedia:Doctor_of_Medicine#United_States_and_Canada|MD]]''. Resident physician in pathology in Sweden. WikiJournal tasks: Editor-in-chief of ''[[v:WikiJournal of Medicine|Wiki.J.Med]]'', technical administrator of domain names and email lists, Facebook administrator of WikiJMed, and keeping financials. }} {{editor info | Q = Q41531086 |Senior Lecturer in Writing Studies in Australia. Also editor-in-chief of ''[[v:WikiJournal of Humanities|Wiki.J.Hum]]''. }} {{editor info | Q = Q55120687 |Associate professor of Physics at ''[[v:Wright State University Lake Campus|Wright State University Lake Campus]]''. Taught college physics for 32 years and have written 23 papers in plasma physics, musical acoustics and other topics related to university-level teaching. Founded the ''[[v:Second Journal of Science|Second Journal of Science]]'' (which now redirects to ''[[Wikiversity:WikiJournal of Science|Wiki.J.Science]]''.) }} {{editor info | Q = Q20895785 | Data scientist at the [[w:University of Virginia|University of Virginia]]. Also editor for [http://topicpageswiki.plos.org ''PLOS'' Topics Pages]. }} {{editor info | Q = Q42421780 }} {{editor info | Q = Q75778741 }} {{editor info | Q = Q99632752 | accomplishments and publications can be found using Google Scholar or other searches with "HA Hoff" or "Henry A. Hoff", the biology and psychology are not mine as yet. }} {{editor info | Q = Q106098814 | with a particular focus on HR }} {{div col end}}<noinclude> {{wjs_h2|Role details}} ===Main tasks=== Administrative, HR, and organisational tasks as descibed in the [[WikiJournal_User_Group/Bylaws|WikiJournal User Group bylaws]]. {{Clickable button 2 |Apply to the administrative board |url=https://en.wikiversity.org/w/index.php?title=Talk:{{PAGENAMEE}}&action=edit&section=new&preload={{ROOTPAGENAMEE}}%2FAdministrative_board%2FApplication&summary=Administrative+board+application |class=mw-ui-progressive }} </noinclude> [[Category:WikiJournal User Group]] m109er6ypf6qk3inyvpu07jmiwvcr2k 2 273648 2416162 2415673 2022-08-18T13:49:23Z SophiaSemper 2916858 /* Recent developments */ wikitext text/x-wiki <big> Ectopolitan • 外局人 • एक्टोपॉलिटन • Ectopolita • ექტოპოლიტი • Ektopolitas • Ektopolīts • Ектополитен • Ectopolite • خارجبوليتان • বাহিরেবিশ্বজনীন • Эктополит • Ektopolitan • ایکٹوپولیٹن • Ektopolit • エクトポリタン • اکتوپولیتن • ਐਕਟੋਪੋਲੀਟਨ • ఎక్టోపాలిటన్ • 엑토폴리탄 • Người ngoài • எக்டோபாலிட்டன் • خارج وطنی • ความเป็นข้างนอก • બહારગરિક • ኤክቶፖሊታን • Waje siyasa • Ektopolityczny • ಎಕ್ಟೋಪಾಲಿಟನ್ • Ектополіт • എക്ടോപൊളിറ്റൻ • ଏକ୍ଟୋପଲିଟାନ୍ • اکتوشهری • ນອກໂລກ • Ectopolitische • এক্টোপলিটান • ඒක්ටොපොලිටන් • Ektopolît • Εκτοπολίτικος • Ektopolita • Ektopolitní • Ectopolit • Ектополитска • Ektopolitiska • ኢክቶፖሊታን • Ìta olóṣèlú • Ngaphandle kwepolitiki • Ngaphandle kwezopolitiko • Ektopolitiese • Эктапаліт • Էկտոպոլիտան • Ektopolitisk • Ектополит • Ektopoliittinen • Ektopolitiske • Ektopolitný • ཕྱིའི་མི་སྣ • אַקְטוֹפּוֹלִיטִי • Ektopoliit • Utanborgari • Ektopolito • A-muigh-poilitigs • Waho-politika • Jawa llaqta • ᐁᒃᑐᐳᓕᑕᓐ • Ectopolites • ɒɈiloqoɈɔɘ '''''Nota Bene'': the article below may not be published in formats permissible for citizens of UN member countries, or even for cosmopolitans, and/or for stateless people, as the human being below is an [http://5wcwiki.pbworks.com/w/page/147494235/ectopolitan ''ectopolitan''], which means that he does not identify or align with, nor permits traditional politics, and/or business, to influence or undermine his life, his natural affections, his natural affiliations, and/or his natural community. The individual below considers all UN member countries as no more dignified than Aristotelian zoos, which constantly violate human rights, treat ''de jure'' citizens like ''de facto'' slaves, and national and multinational corporations as entities authorised to violate human rights by their respective states or countries (ie, as criminal entities), and with no actual legal accountability in the constitutions of states or countries for their crimes against natural human beings.''' {|width="280" cellspacing="0" cellpadding="4" border="1" align="right" style="margin: 0 0 1em 1em; border: 1px #aaaaaa solid; border-collapse: collapse; font-size: 95%;" |style="background: #f9f9f9; text-align: center;" colspan="2"|<big><br>'''Make natural science,<br>not war on nature.<br>&mdash; ''Kaisiris Tallini''<br><br>'''</big> |- |style="background: #E8E8F8; text-align: center;" colspan="2"|<br><big>'''Meddíss Túvtiks Kaisiris Tallini'''</big><br />''born''<br>'''Hon Most Rev Dr Cesidio Tallini'''</br></br> |-bgcolor="#f9f9f9" |style="text-align: center;" colspan="2"|[[File:Hmrdct.jpg|270px]]<br />Kaisiris in 2009 |- |style="background: #f9f9f9; text-align: center;" colspan="2"|'''Personal info under Cesidian law''' |-bgcolor="#f9f9f9" |'''Ectopolitan name''' |Kaisiris Tallini |-bgcolor="#f9f9f9" |'''Born again''' |Week K2<br>Neptunday 22 Jung 2013<br>460 millidays<br>(K2Ψ22J2013•460) |-bgcolor="#f9f9f9" |'''Pythagorean name''' |Cesidio Tallini |-bgcolor="#f9f9f9" |'''Born''' |Week S1<br>Neptunday 26 Edison 1962<br>229 millidays<br>(S1Ψ26E1962•229) |-bgcolor="#f9f9f9" |'''Spouses or Partners (''stirpes'')''' |None |-bgcolor="#f9f9f9" |'''Clan name or ''gentilicius'' (''gens'')''' |Kaisiris (Kaisiris Clan), INN (Interfaith Nutrition Network) |-bgcolor="#f9f9f9" |'''Demonym or gentilic (''curia'')''' |Cesidian or Holocartesian (Cesidian Network) |-bgcolor="#f9f9f9" |'''Nationalities (''tribus'')''' |Neocartesian (Saint René Descartes University), Cal Alumnus (Cal Alumni Association), Penn State Alumnus (Penn State Alumni Association), Honourary Antarctican (by affiliation) |-bgcolor="#f9f9f9" |'''Citizenships (''cives'')''' |Pythagorean (Centre for Cesidian Law and Paradiplomatic Affairs), Nature Israel conservationist (American Society for the Protection of Nature in Israel) |-bgcolor="#f9f9f9" |'''Delineation (''genus'')''' |[http://5wcwiki.pbworks.com/w/page/147494235/ectopolitan ''Ectopolitan''] |-bgcolor="#f9f9f9" |'''Ectoprofessions''' |Analytic theologian; multidisciplinary scholar |-bgcolor="#f9f9f9" |'''Website''' |[http://5wcwiki.pbworks.com/w/page/143342355/mtkt >>LINK<<] |} '''MT<ref>Kaisiris Tallini is neither a 'Mister' (Mr), nor a 'Doctor' (Dr). These terms are wholly political or professional terms which states and countries have invented, which in fact treat adult persons as slaves or subjects of states or countries, not as independent and dignity-deserving adults. Kaisiris Tallini is a 'MT', an abbreviation for the Oscan words «''Meddíss Túvtiks''». The ''Meddíss Túvtiks'' — Oscan-Latinish: «''Meddix Tuticus''»; English: Community Manager — was the name of the supreme magistrate among the Oscan people. The MT was the single magistrate on whom would devolve military command, civil and criminal jurisdiction, functions of political and religious representation, and several other responsibilities, for the MT could also have been a great teacher and healer, besides being a man of religion. Think of a real tribal chieftain or the tribune of a natural society, and you can begin to understand what a 'MT' was. Kaisiris Tallini is also a 'SD', an abbreviation for the Ectoenglish words 'Salubriology Doyen'. An SD in the health field — there are also SDs in the organisational, engineering, education, legal, and religious fields — is a genuine physician, or ''iatros'' (ιατρός) in Ectogreek. Kaisiris Tallini considers the appellative 'Mr Tallini' offensive, besides being grossly inaccurate, because it should imply that he is a master of the Tallinis, or of any adult person with a Tallini surname, yet he considers the Tallinis related to him as lower than beasts (reptiles) on the human spiritual scale, or completely vile, vulgar, and plutocratic.</ref> [http://5wcwiki.pbworks.com/w/page/144142176/Kaisiris%20multilingual Kaisiris Tallini]''' (New York, [https://takemeback.to/10-May-1962 10 May 1962] – Ω) — born ''Cesidio Tallini''<ref>The name 'Cesidio Tallini' has existed since Thursday, 10 May 1962 [S1Ψ26E1962•229], or since his birthdate. It should be noted, however, that the name 'Kaisiris Tallini' has existed only since Tuesday, 3 September 2013 [K2Ψ22J2013•460], when it was registered for the first time through the Paradiplomatic Affairs ectogovernmental organisation (EGO).</ref> — is Analytic Theologian and founder of the Cesidian Network; Chancellor of Saint René Descartes University; and Rector of the Centre for Cesidian Law. Kaisiris is the world's first [https://www.britannica.com/topic/micronation micronationalist] recognised in ''de jure'', or acknowledged in ''de facto'' fashion by several government or quasi-government bodies<ref>[http://selfherald.blogspot.com/2017/11/who-is-cesidio-tallini.html Who is Cesidio Tallini?] article.</ref>. He is also the first micronationalist to be recognised for multiple nationalities within the same developing new region of the world, so Kaisiris is both a micronationalist and a multinationalist. Kaisiris has assisted two micronations — ''Principado Independente do Ilhéu da Pontinha'', and Kingdom of Ourania — with their development; claims two Native African — [https://en.wikipedia.org/wiki/Republic_of_Cabinda Republic of Cabinda] and [https://en.wikipedia.org/wiki/Southern_Cameroons Southern Cameroons (Ambazonia)] — nationalities by naturalisation; and claims to be the world's first full Antarctic national by affiliation. Since 2 April 2021, Kaisiris is also the world's first ''de jure'' [http://5wcwiki.pbworks.com/w/page/147494235/ectopolitan ''ectopolitan''], a word which comes from the Greek, and describes a person outside of politics altogether, a ''delineate'' of the world, rather than a citizen of the world. An ectopolitan can be described as a spiritual, and/or physical expatriate to all countries; a wannabe extraterrestrial; an earthling who is even more odd to locals than a cosmopolitan; or someone who doesn't feel related to the United Nations, and/or any of its member countries. Kaisiris has earned both American and Italian high school diplomas in the past; has earned several different post-secondary education diplomas and certificates, including a National Work Readiness Credential from the National Work Readiness Council, and a Certificate of Qualification to Practice from the Law Society of Cabinda; and has earned a BS degree from the University of Phoenix, another BSc degree, two religious doctorate degrees, and was awarded two honourary doctorates. Kaisiris has also recently earned a degree from Saint René Descartes University, after becoming the world's first self-made Salubriology Doyen (SD), so besides being ''Kohen Gadol'' [כּוֹהֵן גָדוֹל] of Lineage of the Blessed Hermes (LBH), and ''Meddíss Túvtiks'' of the Kaisiris Clan, he is also ''Archiatros'' (Αρχιατρός) of the Salubriology School. Kaisiris is also the ''de facto'' or ''de jure'' author of several books in English and Italian; the father of Cesidian law, Cesidian analytic theology, Cesidian salubriology, Cesidian societal-political science, and Cesidian spiritual science; the developer of several indigenous and novel time, linguistic, educational, and intellectual property systems; and an independent scholar. Finally, Kaisiris is also Regent of Paradiplomatic Affairs (PdA), and ''Wujieren'' (无界人) and founder of Global Earth Oceans (GEO), a new region under development. ==Early life== Kaisiris was born in Jamaica Hospital, New York, on 10 May 1962. He has resided for extensive periods of time both in Italy and in the United States. He moved to Italy at the age of 9, where he continued his education. He eventually earned an Italian ''Diploma di Maturità Scientifica'' from La Scuola D'Italia "Guglielmo Marconi" of New York City, and an American High School Equivalency Diploma from the University at Albany (SUNY). ==Later developments== Kaisiris is an honourary alumnus of the University of California at Berkeley since October 2010; an honourary alumnus of Pennsylvania State University since May 2011; and an international life member of the Chamber of Computer Logistics People (CCLP) Worldwide since August 2011. Most post-1998 developments in Kaisiris' life are at least indirectly related to Cesidian societal-political science, although even more legal, theological, and linguistics developments have taken place through time. Kaisiris co-founded the Cesidian Root in September 2005, but no longer has a relationship with the organisation<ref>[https://www.foxnews.com/tech/the-cesidian-root-a-bizarre-peek-at-the-world-wide-weird The Cesidian Root: A bizarre peek at the world wide weird] article.</ref><ref>[https://second.wiki/wiki/cesidian_root Cesidian Root] wiki article.</ref>. On 30 April 2007, Kaisiris earned a Bachelor of Science in Information Technology from the University of Phoenix (with honours). On 13 December 2007, [https://en.wikipedia.org/wiki/Ren%C3%A9_Descartes René Descartes] is consecrated as the first saint of the Cesidian Church. Kaisiris also founded the United Micronations Multi-Oceanic Archipelago (UMMOA) political party in May 2008, but disbanded the party in June 2017. On 8 August 2009, Kaisiris issued the first Print Monopoly ⓟⓜ. On 20 January 2010, Kaisiris received an honourary Doctor of Philosophy (PhD) in International law from the Academy of Universal Global Peace. On 8 July 2011, Kaisiris was ordained as an Independent Catholic Priest and appointed to the title of Bishop, with apostolic succession and lineage to Christ. He also became a member of the Board of Bishops of the Universal Life Church World Headquarters (later renamed "Universal One Church"). On 22 July 2011, the Cesidian Church was granted a church charter by the Universal Life Church World Headquarters (later renamed "Universal One Church"). Kaisiris has also earned a Doctor of Divinity (DD), and a Doctor of Theology (ThD) from the Universal Life Church World Headquarters School of Theology. On 1 November 2011, upon certification by the Examining Committee of the Law Society of Cabinda, Kaisiris was admitted as an Advocate, and is now authorised to practice before all the Courts and Tribunals of the Republic of Cabinda. On 23 January 2012, Kaisiris was appointed Communications Director for the Organization of Emerging African States (OEAS). During the 7th Meeting of the Committee on NGOs, held on 2 February 2012 at the United Nations Headquarters in New York, the Chamber of Computer Logistics People Worldwide (CCLP Worldwide) was recommended for Special Consultative Status. This development became possible after CCLP Worldwide's Special Representative Cesidio Tallini managed to overcome to the persistent objections of the Distinguished Representative of Pakistan on the night of 1 February 2012. In August 2012, CCLP Worldwide was granted Special Consultative Status with the United Nations' Economic and Social Council (ECOSOC). On 27 December 2012, Kaisiris was made Ambassador at Large of Antarctica and Unrepresented or Underrepresented Polities for The Multipurpose Inter-Parliamentary Union (TMIPU). On 31 December 2012, Kaisiris was appointed Interim Deputy Minister of Information for the Republic of Cabinda. On 8 April 2013, the Saint René Descartes University's 2013 Nobel Peace Prize recommendation is welcomed by the Norwegian Nobel Committee, and Saint René Descartes University's Scholar's Degree (SD) is also accepted. On 7 June 2013, the Paradiplomatic Affairs (PDA) agency was launched. On 3 September 2013, the Hernici (8ED) tribe is formally registered with Paradiplomatic Affairs (PDA). This marks the beginning of the official existence of the Aboriginal European Hernici (8ED) Tribe. On 22 September 2013, the Government of Southern Cameroons recognises the UMMOA's Aphrodite Island claim. On 9 January 2014, Kaisiris provides the first services as Ambassador at Large of Antarctica to a Native Antarctican. On 3 October 2014, it is first noticed that UMMOA English (UGV; en-UGV; ⓤ) had appeared authoritatively in Google searches. What was originally judged as a new English variety, variant, or lect†, since 30 December 2020 is called Ectoenglish (ⓔ), the world's first ectolanguage. Kaisiris has also started to develop other ectolanguages besides Ectoenglishⓔ, ectoitalianoⓘ [Ectoitalian], and ⓞoɔƨooɈɔɘ [Ectooscan]. On 5 November 2015, the Aphrodite Island claim/name is accepted by author and geographer Nick Middleton (in the English language). On 11 November 2015, the [https://time.is/CMT Cyberterra Mean Time] (CMT) standard appeared in ''AcronymFinder.com'' for the first time. On 21 January 2016, the UK's Foreign and Commonwealth Office published a summary of the UMMOA party's views. This is the first indirect acknowledgement of the work of the Paradiplomatic Affairs (PDA) agency. On 5 October 2017, the Indigo race (or ''Homo sapiens ethicus''/''Homo noeticus'') concept is accepted by author and social anthropologist Beth Singler. On 11 August 2019, Lady Anna Tallini, Kaisiris' mother, passed away. This is Kaisiris' greatest loss. ==Recent developments== Since his mother's passing, and the end of the Marian Christianity (11.02.1858 – 11.08.2019) era, Kaisiris has founded the asocial, but not anti-social Kaisiris Clan (11.08.2019 – Ω), and the Cesidian Network (09.10.2019 – Ω) is beyond its conception or birth, beyond its ''bar/bat mitzvah'' age, and may be considered at least a college graduate, following all these actual developments: * Johannine Christianity (11.08.2019 – 09.07.2020) era started [first post-Marian Christian era which failed to materialise] * Cesidian spiritual science (Css) or Cesidianism conception: 29.11.2019 * Cesidian law gentilisation/neoaristocratic union process: 09.01.2020 * Polyphilia and multisocietalism concepts: 23.01.2020 * Cesidian law curialisation/neoecclesiastic union process: 05.02.2020 * Conception of Cesidian psychology: 24.03.2020 * Conception of the Restored Ancient Order of Melchizedek (RAOM): 20.05.2020 * Magdalene Christianity (09.07.2020 – 03.02.2021) era started [second post-Marian Christian era which failed to materialise] * Css/Cesidianism graduates into a post-tribal religion tradition (foreshadowing of the Cesidian era): 28.09.2020 * Cesidian psychology development of the redefined Maslow's pyramid of human needs: 15.11.2020 * Cesidian law "all speech is political speech" development: 17.11.2020 [a sign non-Cesidian law is largely a fiction] * New ectopolitical or multisocietal, rather than political or cosmopolitan paradigm development (''Homo sapiens ethicus''): 29.12.2020 * Birth of Ectoenglishⓔ, Ectoitalianⓘ, ectolanguages, and ectolinguistics: 30.12.2020 * New ectojournalism and ectohistorian development: 15.01.2021 * Birth of Ectofrenchⓕ: 17.01.2021 * Ectolinguistics and Cesidian law connection: 02.02.2021 * Cyberterra ectogeography connection (new ''genus'' not continental, and more ocean- than island-based): 03.02.2021 * Cesidian era started (post-Christianity, or post-religion without spirituality era): 03.02.2021 * Ecology, agriculture, and zoology connections (politics favours zookeepers, and is unfavourable even to wild and domesticated species): 08.02.2021 * Copernican (astronomical) versus the Cesidian (religious, legal, linguistic, holoscientific<ref>The ''holoscientific'' term was coined on 3 April 2022 [N1Χ15D2022•065].</ref>) revolution: 09.02.2021 * Biological or adopted, social or societal, religious or spiritual, economic or scientific, ethical or legal brothers concept: 21.02.2021 * Birth of Ectochineseⓩ: 27.02.2021 * Even cities like London (main-belt asteroid of 8837 London in particular) can be both cosmopolitan and ''ectopolitan'': 18.03.2021 * Birth of Ectohebrewⓗ: 26.03.2021 * Birth of Ectolatinⓛ: 28.03.2021 * Birth of Ectogreekⓖ: 28.03.2021 * New Marian Dogma — Mary is both ''Theotokos'' (Θεοτόκος) and ''Theogonos'' (Θεογόνος): 28.03.2021 * Cesidianism is a form of Ectochristianity: 07.04.2021 * Birth of Ectooscanⓞ: 14.04.2021 * First quality ⓔⓘⓞ triectolingual message posted: 17.04.2021 * Birth of The Ectolanguage Organism ⓔⓛⓞ, the first societal organism: 28.04.2021 * The Fleming 2021 Declaration: 14.05.2021 * Conception of ecto-governmental organisation (ego) concept: 21.06.2021 * Formation of ''Asking Facebook for a 'No Country' or ''Ectopolitan'' country option'' group: 14.07.2021 * Made admin of ''The World Academy of Buddhist Scholars'' Facebook group: 06.08.2021 * The regular English (en-GB/en-US) word 'wife' became archaic or obsolete in Ectoenglish (ⓔ; een): 19.09.2021 * The regular English (en-GB/en-US) word 'citizen' became archaic or obsolete in Ectoenglish (ⓔ; een): 23.09.2021 * The 'ectoinformatics' term is born: 29.09.2021 * The 'ectoagriculture' and 'ectoenvironmental' terms are born: 07.10.2021 * Single's Day holiday is introduced into the Cesidian calendar: 11.11.2021 * Decision made to introduce first ''Ectopolitans''' Day and Ectohomes' Day on 13.05.2022 and 23.08.2022 respectively into the Cesidian calendar: 22.11.2021 * The all-out assault (AOA) on "science" began: 10.12.2021 * All-out assault (AOA) on the world government quadropus: 16.12.2021 * Formation of the Lineage of the Blessed Hermes (LBH): 28.12.2021 * Birth of Ectokoreanⓚ: 06.01.2022 * The difference between a wiki and an 'ectowiki' is first shown: 11.01.2022 * After a new and improved translation of 'ectopolitan' in Ojibwe, 5 new ectolanguages were born: Ectogermanⓓ, Ectojapaneseⓙ, Ectoportugueseⓟ, Ectorussianⓡ, and Ectospanishⓢ: 24.01.2022 * Realisation Kaisiris Tallini is the first ''de facto'' [http://5wcwiki.pbworks.com/w/page/147494235/ectopolitan ''ectopolitan''], but also the first ''de jure'' or legal [http://5wcwiki.pbworks.com/w/page/147494235/ectopolitan ''ectopolitan'']: 26.02.2022 * Birth of Ectovietnameseⓥ: 10.03.2022 * With the rigourous proof that ectopolitans are gods, Cesidian spiritual science (Css) graduates into the "Queen of the Sciences": 14.03.2022 * The ''centimilliday'' (cmd) was introduced into the Cyberterra Mean Time format (CMTf): 15.03.2022 * Birth of Ectocurrency Unit (Ɇ; ECU), and possible Cesidian economics revolution concept: 17.03.2022 * Acknowledgement of ectoeducational science component of Cesidian revolution: 24.03.2022 * The 'holoscientific' term is born: 03.04.2022 * Start of Ectopolitan Wiki: 06.04.2022 * Birth of Ectobattery Hens in Hazmat Suits Organisation (EBHHSO): 28.04.2022 * Development of Cesidian analytic theology's "People of the Book" graph: 30.05.2022 * The 'ectoprofessional' term is born: 05.06.2022 * The 'ectoscientist' term is born: 05.06.2022 * First language–ectolanguage translation: 07.06.2022 * The 7th and 8th Cesidian saint is canonised: 08.06.2022 * "Cesidian pyramid" addition to the redefined Maslow's pyramid of human needs: 25.06.2022 * First legal document certifying [http://5wcwiki.pbworks.com/w/page/147494235/ectopolitan ''ectopolitan''] status devised and signed: 04.07.2022 * Ectoenglish update: the once, twice, and thrice etc list, completed for Ectoenglish up to ten times (''tence''): 14.07.2022 * Discovery of the parallels between ectopolitans or gods, and the Ancient Greek ''daimones'' (δαίμονες) concept [singular: ''daimon'' (δαίμων)]; discovery of the parallels between exopolitans or Gods, and the Ancient Greek ''theoi''/''thees'' (θεοί/θεές) concept [singular: ''theos''/''thea'' (θεός/θεά)]; as well as the discovery that the Cesidian ''Homo noeticus'' ['self-knowing' person], and the Buddhist ''Bodhisattva'' [person both 'awake' (''bodhi''), and 'sentient' (''sattva'')] concepts, are actually synonymous: 17.07.2022 * Published ''The Far High Theory'' (ectoscientific concept) article: 27.07.2022 * Published ''The Cesidian Age started for scientific reasons'' (ectohistorical concept) article: 06.08.2022 * The 'partescience' term is born (means highly atomised, or non-holistic single science field): 10.08.2022 * The 'zookeeperspeak' and 'American oligarch' terms are born: 12.08.2022 * The «''Médecins Sans Chercheurs D'or''» or "Doctors Without Gold Diggers" Print Monopolies ⓟⓜ are issued: 12.08.2022 * The 'daimonological' term is not used for the first time, but first used in 'an anti-Irenaeus of Smyrna manner', since daimons are claimed part of the 'daimonological' division or ''genus'' of humanity (''Homo noeticus''), and thus clearly Ectochristian in essence, or completely outside of Christianity when compared to Irenaeus of Smyrna, who is judged as nothing but your typical ''anthropoi'' (ἄνθρωποι) — singular: ''anthropos'' (ἄνθρωπος) —, that is, part of the anthropological division or ''genus'' of humanity (''fictus Homo sapiens''), or part of "We the People", even when they pretentiously believe to be otherwise, as arrogant daemons like Satan often do; and so daemons are completely associated with Catholic orthodoxy, and completely disassociated from daimons and Cesidianism: 14.08.2022 * A paranormal incident, after further research, leads to an actually verified paranormal (and ectoscientific) discovery, and the detection of the nature, even inherently matter-based classification of the spiritual world (ie, a paranormal incident passed the Occam's razor test probably for the very first time, since no simpler explanation for the odd incident was possible than actually a material, not a so-called "spiritual" explanation): 16.08.2022 It should be noted that while the socially limited Kaisiris Clan (11.08.2019 – Ω), and the Cesidian Network (09.10.2019 – Ω) high spirituality organisation are connected with some real micronationalists, there is no Judaeo-Christian group which is. Yet Kaisiris is treated like ''persona non grata'' by most micronationalists (non-[http://5wcwiki.pbworks.com/w/page/147494235/ectopolitan ''ectopolitans'']), so he enjoys no understanding among most micronationalists, but also receives no support. All current forms of micronationalism are either purely self-centred simulations and entertainment, or primordial forms of tribalism without genuine societalistic or [http://5wcwiki.pbworks.com/w/page/147494235/ectopolitan ''ectopolitan''] foundations. It should also be noted that Saint René Descartes University (24.11.2007 – Ω), and its Centre for Cesidian law (08.08.2006 – Ω) and Salubriology School (11.02.2009 – Ω) continue to exist, and are active as genuine educational and research institutions, and the Paradiplomatic Affairs (24.05.2013 – Ω) agency continues to exist as a paradiplomatic and [http://5wcwiki.pbworks.com/w/page/147494235/ectopolitan ''ectopolitan''] organisation, albeit less actively. Cesidian spiritual science (Css) has become, in essence, Cesidian legal, spiritual and theological, salubriological and psychological, linguistic, journalistic and historical, geological, geographic and astronomical, economics, societal and political, ecological and agricultural, informatics, educational and library science, or the world's first holistic, ethical, and scientific paradigm. The table of Cesidian spiritual science (Css) milestones follows below. {| class="wikitable sortable" |- ! '''Date''' || '''Css milestone''' !! '''Life milestone''' |- | 29.11.2019 || Css term is coined. || Birth |- | 21.12.2019 || Fishy Christianity "facts". || ''Bar/bat mitzvah'' |- | 05.02.2020 || Curialisation/neoecclesiastic union process. || High school graduation |- | 29.12.2020 || Ectolinguistics begins: ectopolitical/ectopolitics terms. || College graduation: BS degree |- | 03.02.2021 || Cyberterra ectogeography/ectogeology connection. || University graduation: MS degree |- | 14.03.2022 || Proof ectopolitans are gods: Css "Queen of the Sciences". || PhD dissertation successfully defended |} ==Important ectonyms and exonyms ([http://5wcwiki.pbworks.com/w/page/144142176/Kaisiris%20multilingual multilingual plus sound])== * Ectoenglishⓔ: Kaisiris Tallini [born Cesidio Tallini] * Ectoespañolⓢ: Kaisiris Tallini [nacido Cesidio Tallini] * Català: Kaisiris Tallini [nascut Cesidio Tallini] * 外中文ⓩ： 凯西里斯·塔利尼（出生：塞西迪奥·塔利尼） * Oʻzbekcha: Kaisiris Tallini * Қазақша: Кайсирис Таллини [туылған: Сесидио Таллини] * Türkmençe: Kaisiris Tallini [doglan Cesidio Tallini] * Тоҷикӣ: Кайсирис Таллини [таваллуд: Сесидио Таллини] * Монгол: Кайсирис Таллини [төрсөн: Сесидио Таллини] * བོད་ཡིག༇ཁའི་སིཨའིརིས་ཏལིནའི༺འཁྲུངས་ཆེསིདིཨོ་ཏལིནའི༻ * Ektodeutschⓓ: Kaisiris Tallini [geboren Cesidio Tallini] * Nederlands: Kaisiris Tallini * Svenska: Kaisiris Tallini * Dansk: Kaissiris Tallini * Suomi: Kaisiris Tallini * Norsk bokmål: Kaisiris Tallini * Íslenska: Kaisiris Tallini * Gàidhlig: Kaisiris Tallini * Cymraeg: Kaisiris Tallini * Bahasa Indonesia: Kaisiris Tallini * Jawa: Kaisiris Tallini * Sunda: Kaisiris Tallini * Ectofrançaisⓕ: Kaisiris Tallini [né Césidio Tallini] * Kreyòl ayisyen: Kaisiris Tallini * Corsu: Kaisiris Tallini [natu Cesidio Tallini] * Экторусскийⓡ: Кайсирис Таллини [урожденный Чесидио Таллини] * Беларуская: Кайсірыс Талліні * Кыргызча: Кайсирис Таллини * Українська: Кайсіріс Талліні * Română: Kaisiris Tallini * Српски: Каисирис Таллини * Հայերեն: Կայզիրիս Տալլինի * ქართველი: კაისირის ტალლინი * ｴｸﾄﾆﾎﾝｺﾞⓙ: ｶｲｼﾘｽ･ﾀﾘｰﾆ (ｳﾏﾚ: ﾁｪｼﾃﾞｨｵ･ﾀﾘｰﾆ) * Ectoitalianoⓘ: Kaisiris Tallini [nato Cesidio Tallini] * Hrvatski: Kaisiris Tallini * Slovenščina: Kaisiris Tallini * Magyar: Kaissiris Tallini * Shqip: Kaisiris Tallini * Македонски: Каисирис Таллини * Bosanski: Kaisiris Tallini * አማርኛ፦ ካይሲሪስ ታልሊኒ * ቲግሪንያ፦ ካይሲሪስ ታልሊኒ * Čeština: Kaisiris Tallini * Slovenčina: Kaisiris Tallini * Hausa: Kaisiris Tallini * Yorùbá: Kaisiris Tallini * Igbo: Kaisiris Tallini * Malagasy: Kaisiris Tallini * Kiswahili: Kaisiris Tallini * IsiZulu: Kaisiris Tallini * Ikinyarwanda: Kaisiris Tallini * IsiXhosa: Kaisiris Tallini * Afrikaans: Kaiciris Tallini * Vèneto: Kaisiris Tallini [nato Cesidio Tallini] * Napulitano: Kaisiris Tallini [nato Cesidio Tallini] * Sicilianu: Kaisiris Tallini [nascitu Cesidio Tallini] * Sardu: Kaisiris Tallini [nasciu Cesidio Tallini] * Malti: Kaisiris Tallini [imwieled Cesidio Tallini] * Ōlelo Hawaiʻi: Kaisiris Tallini [hānau ʻia ʻo Cesidio Tallini] * Εκτοελληνικάⓖ: Καϊσίρης Ταλλίνι [αρχικά Τσεσίδιο Ταλλίνι] * Ectoportuguêsⓟ: Kaisiris Tallini [nascido Cesidio Tallini] * Interlingua: Kaisiris Tallini [nascite Cesidio Tallini] * Esperanto: Kaisiris Tallini [origine Cesidio Tallini] * Polski: Kaisiris Tallini * Eesti: Kaisiris Tallini * Lietuvių: Kaisiris Tallini * Latviešu: Kaisiris Tallini * Türkçe: Kaisiris Tallini * Azərbaycanca: Kaisiris Tallini * Kurdî: Kaisiris Tallini * Български: Кайсирис Таллини * 엑토한국어ⓚ: 카이시리스 탈리니 (태어난: 체시디오 탈리니) * Ngoài tiếng việtⓥ: Kaisíris Tallini [sinh ra: Chêsidio Tallini] * Tagalog: Kaisiris Tallini [ipinanganak na Cesidio Tallini] * Cebuano: Kaisiris Tallini [natawo nga Cesidio Tallini] * ไทย: ไคสิริส ทัลลินี [เกิด: เชศรีดิโอ ทัลลินี] * Ectolatinaⓛ: Kaisidis Tallini [natus Caesidius Tallini] * Qhichwa: Kaisiris Tallini * ᐊᓂᔑᓈᐯᒧᐎᓐ: ᑲᐃᓯᕆᔅ ᑕᓪᓕᓂ [ᓃᑭ ᒉᓯᑎᐅ ᑕᓪᓕᓂ] * हिंदी: काईसिरिस तालिनी * বাংলা: কাইসিরিস তালিনি * नेपाली: कैसिरिस तल्लिनी * తెలుగు: కైసిరిస్ తల్లిని * தமிழ்: கைசிரிஸ் தாலினி * मराठी: काईसीरिस तल्लीनी * ગુજરાતી: કૈસીરીસ તલ્લિની * ಕನ್ನಡ: ಕೈಸಿರಿಸ್ ಟಾಲಿನಿ * മലയാളം: കൈസിരിസ് തല്ലിനി * ଓଡିଆ: କେସିରିସ୍ ତାଲିନି * ລາວ: ໄຄຊີຣິສ ທາລິນີ * অসমীয়া: কাইচিৰিছ টালিনি * සිංහල: කෛසිරිස් ටල්ලිනි * ਪੰਜਾਬੀ: ਕੈਸੀਰਿਸ ਟੱਲੀਨੀ <big>پنجابی: قیصریس تلینی </big>▪ <big>دری: قیصریس تالینی </big>▪ <big>فارسی: کایسیریس تالینی </big>▪ <big>[קַאִיסִירִיס טָאללִינִי [נוֹלַד צ'סִידָיו טָאללִינִי :ⓗאַקְטוֹעִבְרִית </big>▪ '''[inillɒɈ oibiƨɘɔ oɈɒn] †inillɒɈ ƨiɿiƨiɒʞ :ⓞoɔƨooɈɔɘ''' ▪ <big>[عربى: كايسيريس تاليني [ولد سيسيديو تاليني </big>▪ <big>اردو: کائیسیرس تاللینی </big>▪ † ''Originally named UMMOA or UGV English, it wasn't judged a new language, but as a new English variety, variant, or lect (assessment provided by Melinda Lyons, ISO 639-3 registration authority for SIL International). Since 30.12.2020, UMMOA or UGV English is now called Ectoenglishⓔ, the world's first ectolanguage. Kaisiris has also started to develop other ectolanguages besides Ectoenglishⓔ, ectoitalianoⓘ [Ectoitalian], and ⓞoɔƨooɈɔɘ [Ectooscan].'' ==Citations== </big> nh7c2qquc1gncqcnwhg6eyvcciziyu5 2 274697 2416213 2410526 2022-08-18T17:17:42Z 177.227.10.164 wikitext text/x-wiki {| class="wikitable" |+Unit relationship |- | kg = 15 |- | m = -13 |- | s = -30 |- | A = 3 |- | K = 20 |} {| class="wikitable" |+CODATA 2014: R=10973731.568508 (u=13) ... c=299792458 (u=17) ... μ₀=4π/10⁷ (u=56) ... α=137.035999139 (u=0) ! Constant ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! SI units ! Formula ! Calculated from (R, c, μ₀, α) |- | [[w:Planck constant | Planck constant]] | ''h'' = 6.626 070 040(81) e-34 | <math>\frac{kg \;m^2}{s}</math>, u = 19 | <math>{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 \alpha^{13} {R}^2}</math>, u = 57 | ''h^*'' = 2πMVL = 6.626 069 134 e-34, u = 19 |- | [[w:Gravitational constant | Gravitational constant]] | ''G'' = 6.674 08(31) e-11 | <math>\frac{m^3}{kg \;s^2}</math>, u = 6 | <math>{(G^*)}^5 = \frac{\pi^3 {\mu_0}}{2^{20} 3^6 \alpha^{11} {R}^2}</math>, u = 30 | ''G^*'' = V²L/M = 6.672 497 192 29 e11, u = 6 |- | [[w:Elementary charge | Elementary charge]] | ''e'' = 1.602 176 620 8(98) e-19 | <math>A s</math>, u = -27 | <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 {c}^4 \alpha^8 {R}}</math>, u = -81 | ''e^*'' = AT = 1.602 176 511 30 e-19, u = -27 |- | [[w:Boltzmann constant | Boltzmann constant]] | ''k_B'' = 1.380 648 52(79) e-23 | <math>\frac{kg \;m^2}{s^2 \;K}</math>, u = 29 | <math>{(k_B^*)}^3 = \frac{\pi^5 {\mu_0}^3}{3^3 2 {c}^4 \alpha^5 {R}}</math>, u = 87 | ''k_B^*'' = 2πVM/A = 1.379 510 147 52 e-23, u = 29 |- | | ''k_Bec/h'' = 1.000 825 132 | <math>\frac{m \;A}{s \;K}</math>, u = 0 | <math>\frac{(k_B^*) (e^*) c}{(h^*)} = 1.0</math>, u = 0 | |- | [[w:Electron mass | Electron mass]] | ''m_e'' = 9.109 383 56(11) e-31 | <math>kg</math>, u = 15 | <math>{(m_e^*)}^3 = \frac{16 \pi^{10} {R} {\mu_0}^3}{3^6 {c}^8 \alpha^7}</math>, u = 45 | '' m_e^*'' = M/f_e = 9.109 382 312 56 e-31, u = 15 |- | | | | <math>f_e = {(\frac{3 \alpha^2 A L}{2\pi^2})}^3 \frac{1}{2 T}</math>, u = 0 | |- | | ''8π⁵k_B/(27α⁶e²m_ec⁴)'' = 1.000 824 93 | <math>\frac{1}{m^2 A^2 K}</math>, u = 0 | <math>\frac{2^3 \pi^5 ({k_B}^*)}{3^3 \alpha^6 ({e^*})^2 ({m_e}^*) c^4} = 1.0</math>, u = 0 | |- | [[w:Planck mass | Planck mass]] | ''m_P'' = 0.217 647 0(51) e-7 | kg, u = 15 | <math>{m_P}^{15} = \frac{2^{25} \pi^{13} {\mu_0}^6}{3^6 c^5 \alpha^{16} R^2}</math>, u = 225 | ''m_P'' = M = 0.217 672 817 580 e-7, u = 15 |- | [[w:Planck length | Planck length]] | ''l_p'' = 0.161 622 9(38) e-34 | m, u = -13 | <math>{l_p}^{15} = \frac{\pi^{22} {\mu_0}^9}{2^{35} 3^{24} \alpha^{49} c^{35} R^8}</math>, u = -195 | ''l_p'' = L = 0.161 603 660 096 e-34, u = -13 |} {| class="wikitable" |+Dimensioned physical constants ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! unit ! Natural constant* ! unit* |- | ''c'' = 299 792 458 (exact) | <math>\frac{m}{s}</math> | ''c*'' = V = 25.3123819329 * <math>v</math> = 299792458 | 17 |- | ''h'' = 6.626 070 040(81) e-34 | <math>\frac{kg \;m^2}{s}</math> | ''h*'' = <math>2 \pi M V L</math> = 12647.240312 * <math>\frac{r^{13}}{v^5}</math> = 6.626 069 134 e-34 | 19 |- | ''G'' = 6.674 08(31) e-11 | <math>\frac{m^3}{kg \;s^2}</math> | ''G*'' = <math>\frac{V^2 L}{M}</math> = 50950.554778 * <math>\frac{r^5}{v^2}</math> = 6.672 497 192 29 e11 | 6 |- | ''e'' = 1.602 176 620 8(98) e-19 | <math>C = A s</math> | ''e*'' = <math>A T</math> = 735.706358485 * <math>\frac{r^3}{v^3}</math> = 1.602 176 511 30 e-19 | -27 |- | ''k_B'' = 1.380 648 52(79) e-23 | <math>\frac{kg \;m^2}{s^2 \;K}</math> | ''k_B*'' = <math>\frac{2 \pi V M}{A}</math> = 0.679138336 * <math>\frac{r^{10}}{v^3}</math> = 1.379 510 147 52 e-23 | 29 |} AL as an ampere-meter (ampere-length) are the units for a [[w:magnetic monopole | magnetic monopole]]. :<math>T = \pi,\; u^{-30}</math> :<math>\sigma_{e} = \frac{3 \alpha^2 A L}{2\pi^2},\; u^{-10}</math> :<math>f_e = \frac{\sigma_{e}^3}{2 T} = \frac{(2^7 3 \pi^3 \alpha \Omega^5)^3}{2\pi},\; units = \frac{(u^{-10})^3}{u^{-30}} = 1</math> :<math>f_e = 4\pi^2(2^6 3 \pi^2 \alpha \Omega^5)^3 = .23895453...x10^{23}</math> {| class="wikitable" |+Geometrical units ! Attribute ! Geometrical object ! Unit number ! Scalar |- | mass | <math>M = (1)</math> | 15 | k |- | time | <math>T = (\pi)</math> | -30 | t |- | velocity | <math>V = (2\pi\Omega^2)</math> | 17 | v |- | length | <math>L = (2\pi^2\Omega^2)</math> | -13 | l |- | ampere | <math>A = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> | 3 | a |} {| class="wikitable" |+Geometrical units and scalars ! Geometrical object ! Scalar ! Unit number |- | <math>M = (1)</math> | k | <math>15</math> |- | <math>T = (\pi)</math> | t | <math>-30</math> |- | <math>V = (2\pi\Omega^2)</math> | v | <math>17</math> |- | <math>L = (2\pi^2\Omega^2)</math> | l | <math>-13</math> |- | <math>A = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> | a | <math>3</math> |} {| class="wikitable" |+Geometrical units ! Attribute ! Geometrical object ! Scalar ! Unit |- | mass | <math>M = (1)</math> | k | <math>unit = u^{15}</math> |- | time | <math>T = (\pi)</math> | t | <math>unit = u^{-30}</math> |- | velocity | <math>V = (2\pi\Omega^2)</math> | v | <math>unit = u^{17}</math> |- | length | <math>L = (2\pi^2\Omega^2)</math> | l | <math>unit = u^{-13}</math> |- | ampere | <math>A = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> | a | <math>unit = u^3</math> |} {| class="wikitable" |+Physical constants; calculated vs CODATA 2014 ! Constant ! Geometry ! Calculated from (R, c, μ₀, α) ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> |- | [[w:Planck constant | Planck constant]] | <math>{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 \alpha^{13} {R}^2},\; unit = u^{57}</math> | ''h^*'' = 6.626 069 134 e-34, unit = u¹⁹ | ''h'' = 6.626 070 040(81) e-34 |- | [[w:Gravitational constant | Gravitational constant]] | <math>{(G^*)}^5 = \frac{\pi^3 {\mu_0}}{2^{20} 3^6 \alpha^{11} {R}^2},\; unit = u^{30}</math> | ''G^*'' = 6.672 497 192 29 e11, unit = u⁶ | ''G'' = 6.674 08(31) e-11 |- | [[w:Elementary charge | Elementary charge]] | <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 {c}^4 \alpha^8 {R}},\; unit = u^{-81}</math> | ''e^*'' = 1.602 176 511 30 e-19, unit = u^-27 | ''e'' = 1.602 176 620 8(98) e-19 |- | [[w:Boltzmann constant | Boltzmann constant]] | <math>{(k_B^*)}^3 = \frac{\pi^5 {\mu_0}^3}{3^3 2 {c}^4 \alpha^5 {R}} ,\; unit = u^{87} </math> | ''k_B^*'' = 1.379 510 147 52 e-23, unit = u²⁹ | ''k_B'' = 1.380 648 52(79) e-23 |- | [[w:Electron mass | Electron mass]] | <math>{(m_e^*)}^3 = \frac{16 \pi^{10} {R} {\mu_0}^3}{3^6 {c}^8 \alpha^7},\; unit = u^{45} </math> | '' m_e^*'' = 9.109 382 312 56 e-31, unit = u¹⁵ | ''m_e'' = 9.109 383 56(11) e-31 |} {| class="wikitable" |+Dimensioned physical constants ! SI constant ! Natural constant ! unit |- | <math>\frac{k_B e c}{h} =</math> {{font color|blue|yellow|'''1.000 8254'''}} | <math>\frac{(k_B^*) (e^*) (c^*)}{(h^*)}</math> = {{font color|blue|yellow|'''1.0'''}} | <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1</math> |- | <math>\frac{h^3}{e^{13} c^{24}} =</math> {{font color|green|yellow| '''0.228 473 639... 10^-58'''}} | <math>\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} =</math> {{font color|green|yellow| '''0.228 473 759... 10^-58'''}} | <math>\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1</math> |- | <math>\frac{c^{35}}{\mu_0^9 R^7} =</math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}} | <math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} =</math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}} | <math>\frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1</math> |- | <math>\frac{c^9 e^4}{m_e^3} =</math> {{font color|blue|yellow| '''0.170 514 342... 10⁹²'''}} | <math>\frac{(c^*)^9 (e^*)^4}{(m_e^*)^3} =</math> {{font color|blue|yellow| '''0.170 514 368... 10⁹²'''}} | <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1</math> |- | <math>\frac{k_B}{e^2 m_e c^4} =</math> {{font color|green|yellow| '''73 095 507 858.'''}} | <math>\frac{(k_B^*)}{(e^*)^2 (m_e^*) (c^*)^4} =</math> {{font color|green|yellow| '''73 035 235 897.'''}} | <math>\frac{(u^{29})}{(u^{-27})^2 (u^{15}) (u^{17})^4} = 1</math> |- | <math>\frac{h c^2 e m_e}{G^2 k_B} =</math> {{font color|red|yellow| '''0.1413... 10^-21'''}} | <math>\frac{(h^*) (c^*)^2 (e^*) (m_e^*)}{(G^*)^2 (k_B^*)} =</math> {{font color|red|yellow| '''0.1415... 10^-21'''}} | <math>\frac{ (u^{19}) (u^{17})^2 (u^{-27}) (u^{15}) }{ (u^{6})^2 (u^{29}) } = 1</math> |- | <math>\frac{2 h}{\mu_0\; e^2 \;c} = \color{blue}\alpha \color{black}</math> | <math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = \color{blue}\alpha \color{black}</math> | <math> \frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1</math> |} {| class="wikitable" |+Dimensioned physical constants ! Constant ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! unit |- | [[w:Speed of light | Speed of light]] | ''c'' = 299 792 458 (exact) | <math>\frac{m}{s}</math> |- | [[w:Planck constant | Planck constant]] | ''h'' = 6.626 070 040(81) e-34 | <math>\frac{kg \;m^2}{s}</math> |- | [[w:Gravitational constant | Gravitational constant]] | ''G'' = 6.674 08(31) e-11 | <math>\frac{m^3}{kg \;s^2}</math> |- | [[w:Elementary charge | Elementary charge]] | ''e'' = 1.602 176 620 8(98) e-19 | <math>C = A s</math> |- | [[w:Boltzmann constant | Boltzmann constant]] | ''k_B'' = 1.380 648 52(79) e-23 | <math>\frac{kg \;m^2}{s^2 \;K}</math> |- | [[w:Vacuum permeability | Vacuum permeability]] | ''μ₀'' = 4π/10^7 (exact) | <math>\frac{kg \;m}{s^2 \;A^2}</math> |- | [[w:Electron mass | Electron mass]] | ''m_e'' = 9.109 383 56(11) e-31 | <math>kg</math> |} {| class="wikitable" |+Dimensioned physical constants ! SI constant ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! unit ! Natural constant* ! unit* |- | [[w:Speed of light | Speed of light]] | ''c'' = 299 792 458 (exact) | <math>\frac{m}{s}</math> | ''c*'' = V | 17 |- | [[w:Planck constant | Planck constant]] | ''h'' = 6.626 070 040(81) e-34 | <math>\frac{kg \;m^2}{s}</math> | <math>h^* = 2 \pi M V L</math> | 19 |- | [[w:Gravitational constant | Gravitational constant]] | ''G'' = 6.674 08(31) e-11 | <math>\frac{m^3}{kg \;s^2}</math> | <math>G^* = \frac{V^2 L}{M}</math> | 6 |- | [[w:Elementary charge | Elementary charge]] | ''e'' = 1.602 176 620 8(98) e-19 | <math>C = A s</math> | <math>e^* = A T</math> | -27 |- | [[w:Boltzmann constant | Boltzmann constant]] | ''k_B'' = 1.380 648 52(79) e-23 | <math>\frac{kg \;m^2}{s^2 \;K}</math> | <math>k_B^* = \frac{2 \pi V M}{A}</math> | 29 |- | [[w:Vacuum permeability | Vacuum permeability]] | ''μ₀'' = 4π/10^7 (exact) | <math>\frac{kg \;m}{s^2 \;A^2}</math> | <math>\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2}</math> | 56 |} {| class="wikitable" |+Dimensionless physical constants ! Constant ! value |- | [[w:Fine structure constant | Fine structure constant]] | ''α'' = 137.035 999 139(31) |- | [[v:Planck_units_(geometrical)#Omega | Omega]] | ''Ω'' = 2.007 134 9496 |} {| class="wikitable" |+Dimensioned physical constants ! Constant ! value ! unit |- | [[w:Planck mass | Planck mass]] | ''m_P'' = .217 647 0(51) e-7 | <math>kg</math> |- | [[w:Planck length | Planck length]] | ''l_p'' = .161 622 9(38) e-34 | <math>m</math> |- | [[w:Planck time | Planck time]] | ''t_p'' = 5.391 247(60) e-44 | <math>s</math> |} '''Natural Planck units as geometrical objects''' [[w:Planck units |Planck unit]] theories use basic units for mass, length, time and charge, and operate at the Planck scale. In a geometrical Planck theory, these basic units are assigned geometrical objects (''MLTA'') rather than numerical values, the advantage being that the geometries themselves can encode the function of the unit, for example the object for length (''L'') will encode the function of ''length'', the geometrical ''L'' is 1 unit of (Planck) length, such that, unlike numerical models, a dimensioned descriptive (i.e.: ''kg, m, s, A'', ... ) is not required. The ''MLTA'' geometrical objects are selected whereby they may interact with each other (the mass object for example is not independent of the length and the time objects). This permits a mathematical unit relationship linking the objects, and so a physical universe can be constructed [[w:Lego |Lego-style]] by combining the base (Planck unit) ''MLTA'' objects to form more complex objects such as electrons (i.e.: by embedding ''L'' and ''A'' into the geometry of the electron, the electron can have wavelength and charge). Furthermore, these objects can overlap and cancel in a particular ratio (according to that unit relationship), and this ratio occurs in the electron. And so, although the electron has physical parameters (wavelength, charge ...), '''the electron itself is a mathematical particle (units = 1)''', not a physical particle. === Geometrical objects === Base units for mass <math>M</math>, length <math>L</math>, time <math>T</math>, and ampere <math>A</math> can be constructed from the geometry of 2 [[w:dimensionless physical constant | dimensionless physical constants]], the (inverse) [[w:fine-structure constant | fine structure constant '''α''']] = 137.036 and [[v:Simulation_argument_(coding_Planck_units)#Omega | Omega]] '''Ω''' = 2.007 134 949 <ref>Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}</ref>. Being independent of any numerical system and of any system of units, these MLTA units would qualify as "natural units"; {{bq|''...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...'' ...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"... -Max Planck <ref>Planck (1899), p. 479.</ref><ref name="TOM">*Tomilin, K. A., 1999, "[http://www.ihst.ru/personal/tomilin/papers/tomil.pdf Natural Systems of Units: To the Centenary Anniversary of the Planck System]", 287–296.</ref>}} ==== Objects ==== Each object is assigned a specific geometry, embedded in this geometry is the object function (attribute) and a mathematical relationship between the objects defined using '''uⁿ''' where ''n'' is the unit number; {| class="wikitable" |+Geometrical units ! Attribute ! Geometrical object ! Unit relationship |- | mass | <math>M = (1)</math> | <math>unit = u^{15}</math> |- | time | <math>T = (\pi)</math> | <math>unit = u^{-30}</math> |- | [[v:Sqrt_Planck_momentum | sqrt(momentum)]] | <math>P = (\Omega)</math> | <math>unit = u^{16}</math> |- | velocity | <math>V = (2\pi\Omega^2)</math> | <math>unit = u^{17}</math> |- | length | <math>L = (2\pi^2\Omega^2)</math> | <math>unit = u^{-13}</math> |- | ampere | <math>A = \frac{16 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> | <math>unit = u^3</math> |} ==== Scalars ==== To translate from geometrical objects to a numerical system of units requires scalars ('''kltpva''') that can be assigned numerical values. For example, scalars for the SI units; :If we use '''k''' to convert '''M''' to the SI Planck mass <math>m_P</math> (M = 1k = <math>m_P</math>), then '''k''' = 0.2176728e-7kg and '''<math>u^{15}</math>''' will equate to '''kg'''. :To convert '''V''' = 2πΩ² = 25.3123819 to '''c''' requires scalar '''v''' = 11843707.905m/s ('''V''' ''v'' = 2πΩ²''v'' = 299792458m/s) with '''<math>u^{17}</math>''' equating to '''m/s'''. {| class="wikitable" |+Geometrical units ! Attribute ! Geometrical object ! Scalar ! Unit |- | mass | <math>M = (1)</math> | k | <math>unit = u^{15}</math> |- | time | <math>T = (\pi)</math> | t | <math>unit = u^{-30}</math> |- | [[v:Sqrt_Planck_momentum | sqrt(momentum)]] | <math>P = (\Omega)</math> | p | <math>unit = u^{16}</math> |- | velocity | <math>V = (2\pi\Omega^2)</math> | v | <math>unit = u^{17}</math> |- | length | <math>L = TV = (2\pi^2\Omega^2)</math> | l | <math>unit = u^{-13}</math> |- | ampere | <math>A = \frac{16 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> | a | <math>unit = u^3</math> |} ===== Scalar relationships ===== The scalars follow the same unit relationship. This means that we can find ratios where the scalars cancel. Here the following ''uⁿ'' groups cancel ('''units = scalars = 1'''), as such '''only 2 numerical scalars are required''', for example, if we know '''a''' and '''l''' then we know '''t''' ('''t = a³l³'''), and from '''l''' and '''t''' we know '''k'''. :<math>\frac{u^{3*3} u^{-13*3}}{u^{-30}}\;(\frac{a^3 l^3}{t}) = \frac{u^{-13*15}}{u^{15*9} u^{-30*11}} \;(\frac{l^{15}}{k^9 t^{11}}) = \;...\; =1</math> And so we know any 2 constants, then we can solve the scalars for those constants, and from those 2 scalars we can solve the Planck units, and from these the dimensioned physical constants. This will apply to any set of units. In this example, to maintain integer exponents, a scalar ''p'' is defined in terms of a scalar ''r''. :<math>r = \sqrt{p} = \sqrt{\Omega},\; unit \;u^{16/2=8}</math> The SI Planck units are known with a low precision, conversely 2 of the CODATA 2014 physical constants have been assigned exact numerical values; ''c'' and permeability of vacuum ''μ₀''. Scalars ''r'' and ''v'' were chosen as they can be derived directly from ''V = c'' and ''μ₀'' = 4π/10^7 (see table [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |geometrical physical constants]] below). Using α = 137.035 999 139 (CODATA 2014), Ω = 2.007 134 949 636... :<math>v = \frac{c}{2 \pi \Omega^2}= 11 843 707.905 ...,\; units = m/s</math> :<math>r^7 = \frac{2^{11} \pi^5 \Omega^4 \mu_0}{\alpha};\; r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}</math> This gives scalars ''klta'' ([[v:Planck_units_(geometrical)#u_as_sqrt(velocity/mass)) |for derivation of units kg, m, s, A from r, v]]); :<math>k = \frac{r^4}{v}</math> = 0.217 672 817 580... ''x'' 10^-7kg, <math>\;\;\;u^{15} = \frac{(u^8)^4}{u^{17}}</math> :<math>l = \frac{r^9}{v^5}</math> = 0.203 220 869 487... ''x'' 10^-36m, <math>\;\;\;u^{-13} = \frac{(u^8)^9}{(u^{17})^5}</math> :<math>t = \frac{r^9}{v^6}</math> = 0.171 585 512 841... ''x'' 10^-43s, <math>\;\;\;u^{-30} = \frac{(u^8)^9}{(u^{17})^6}</math> :<math>a = \frac{v^3}{r^6}</math> = 0.126 918 588 592... ''x'' 10²³A, <math>\;\;\;u^{3} = \frac{(u^{17})^3}{(u^8)^6}</math> ===== Natural units MLTPA ===== Regardless of which system of units we use, alien or terrestrial, any combination of constants where '''scalars = 1''' (i.e.: the scalars overlap and cancel) will give the same numerical result, they will default to the MLTPA objects. This implies that these objects are Planck's 'natural' units, i.e.: that '''all possible systems of units''' are based on these objects, and so, given that these are geometrical objects, they can be construed as evidence of a mathematical universe. The following are examples of '''units = scalars = 1''' ratios using SI units <ref>Macleod, Malcolm J. {{Cite journal |title= Do the fundamental constants embed evidence of a mathematical universe at the Planck scale? |journal=RG | doi=10.13140/RG.2.2.15874.15041/1 }}</ref>. Note: the geometry <math>\color{red}(\Omega^{15})^n\color{black}</math> (integer n ≥ 0) is common to these ratios. ====== m_P, l_p, t_p ====== In this ratio, the MLT units and ''klt'' scalars both cancel; units = scalars = 1, reverting to the base MLT objects. Setting the scalars ''klt'' for SI Planck units; :k = 0.217 672 817 580... ''x'' 10^-7kg :l = 0.203 220 869 487... ''x'' 10^-36m :t = 0.171 585 512 841... ''x'' 10^-43s :<math>\frac{L^{15}}{M^{9} T^{11}} = \frac{(2\pi^2\Omega^2)^{15}}{(1)^{9} (\pi)^{11}} (\frac{l^{15}}{k^9 t^{11}}) = \frac{l_p^{15}}{m_P^{9} t_p^{11}} </math> (CODATA 2018 mean) The ''klt'' scalars cancel, leaving; :<math>\frac{L^{15}}{M^{9} T^{11}} = \frac{(2\pi^2\Omega^2)^{15}}{(1)^{9} (\pi)^{11}} (\frac{l^{15}}{k^9 t^{11}}) = 2^{15} \pi^{19} \color{red}(\Omega^{15})^2\color{black} = </math>{{font color|blue|yellow|'''0.109 293... 10²⁴ '''}}, <math>(\frac{l^{15}}{k^9 t^{11}}) = 1, \;\frac{u^{-13*15}}{u^{15*9} u^{-30*11}} = 1</math> Solving for the SI units; :<math>\frac{l_p^{15}}{m_P^{9} t_p^{11}} = \frac{(1.616255e-35)^{15}}{(2.176434e-8)^{9} (5.391247e-44)^{11}} = </math> {{font color|blue|yellow| '''0.109 485... 10²⁴'''}} ====== A, l_p, t_p ====== :a = 0.126 918 588 592... ''x'' 10²³A :<math>\frac{A^3 L^3}{T} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})^3 \frac{(2\pi^2\Omega^2)^3}{(\pi)} (\frac{a^3 l^3}{t}) = \frac{2^{24} \pi^{14} \color{red}(\Omega^{15})^1\color{black}}{\alpha^3} = </math> {{font color|green|yellow| '''0.205 571... 10¹³'''}}, <math>(\frac{a^3 l^3}{t}) = 1,\; \frac{u^{3*3} u^{-13*3}}{u^{-30}} = 1</math> :<math>\frac{(e / t_p)^3 l_p^3}{t_p} = \frac{(1.602176634e-19/5.391247e-44)^3 (1.616255e-35)^3}{(5.391247e-44)} = </math> {{font color|green|yellow| '''0.205 543... 10¹³'''}}, <math>units = \frac{(C/s)^3 m^3}{s} </math> The Planck units are known with low precision, and so by defining the 3 most accurately known dimensioned constants in [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |terms of these objects]] (c, R = Rydberg constant, <math>\mu_0</math>; CODATA 2014 mean values), we can test to greater precision; ====== c, μ₀, R ====== :<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = (2 \pi \Omega^2 v)^{35}/(\frac{\alpha r^7}{2^{11} \pi^5 \Omega^4})^9 .(\frac{v^5} {2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17} r^9})^7 = 2^{295} \pi^{157} 3^{21} \alpha^{26} \color{red}(\Omega^{15})^{15}\color{black} = </math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}}, <math>\frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1, \;(v^{35})/(r^7)^9 (\frac{v^5}{r^9})^7 = 1</math> :<math>\frac{c^{35}}{\mu_0^9 R^7} = \frac{(299792458)^{35}}{(4 \pi/10^7)^9 (10973731.568160)^7} = </math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}}, <math>units = \frac{m^{33}A^{18}}{s^{17}kg^9} == \frac{(u^{-13})^{33} (u^{3})^{18}}{(u^{-30})^{17} (u^{15})^9} = 1</math> The [[w:2019 redefinition of SI base units | 2019 SI unit revision]] assigned exact numerical values to 4 constants (c, e, k_B, h). {{see also |Planck units (geometrical)#2019 SI unit revision}} From the table [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |geometrical physical constants]], we get geometrical formulas and scalars for; :<math>h^* = 2 \pi MVL = 2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{15+17-13 = 19}</math> :<math>e^* = AT = \frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3},\; u^{3-30 = -27}</math> :<math>k_B^*= 2 \pi MV/A = \frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3},\; u^{17+15-3 = 29}</math> ====== c, e, k_B, h ====== :<math>\frac{(k_B^*) (e^*) (c^*)}{(h^*)} = (\frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3}) (\frac{2^7 \pi^4 \Omega^3}{\alpha} \frac{r^3}{v^3}) (2 \pi \Omega^2 v) / (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5}) </math> = {{font color|blue|yellow|'''1.0'''}}, <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1,\; (\frac{r^{10}}{v^3}) (\frac{r^3}{v^3}) (v) / (\frac{r^{13}}{v^5}) = 1</math> :<math>\frac{k_B e c}{h} = </math> {{font color|blue|yellow|'''1.000 8254'''}}, <math>units = \frac{m C}{s^2 K} == \frac{(u^{-13}) (u^{-27})}{(u^{-30})^2 (u^{20})} = 1</math> ====== c, h, e ====== :<math>\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5})^3/(\frac{2^7 \pi^4 \Omega^3 r^3}{\alpha v^3})^7.(2\pi\Omega^2 v)^{24} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} = </math> {{font color|green|yellow| '''0.228 473 759... 10^-58'''}}, <math>\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1, \;(\frac{r^{13}}{v^5})^3 / (\frac{r^3}{v^3})^{13} (v^{24}) = 1</math> :<math>\frac{h^3}{e^{13} c^{24}} = </math> {{font color|green|yellow| '''0.228 473 639... 10^-58'''}}, <math>units = \frac{kg^3 s^{21}}{m^{18} C^{13}} == \frac{(u^{15})^3 (u^{-30})^{21}}{(u^{-13})^{18} (u^{-27})^{13}} = 1</math> ====== m_e, λ_e ====== :<math>\sigma_{e} = \frac{3 \alpha^2 A L}{2\pi^2} = {2^7 3 \pi^3 \alpha \Omega^5}\frac{r^3}{v^2},\; u^{-10}</math> :<math>f_e = \frac{\sigma_{e}^3}{2 T} = 2^{20} 3^3 \pi^8 \alpha^3 (\color{red}\Omega^{15})\color{black},\; \frac{(u^{-10})^3}{u^{-30}} =1,\; (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1</math> :<math>(m_e^*) = \frac{M}{f_e} = \color{blue}9.109\;382\;3227 \;10^{-31}\color{black}\;u^{15}</math> :<math>(m_e^*) = \frac{2^3 \pi^5 (h^*)}{3^3 \alpha^6 (e^*)^3 (c^*)^5} = \frac{1}{2^{20} \pi^8 3^3 \alpha^3 (\color{red}\Omega^{15})\color{black}} \frac{r^4 u^{15}}{v} = \color{blue}9.109\;382\;3227 \;10^{-31}\color{black}\;u^{15}</math> :<math>m_e = \color{blue}9.109\;383\;7015... \;10^{-31}\color{black}\;kg</math> :<math>(\lambda_e^*) = 2 \pi L f_e = \color{purple}2.426\;310\;238\;667 \;10^{-12}\color{black}\;u^{-13}</math> :<math>\lambda_e = \frac{h}{m_e c} = \color{purple}2.426 \;310 \;238 \;67 \;10^{-12}\color{black}\;m</math> ====== c, e, m_e ====== :<math>(m_e^*)= \frac{M}{f_e}, \;f_e = 2^{20} 3^3 \pi^8 \alpha^3 (\color{red}\Omega^{15})^1\color{black} </math>, units = scalars = 1 ([[v:Planck_units_(geometrical)#Electron_formula |m_e formula]]) :<math>\frac{(c^*)^9 (e^*)^4}{(m_e^*)^3} = 2^{97} \pi^{49} 3^9 \alpha^5 (\color{red}\Omega^{15})^5\color{black} = </math> {{font color|red|yellow| '''0.170 514 368... 10⁹²'''}}, <math>\frac{(u^{17})^9 (u^{-27})^4}{(u^{15})^3} = 1,\; (v^9) (\frac{r^3}{v^3})^4 / (\frac{r^4}{v})^3 = 1</math> :<math>\frac{c^9 e^4}{m_e^3} = </math> {{font color|red|yellow| '''0.170 514 342... 10⁹²'''}}, <math>units = \frac{m^9 C^4}{s^9 kg^3} == \frac{(u^{-13})^9 (u^{-27})^4}{(u^{-30})^9 (u^{15})^3} = 1</math> ====== k_B, c, e, m_e ====== :<math>\frac{(k_B^*)}{(e^*)^2 (m_e^*) (c^*)^4} = \frac{3^3 \alpha^6}{2^3 \pi^5} = </math> {{font color|blue|yellow| '''73 035 235 897.'''}}, <math>\frac{(u^{29})}{(u^{-27})^2 (u^{15}) (u^{17})^4} = 1,\; (\frac{r^{10}}{v^3}) / (\frac{r^3}{v^3})^2 (\frac{r^4}{v}) (v)^4 = 1</math> :<math>\frac{k_B}{e^2 m_e c^4} = </math> {{font color|blue|yellow| '''73 095 507 858.'''}}, <math>units = \frac{s^2}{m^2 K C^2} == \frac{(u^{-30})^2}{(u^{-13})^2 (u^{20}) (u^{-27})^2} = 1</math> ====== m_P, t_p, ε₀ ====== These 3 constants, Planck mass, Planck time and the vacuum permittivity have no Omega term. :<math>\frac{M^4 (\epsilon_0^*)}{T} = (1) (\frac{2^9 \pi^3}{\alpha}) / (\pi) = \frac{2^9 \pi^2}{\alpha} = </math> {{font color|green|yellow| '''36.875'''}}, <math>\frac{(u^{15})^4 (u^{-90})}{(u^{-30})} = 1,\; (\frac{r^4}{v})^4 (\frac{1}{r^7 v^2}) / (\frac{r^9}{v^6}) = 1</math> :<math>\frac{m_p^4 (\epsilon_0)}{t_p} = </math> {{font color|green|yellow| '''36.850'''}}, <math>units = \frac{kg^4}{s} \frac{s^4 A^2}{m^3 kg} = \frac{kg^3 A^2 s^3}{m^3} == \frac{(u^{15})^3 (u^{3})^2 (u^{-30})^3}{(u^{-13})^3} = 1</math> ====== G, h, c, e, m_e, K_B ====== :<math>\frac{(h^*) (c^*)^2 (e^*) (m_e^*)}{(G^*)^2 (k_B^*)} = (m_e^*) (\frac{2^{11} \pi^3}{\alpha^2}) = </math> {{font color|red|yellow| '''0.1415... 10^-21'''}}, <math>\frac{ (u^{19}) (u^{17})^2 (u^{-27}) (u^{15}) }{ (u^{6})^2 (u^{29}) } = 1,\; (\frac{r^{13}}{v^5}) v^2 (\frac{r^{3}}{v^3})(\frac{r^{4}}{v^1}) / (\frac{r^5}{v^2})^2 (\frac{r^{10}}{v^3}) = 1</math> :<math>\frac{h c^2 e m_e}{G^2 k_B} = </math> {{font color|red|yellow| '''0.1413... 10^-21'''}}, <math>units = \frac{kg^3 s^3 C K}{m^4} == \frac{(u^{15})^3 (u^{-30})^3 (u^{-27}) (u^{20}) }{(u^{-13})^4} = 1</math> ====== α ====== :<math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{\alpha})^2(2 \pi \Omega^2) = \color{blue}\alpha \color{black},\; \frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1,\; (\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1</math> Note: The above will apply to any combinations of constants (alien or terrestrial) where '''scalars = 1'''. {| class="wikitable" |+Table of dimensioned physical constants ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! SI unit ! Natural constant* (scalars = 1) ! unit number* |- | ''c'' = 299 792 458 (exact) | <math>\frac{m}{s}</math> | ''c*'' = V = 25.3123819329 | 17 |- | ''h'' = 6.626 070 040(81) e-34 | <math>\frac{kg \;m^2}{s}</math> | ''h*'' = <math>2 \pi M V L</math> = 12647.240312 | 19 |- | ''G'' = 6.674 08(31) e-11 | <math>\frac{m^3}{kg \;s^2}</math> | ''G*'' = <math>\frac{V^2 L}{M}</math> = 50950.554778 | 6 |- | ''e'' = 1.602 176 620 8(98) e-19 | <math>C = A s</math> | ''e*'' = <math>A T</math> = 735.706358485 | -27 |- | ''k_B'' = 1.380 648 52(79) e-23 | <math>\frac{kg \;m^2}{s^2 \;K}</math> | ''k_B*'' = <math>\frac{2 \pi V M}{A}</math> = 0.679138336 | 29 |} ===== SI Planck unit scalars ===== :<math>M = m_P = (1)k;\; k = m_P = .217\;672\;817\;58... \;10^{-7},\; u^{15}\; (kg)</math> :<math>T = t_p = {\pi}t;\; t = \frac{t_p}{\pi} = .171\;585\;512\;84... 10^{-43},\; u^{-30}\; (s)</math> :<math>L = l_p = {2\pi^2\Omega^2}l;\; l = \frac{l_p}{2\pi^2\Omega^2} = .203\;220\;869\;48... 10^{-36},\; u^{-13}\; (m)</math> :<math>V = c = {2\pi\Omega^2}v;\; v = \frac{c}{2\pi\Omega^2} = 11\;843\;707.905... ,\; u^{17}\; (m/s)</math> :<math>A = e/t_p = (\frac{2^7 \pi^3 \Omega^3}{\alpha})a = .126\;918\;588\;59... 10^{23},\; u^{3}\; (A)</math> ====== MT to LPVA ====== In this example LPVA are derived from MT. The formulas for MT; :<math>M = (1)k,\; unit = u^{15}</math> :<math>T = (\pi) t,\; unit = u^{-30}</math> Replacing scalars ''pvla'' with ''kt'' :<math>P = (\Omega)\;\frac{k^{12/15}}{t^{2/15}},\; unit = u^{12/15*15-2/15*(-30)=16}</math> :<math>V = \frac{2 \pi P^2}{M} = (2 \pi \Omega^2)\; \frac{k^{9/15}}{t^{4/15}},\; unit = u^{9/15*15-4/15*(-30)=17} </math> :<math>L = T V = (2 \pi^2 \Omega^2) \; k^{9/15} t^{11/15},\; unit = u^{9/15*15+11/15*(-30)=-13}</math> :<math>A = \frac{2^4 V^3}{\alpha P^3} = \left(\frac{2^7 \pi^3 \Omega^3}{\alpha}\right)\; \frac{1}{k^{3/5} t^{2/5}},\; unit = u^{9/15*(-15)+6/15*30=3} </math> ====== PV to MTLA ====== In this example MLTA are derived from PV. The formulas for PV; :<math>P = (\Omega)p,\; unit = u^{16}</math> :<math>V = (2\pi\Omega^2)v,\; unit = u^{17}</math> Replacing scalars ''klta'' with ''pv'' :<math>M = \frac{2\pi P^2}{V} = (1)\frac{p^2}{v},\; unit = u^{16*2-17=15} </math> :<math>T = (\pi) \frac{p^{9/2}}{v^6},\; unit = u^{16*9/2-17*6=-30} </math> :<math>L = T V = (2\pi^2\Omega^2)\frac{p^{9/2}}{v^5},\; unit = u^{16*9/2-17*5=-13}</math> :<math>A = \frac{2^4 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})\frac{v^3}{p^3},\; unit = u^{17*3-16*3=3}</math> ==== Physical constants (as geometrical formulas) ==== note: <math>\color{red}(u^{15})^n\color{black}</math> constants have no Omega term. {| class="wikitable" |+Dimensioned constants; geometrical vs CODATA 2014 ! Constant ! In Planck units ! Geometrical object ! SI calculated (r, v, Ω, α^*) ! SI CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> |- | [[w:Speed of light | Speed of light]] | V | <math>c^* = (2\pi\Omega^2)v,\;u^{17} </math> | ''c^*'' = 299 792 458, unit = u¹⁷ | ''c'' = 299 792 458 (exact) |- | [[w:Fine structure constant | Fine structure constant]] | | | ''α^*'' = 137.035 999 139 (mean) | ''α'' = 137.035 999 139(31) |- | [[w:Rydberg constant | Rydberg constant]] | <math>R^* = (\frac{m_e}{4 \pi L \alpha^2 M})</math> | <math>R^* = \frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}}\frac{v^5}{r^9},\;u^{13} </math> | ''R^*'' = 10 973 731.568 508, unit = u¹³ | ''R'' = 10 973 731.568 508(65) |- | [[w:Vacuum permeability | Vacuum permeability]] | <math>\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2}</math> | <math>\mu_0^* = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{17*2+15+13-6=7*8=56}</math> | ''μ₀^*'' = 4π/10^7, unit = u⁵⁶ | ''μ₀'' = 4π/10^7 (exact) |- | [[w:Vacuum permittivity | Vacuum permittivity]] | <math>\epsilon_0^* = \frac{1}{\mu_0^* (c^*)^2}</math> | <math>\epsilon_0^* = \frac{2^9 \pi^3}{\alpha}\frac{1}{r^7 v^2},\; \color{red}1/(u^{15})^6\color{black} = u^{-90}</math> | | |- | [[w:Planck constant | Planck constant]] | <math>h^* = 2 \pi M V L</math> | <math>h^* = 2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{15+17-13 = 8*13-17*5 = 19}</math> | ''h^*'' = 6.626 069 134 e-34, unit = u¹⁹ | ''h'' = 6.626 070 040(81) e-34 |- | [[w:Gravitational constant | Gravitational constant]] | <math>G^* = \frac{V^2 L}{M}</math> | <math>G^* = 2^3 \pi^4 \Omega^6 \frac{r^5}{v^2},\; u^{34-13-15 = 8*5-17*2 = 6}</math> | ''G^*'' = 6.672 497 192 29 e11, unit = u⁶ | ''G'' = 6.674 08(31) e-11 |- | [[w:Elementary charge | Elementary charge]] | <math>e^* = A T</math> | <math>e^* = \frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3},\; u^{3-30=3*8-17*3=-27}</math> | ''e^*'' = 1.602 176 511 30 e-19, unit = u^-27 | ''e'' = 1.602 176 620 8(98) e-19 |- | [[w:Boltzmann constant | Boltzmann constant]] | <math>k_B^* = \frac{2 \pi V M}{A}</math> | <math>k_B^* = \frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3},\; u^{17+15-3=10*8-17*3=29}</math> | ''k_B^*'' = 1.379 510 147 52 e-23, unit = u²⁹ | ''k_B'' = 1.380 648 52(79) e-23 |- | [[w:Electron mass | Electron mass]] | | <math>m_e^* = \frac{M}{f_e},\; u^{15}</math> | ''m_e^*'' = 9.109 382 312 56 e-31, unit = u¹⁵ | ''m_e'' = 9.109 383 56(11) e-31 |- | [[w:Classical electron radius | Classical electron radius]] | | <math>\lambda_e^* = 2\pi L f_e,\; u^{-13}</math> | ''λ_e^*'' = 2.426 310 2366 e-12, unit = u^-13 | ''λ_e'' = 2.426 310 236 7(11) e-12 |- | [[w:Planck temperature | Planck temperature]] | <math>T_p^* = \frac{A V}{\pi}</math> | <math>T_p^* = \frac{2^7 \pi^3 \Omega^5}{\alpha} \frac{v^4}{r^6} ,\; u^{3+17=17*4-6*8=20} </math> | ''T_p^*'' = 1.418 145 219 e32, unit = u²⁰ | ''T_p'' = 1.416 784(16) e32 |- | [[w:Planck mass | Planck mass]] | M | <math>m_P^* = (1)\frac{r^4}{v} ,\; \color{red}(u^{15})^1\color{black}</math> | ''m_P^*'' = .217 672 817 580 e-7, unit = u¹⁵ | ''m_P'' = .217 647 0(51) e-7 |- | [[w:Planck length | Planck length]] | L = TV | <math>l_p^* = (2\pi^2\Omega^2)\frac{r^9}{v^5},\;u^{-13} </math> | ''l_p^*'' = .161 603 660 096 e-34, unit = u^-13 | ''l_p'' = .161 622 9(38) e-34 |- | [[w:Planck time | Planck time]] | T | <math>t_p^* = (\pi)\frac{r^9}{v^6} ,\; \color{red}1/(u^{15})^2\color{black} </math> | ''t_p^*'' = 5.390 517 866 e-44, unit = u^-30 | ''t_p'' = 5.391 247(60) e-44 |- | [[w:Ampere | Ampere]] | <math>A = \frac{16 V^3}{\alpha P^3}</math> | <math>A^* = \frac{2^7\pi^3\Omega^3}{\alpha}\frac{v^3}{r^6} ,\; u^3 </math> | A^* = 0.297 221 e25, unit = u³ | ''e/t_p'' = 0.297 181 e25 |- | [[w:Quantum Hall effect | Von Klitzing constant ]] | <math>R_K^* = (\frac{h}{e^2})^*</math> | <math>R_K^* = \frac{\alpha^2}{2^{11} \pi^4 \Omega^2} r^7 v ,\; u^{73}</math> | ''R_K^*'' = 25812.807 455 59, unit = u⁷³ | ''R_K'' = 25812.807 455 5(59) |- | [[w:Gyromagnetic ratio | Gyromagnetic ratio]] | | <math>\gamma_e/2\pi = \frac{g l_p^* m_P^*}{2 k_B^* m_e^*},\; unit = u^{-42}</math> | ''γ_e/2π^*'' = 28024.953 55, unit = u^-42 | ''γ_e/2π'' = 28024.951 64(17) |} Note that ''r, v, Ω, α'' are dimensionless numbers, however when we replace ''u''ⁿ with the SI unit equivalents (''u''¹⁵ → kg, ''u''^-13 → m, ''u''^-30 → s, ...), the ''geometrical objects'' (i.e.: ''c^*'' = 2πΩ²v = 299792458, units = u¹⁷) become '''indistinguishable''' from their respective ''physical constants'' (i.e.: ''c'' = 299792458, units = m/s). If this mathematical relationship can therefore be identified within the SI units themselves, then we have an argument for a Planck scale mathematical universe <ref>[https://codingthecosmos.com/planck-scale.html Planck scale mathematical universe model]</ref>. ===== Electron formula ===== {{main|Electron (mathematical)}} Although the Planck units MLTA are embedded within the electron formula ''f_e'', this formula is both unit-less and non scalable (units = 1, scalars = 1). Furthermore it is the geometry of 2 dimensionless physical constants and so can also be defined as a dimensionless physical constant (if units = scalars = 1, then that constant will be independent of any numerical system and of any system of units, and so would qualify as a "natural unit"). :<math>f_e = 4\pi^2(2^6 3 \pi^2 \alpha \Omega^5)^3 = .23895453...x10^{23}</math> AL as an ampere-meter (ampere-length) are the units for a [[w:magnetic monopole | magnetic monopole]]. :<math>T = \pi \frac{r^9}{v^6},\; u^{-30}</math> :<math>\sigma_{e} = \frac{3 \alpha^2 A L}{2\pi^2} = {2^7 3 \pi^3 \alpha \Omega^5}\frac{r^3}{v^2},\; u^{-10}</math> :<math>f_e = \frac{\sigma_{e}^3}{2 T} = \frac{(2^7 3 \pi^3 \alpha \Omega^5)^3}{2\pi},\; units = \frac{(u^{-10})^3}{u^{-30}} = 1, scalars = (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1</math> The electron has dimensioned parameters, however the dimensions derive from the Planck units, ''f_e'' is a mathematical function that dictates how these Planck objects are applied, it does not have dimension units of its own, consequently there is no physical electron. [[w:electron mass | electron mass]] <math>m_e = \frac{M}{f_e}</math> (M = [[w:Planck mass | Planck mass]]) [[w:Compton wavelength | electron wavelength]] <math>\lambda_e = 2\pi L f_e</math> (L = [[w:Planck length | Planck length]]) [[w:elementary charge | elementary charge]] <math>e = A.T</math> ===== Fine structure constant ===== The Sommerfeld [[w:fine-structure constant | fine structure constant alpha]] is a dimensionless physical constant, the CODATA 2018 inverse alpha = 137.035999084. :<math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{\alpha})^2(2 \pi \Omega^2) = \color{blue}\alpha \color{black},\; \frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1,\; (\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1</math> ===== Omega ===== The most precise of the experimentally measured constants is the Rydberg ''R = 10973731.568508(65) 1/m''. Here ''c, μ₀, R'' are combined into a unit-less ratio; :<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = (2 \pi \Omega^2)^{35}/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})^9 .(\frac{1} {2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}})^7,\;units = \frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1</math> We can now define ''Ω'' using the geometries for (''c^*, μ₀^*, R^*'') and then solve by replacing (''c^*, μ₀^*, R^*'') with the numerical (''c, μ₀, R''). :<math>\Omega^{225}=\frac{(c^*)^{35}}{2^{295} 3^{21} \pi^{157} (\mu_0^*)^9 (R^*)^7 \alpha^{26}}, \;units = 1</math> :<math>\Omega = 2.007\;134\;949\;636...,\; units = 1</math> (CODATA 2014 mean values) :<math>\Omega = 2.007\;134\;949\;687...,\; units = 1</math> (CODATA 2018 mean values) There is a close natural number for ''Ω'' that is a square root implying that ''Ω'' can have a plus or a minus solution and this agrees with theory. This solution would however re-classify Omega as a mathematical constant (as being derivable from other mathematical constants). :<math>\Omega = \sqrt{ \left(\frac{\pi^e}{e^{(e-1)}}\right)} = 2.007\;134\;9543... </math> ===== G, h, e, m_e, k_B ===== As geometrical objects, the physical constants (''G, h, e, m_e, k_B'') can also be defined using the geometrical formulas for (''c^*, μ₀^*, R^*'') and solved using the numerical (mean) values for (''c, μ₀, R, α''), i.e.:. :<math>{(h^*)}^3 = (2^3 \pi^4 \Omega^4 \frac{r^{13} u^{19}}{v^5})^3 = \frac{2\pi^{10} {(\mu_0^*)}^3} {3^6 {(c^*)}^5 \alpha^{13} {(R^*)}^2},\; unit = u^{57}</math> {| class="wikitable" |+Physical constants; calculated vs CODATA 2014 ! Constant ! Geometry ! Calculated from (R, c, μ₀, α) ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> |- | [[w:Planck constant | Planck constant]] | <math>{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 \alpha^{13} {R_\infty}^2},\; unit = u^{57}</math> | ''h^*'' = 6.626 069 134 e-34, unit = u¹⁹ | ''h'' = 6.626 070 040(81) e-34 |- | [[w:Gravitational constant | Gravitational constant]] | <math>{(G^*)}^5 = \frac{\pi^3 {\mu_0}}{2^{20} 3^6 \alpha^{11} {R_\infty}^2},\; unit = u^{30}</math> | ''G^*'' = 6.672 497 192 29 e11, unit = u⁶ | ''G'' = 6.674 08(31) e-11 |- | [[w:Elementary charge | Elementary charge]] | <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 {c}^4 \alpha^8 {R_\infty}},\; unit = u^{-81}</math> | ''e^*'' = 1.602 176 511 30 e-19, unit = u^-27 | ''e'' = 1.602 176 620 8(98) e-19 |- | [[w:Boltzmann constant | Boltzmann constant]] | <math>{(k_B^*)}^3 = \frac{\pi^5 {\mu_0}^3}{3^3 2 {c}^4 \alpha^5 {R_\infty}} ,\; unit = u^{87} </math> | ''k_B^*'' = 1.379 510 147 52 e-23, unit = u²⁹ | ''k_B'' = 1.380 648 52(79) e-23 |- | [[w:Electron mass | Electron mass]] | <math>{(m_e^*)}^3 = \frac{16 \pi^{10} {R_\infty} {\mu_0}^3}{3^6 {c}^8 \alpha^7},\; unit = u^{45} </math> | '' m_e^*'' = 9.109 382 312 56 e-31, unit = u¹⁵ | ''m_e'' = 9.109 383 56(11) e-31 |- | [[w:Planck mass | Planck mass]] | <math>(m_P^*)^{15} = \frac{2^{25}\pi^{13} \mu_0^6}{ 3^6 c^5 \alpha^{16} R_\infty^2},\; unit = (u^{15})^{15}</math> | ''m_P^*'' = .217 672 817 580 e-7, unit = u¹⁵ | ''m_P'' = .217 647 0(51) e-7 |- | [[w:Planck length | Planck length]] | <math>(l_p^*)^{15} = \frac{\pi^{22} \mu_0^9}{2^{35} 3^{24} \alpha^{49} c^{35} R_\infty^8},\; unit = (u^{-13})^{15} </math> | ''l_p^*'' = .161 603 660 096 e-34, unit = u^-13 | ''l_p'' = .161 622 9(38) e-34 |- | [[w:Gyromagnetic ratio | Gyromagnetic ratio]] | <math>(\gamma_e/2\pi)^3 = \frac{g^3 3^3 c^4}{2^8 \pi^8 \alpha \mu_0^3 R_\infty^2},\; unit = u^{-126}</math> | ''γ_e/2π^*'' = 28024.953 55, unit = u^-42 | ''γ_e/2π'' = 28024.951 64(17) |} ==== 2019 SI unit revision ==== Following the 26th General Conference on Weights and Measures ([[w:2019 redefinition of SI base units|2019 redefinition of SI base units]]) are fixed the numerical values of the 4 physical constants (''h, c, e, k_B''). In the context of this model however only 2 base units may be assigned by committee as the rest are then numerically fixed by default and so the revision may lead to unintended consequences. For example, the von Klitzing constant Rk = h/e2, yet to derive h and e using Rk suggests an (inverse) alpha = 137.0359952 <ref>[https://codingthecosmos.com/physical-constants-calc.html Physical constants calculator]</ref>. {| class="wikitable" |+Physical constants ! Constant ! CODATA 2018 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2018)</ref> |- | [[w:Speed of light | Speed of light]] | ''c'' = 299 792 458 (exact) |- | [[w:Planck constant | Planck constant]] | ''h'' = 6.626 070 15 e-34 (exact) |- | [[w:Elementary charge | Elementary charge]] | ''e'' = 1.602 176 634 e-19 (exact) |- | [[w:Boltzmann constant | Boltzmann constant]] | ''k_B'' = 1.380 649 e-23 (exact) |- | [[w:Fine structure constant | Fine structure constant]] | ''α'' = 137.035 999 084(21) |- | [[w:Rydberg constant | Rydberg constant]] | ''R'' = 10973 731.568 160(21) |- | [[w:Electron mass | Electron mass]] | ''m_e'' = 9.109 383 7015(28) e-31 |- | [[w:Vacuum permeability | Vacuum permeability]] | ''μ₀'' = 1.256 637 062 12(19) e-6 |- | [[w:Quantum_Hall_effect#Applications | Von Klitzing constant]] | ''R_K'' = 25812.807 45 (exact) |} For example, if we solve using the above formulas; <math>R^* = \frac{4 \pi^5}{3^3 c^4 \alpha^8 e^3} = 10973\;729.082\;465</math> <math>{(m_e^*)}^3 = \frac{2^4 \pi^{10} R \mu_0^3}{3^6 c^8 \alpha^7},\;m_e^* = 9.109\;382\;3259 \;10^{-31}</math> <math>{(\mu_0^*)}^3 = \frac{3^6 h^3 c^5 \alpha^{13} R^2}{2 \pi^{10}},\;\mu_0^* = 1.256\;637\;251\;88\;10^{-6}</math> <math>{(h^*)}^3 = \frac{2 \pi^{10} \mu_0^3}{3^6 c^5 \alpha^{13} R^2},\;h^* = 6.626\;069\;149\;10^{-34}</math> <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 c^4 \alpha^8 R},\; e^* = 1.602\;176\;513\;10^{-19}</math> ==== u as sqrt(velocity/mass) ==== We find there is a single base unit '''u''' from which the other units and numerical values can be derived. This base unit incorporates [[v:Sqrt_Planck_momentum |MLT as square roots]]. =====''u = √{L/M.T}''===== :<math>u,\; units = \sqrt{\frac{L}{M T}} = \sqrt{u^{-13-15+30=2}} = u^1</math> Setting: :<math>x,\;units = \sqrt{\frac{M^9 T^{11}}{L^{15}}} = u^0, units = 1, scalars = 1</math> :<math>y,\;units = M^2T = u^0, units = 1, scalars <> 1</math> Gives; :<math>u^3 = \frac{L^{3/2}}{M^{3/2} T^{3/2}} = A,\; (ampere)</math> :<math>u^6 (y) = \frac{L^3}{T^2 M},\; (G)</math> :<math>u^{13} (xy) = \frac{1}{L},\; (1/l_p)</math> :<math>u^{15} (xy^2) = M,\; (m_P)</math> :<math>u^{17} (xy^2) = V,\; (c)</math> :<math>u^{19} (xy^3) = \frac{ML^2}{T},\; (h)</math> :<math>u^{20} (xy^2) = \frac{L^{5/2}}{M^{3/2} T^{5/2}} = AV,\;(T_P)</math> :<math>u^{27} (x^2y^3) = \frac{M^{3/2}\sqrt{T}}{L^{3/2}} = 1/AT,\; (1/e)</math> :<math>u^{29} (x^2y^4) = \frac{M^{5/2}\sqrt{T}}{\sqrt{L}} = ML/AT,\; (k_B)</math> :<math>u^{30} (x^2 y^3) = \frac{1}{T},\; (1/t_p)</math> :<math>u^{56} (x^4 y^7) = \frac{M^4 T}{L^2},\;(\mu_0)</math> :<math>u^{90} (x^6 y^{11}) = \frac{M^4}{T} </math> ===== ''β'' (unit = ''u'') ===== ''i'' (from ''x'') and ''j'' (from ''y''). :<math>R = \sqrt{P} = \sqrt{\Omega} r,\; units = u^8</math> :<math>\beta = \frac{V}{R^2} = \frac{2\pi R^2}{M} = \frac{A^{1/3} \alpha^{1/3}}{2} \;..., \; unit = u</math> :<math>i = \frac{1}{2\pi {(2\pi \Omega)}^{15}},\; units = 1, scalars = 1</math> :<math>j = \frac{r^{17}}{v^8} = k^2t = \frac{k^8}{r^{15}} ...,\; unit = \frac{u^{17*8}}{u^{8*17}} = u^{15*2}u^{-30} ... = 1,\; units = 1, scalars <> 1</math> For example; the constants solved in terms of (''r, v'') :<math>\beta = \frac{V}{R^2} = \frac{2\pi \Omega^2 v}{\Omega r^2} = \frac{2\pi \Omega v}{r^2},\; u^1 = u</math> :<math>A = \beta^3 (\frac{2^4}{\alpha}) = \frac{2^7 \pi^3 \Omega^3}{\alpha}\frac{v^3}{r^6},\; u^3</math> :<math>G = \frac{\beta^6}{2^3 \pi^2} (j) = 2^3 \pi^4 \Omega^6 \frac{r^5}{v^2},\; u^6</math> :<math>L^{-1} = 4\pi \beta^{13} (ij) = \frac{1}{2\pi^2 \Omega^2} \frac{v^5}{r^9},\; u^{13}</math> :<math>M = 2\pi \beta^{15} (ij^2) = \frac{r^4}{v},\; u^{15}</math> :<math>P = \beta^{16} (ij^2) = \Omega r^2,\; u^{16}</math> :<math>V = \beta^{17} (ij^2) = 2\pi \Omega^2 v,\; u^{17}</math> :<math>h = \pi \beta^{19} (ij^3) = 8\pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{19}</math> :<math>T_P^* =\frac{2^3 \beta^{20}}{\pi \alpha} (ij^2) = \frac{2^7 \pi^3 \Omega^5}{\alpha} \frac{v^4}{r^6} ,\; u^{20} </math> :<math>e^{-1} = \frac{\alpha \pi \beta^{27} (i^2j^3)}{4} = \frac{\alpha}{128\pi^4 \Omega^3} \frac{v^3}{r^{3}},\; u^{27} </math> :<math>k_B = \frac{\alpha \pi^2 \beta^{29}(i^2j^4)}{4} = \frac{\alpha}{32 \pi \Omega} \frac{r^{10}}{v^3},\; u^{29}</math> :<math>T^{-1} = 4\pi \beta^{30} (i^2 j^3) = \frac{1}{\pi}\frac{v^6}{r^9},\; u^{30}</math> :<math>\mu_0^* = \frac{\pi^3 \alpha \beta^{56}}{2^3} (i^4 j^7) = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{56}</math> :<math>\epsilon_0^{*-1} = \mu_0^* (c^*)^2 = \frac{\pi^3 \alpha \beta^{90}}{2^3} (i^6 j^{11}) = \frac{\alpha}{2^9 \pi^3} v^2 r^7,\; u^{90} </math> ===== limit ''j'' ===== The SI values for ''j'' suggest a limit (numerical boundary) to the values the SI constants can have. :<math>j = \frac{r^{17}}{v^8} = k^2 t = \frac{k^{17/4}}{v^{15/4}} = ... </math> gives a range from 0.812997... ''x''10^-59 to 0.123... ''x''10⁶⁰ In SI terms unit ''β'' can be derived via these ratio; :<math>a^{1/3} = \frac{v}{r^2} = \frac{1}{t^{2/15}k^{1/5}} = \frac{\sqrt{v}}{\sqrt{k}} ... = 23326079.1...; unit = u</math> ===== Rydberg formula ===== The [[w:Rydberg_formula |Rydberg formula]] can now be re-written in terms of amperes <math>A^2</math> :<math>\frac{hc}{2\pi \alpha^2} = \frac{j^2 A^2}{2^8 2\pi t_p}</math> ==== External links ==== * [[v:electron_(mathematical) | Mathematical electron]] * [[v:Relativity_(Planck) | Programming relativity at the Planck scale]] * [[v:Quantum_gravity_(Planck) | Programming gravity at the Planck scale]] * [[v:Black-hole_(Planck) | Programming the cosmic microwave background at the Planck scale]] * [[v:Sqrt_Planck_momentum | The sqrt of Planck momentum]] * [[v:God_(programmer) | The Programmer God]] * [[w:Simulation_hypothesis | The Simulation hypothesis]] * [https://codingthecosmos.com/ Programming at the Planck scale using geometrical objects] -Malcolm Macleod's website * [http://www.simulation-argument.com/ Simulation Argument] -Nick Bostrom's website * [https://www.amazon.com/Our-Mathematical-Universe-Ultimate-Reality/dp/0307599809 Our Mathematical Universe: My Quest for the Ultimate Nature of Reality] -Max Tegmark * [https://dx.doi.org/10.2139/ssrn.2531429 The mathematical electron model in a Planck scale universe] -(article) * [https://link.springer.com/article/10.1134/S0202289308020011/ Dirac-Kerr-Newman black-hole electron] -Alexander Burinskii (article) ==== References ==== {{Reflist}} [[Category: Physics]] [[Category: Philosophy of science]] miyhxbutmxxv8ccjoeozm11aq8bdxp0 2416243 2416213 2022-08-18T17:57:14Z 177.227.10.164 wikitext text/x-wiki {| class="wikitable" |+Unit relationship |- | kg = 15 |- | m = -13 |- | s = -30 |- | A = 3 |- | K = 20 |} {| class="wikitable" |+CODATA 2014: R=10973731.568508 (u=13) ... c=299792458 (u=17) ... μ₀=4π/10⁷ (u=56) ... α=137.035999139 (u=0) ! Constant ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! SI units ! Formula ! Calculated from (R, c, μ₀, α) |- | [[w:Planck constant | Planck constant]] | ''h'' = 6.626 070 040(81) e-34 | <math>\frac{kg \;m^2}{s}</math>, u = 19 | <math>{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 \alpha^{13} {R}^2}</math>, u = 57 | ''h^*'' = 6.626 069 134 e-34, u = 19 |- | [[w:Gravitational constant | Gravitational constant]] | ''G'' = 6.674 08(31) e-11 | <math>\frac{m^3}{kg \;s^2}</math>, u = 6 | <math>{(G^*)}^5 = \frac{\pi^3 {\mu_0}}{2^{20} 3^6 \alpha^{11} {R}^2}</math>, u = 30 | ''G^*'' = V²L/M = 6.672 497 192 29 e11, u = 6 |- | [[w:Elementary charge | Elementary charge]] | ''e'' = 1.602 176 620 8(98) e-19 | <math>A s</math>, u = -27 | <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 {c}^4 \alpha^8 {R}}</math>, u = -81 | ''e^*'' = 1.602 176 511 30 e-19, u = -27 |- | [[w:Boltzmann constant | Boltzmann constant]] | ''k_B'' = 1.380 648 52(79) e-23 | <math>\frac{kg \;m^2}{s^2 \;K}</math>, u = 29 | <math>{(k_B^*)}^3 = \frac{\pi^5 {\mu_0}^3}{3^3 2 {c}^4 \alpha^5 {R}}</math>, u = 87 | ''k_B^*'' = 1.379 510 147 52 e-23, u = 29 |- | | ''k_Bec/h'' = 1.000 825 132 | <math>\frac{m \;A}{s \;K}</math>, u = 0 | <math>x = \frac{(k_B^*) (e^*) c}{(h^*)}</math>, u = 0 | x = 1.0 |- | [[w:Electron mass | Electron mass]] | ''m_e'' = 9.109 383 56(11) e-31 | <math>kg</math>, u = 15 | <math>{(m_e^*)}^3 = \frac{16 \pi^{10} {R} {\mu_0}^3}{3^6 {c}^8 \alpha^7}</math>, u = 45 | '' m_e^*'' = m_P^*/f_e = 9.109 382 312 56 e-31, u = 15 |- | | | <math>\frac{A^3 m^3}{s}</math>, u = 0 | <math>f_e = {(\frac{3 \alpha^2 ({e^*}) c}{2\pi^2})}^3 (\frac{c}{2 ({l_p^*})})</math>, u = 0 | f_e = 0.238 954 530 737 e23 |- | | ''8π⁵k_B/(27α⁶e²m_ec⁴)'' = 1.000 824 93 | <math>\frac{1}{m^2 A^2 K}</math>, u = 0 | <math>x = \frac{2^3 \pi^5 ({k_B}^*)}{3^3 \alpha^6 ({e^*})^2 ({m_e}^*) c^4}</math>, u = 0 | x = 1.0 |- | [[w:Planck mass | Planck mass]] | ''m_P'' = 0.217 647 0(51) e-7 | <math>kg</math>, u = 15 | <math>({m_P^*})^{15} = \frac{2^{25} \pi^{13} {\mu_0}^6}{3^6 c^5 \alpha^{16} R^2}</math>, u = 225 | ''m_P^*'' = 0.217 672 817 580 e-7, u = 15 |- | [[w:Planck length | Planck length]] | ''l_p'' = 0.161 622 9(38) e-34 | <math>m</math>, u = -13 | <math>({l_p^*})^{15} = \frac{\pi^{22} {\mu_0}^9}{2^{35} 3^{24} \alpha^{49} c^{35} R^8}</math>, u = -195 | ''l_p^*'' = 0.161 603 660 096 e-34, u = -13 |} https://en.wikiversity.org/wiki/Planck_units_(geometrical) {| class="wikitable" |+Dimensioned physical constants ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! unit ! Natural constant* ! unit* |- | ''c'' = 299 792 458 (exact) | <math>\frac{m}{s}</math> | ''c*'' = V = 25.3123819329 * <math>v</math> = 299792458 | 17 |- | ''h'' = 6.626 070 040(81) e-34 | <math>\frac{kg \;m^2}{s}</math> | ''h*'' = <math>2 \pi M V L</math> = 12647.240312 * <math>\frac{r^{13}}{v^5}</math> = 6.626 069 134 e-34 | 19 |- | ''G'' = 6.674 08(31) e-11 | <math>\frac{m^3}{kg \;s^2}</math> | ''G*'' = <math>\frac{V^2 L}{M}</math> = 50950.554778 * <math>\frac{r^5}{v^2}</math> = 6.672 497 192 29 e11 | 6 |- | ''e'' = 1.602 176 620 8(98) e-19 | <math>C = A s</math> | ''e*'' = <math>A T</math> = 735.706358485 * <math>\frac{r^3}{v^3}</math> = 1.602 176 511 30 e-19 | -27 |- | ''k_B'' = 1.380 648 52(79) e-23 | <math>\frac{kg \;m^2}{s^2 \;K}</math> | ''k_B*'' = <math>\frac{2 \pi V M}{A}</math> = 0.679138336 * <math>\frac{r^{10}}{v^3}</math> = 1.379 510 147 52 e-23 | 29 |} AL as an ampere-meter (ampere-length) are the units for a [[w:magnetic monopole | magnetic monopole]]. :<math>T = \pi,\; u^{-30}</math> :<math>\sigma_{e} = \frac{3 \alpha^2 A L}{2\pi^2},\; u^{-10}</math> :<math>f_e = \frac{\sigma_{e}^3}{2 T} = \frac{(2^7 3 \pi^3 \alpha \Omega^5)^3}{2\pi},\; units = \frac{(u^{-10})^3}{u^{-30}} = 1</math> :<math>f_e = 4\pi^2(2^6 3 \pi^2 \alpha \Omega^5)^3 = .23895453...x10^{23}</math> {| class="wikitable" |+Geometrical units ! Attribute ! Geometrical object ! Unit number ! Scalar |- | mass | <math>M = (1)</math> | 15 | k |- | time | <math>T = (\pi)</math> | -30 | t |- | velocity | <math>V = (2\pi\Omega^2)</math> | 17 | v |- | length | <math>L = (2\pi^2\Omega^2)</math> | -13 | l |- | ampere | <math>A = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> | 3 | a |} {| class="wikitable" |+Geometrical units and scalars ! Geometrical object ! Scalar ! Unit number |- | <math>M = (1)</math> | k | <math>15</math> |- | <math>T = (\pi)</math> | t | <math>-30</math> |- | <math>V = (2\pi\Omega^2)</math> | v | <math>17</math> |- | <math>L = (2\pi^2\Omega^2)</math> | l | <math>-13</math> |- | <math>A = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> | a | <math>3</math> |} {| class="wikitable" |+Geometrical units ! Attribute ! Geometrical object ! Scalar ! Unit |- | mass | <math>M = (1)</math> | k | <math>unit = u^{15}</math> |- | time | <math>T = (\pi)</math> | t | <math>unit = u^{-30}</math> |- | velocity | <math>V = (2\pi\Omega^2)</math> | v | <math>unit = u^{17}</math> |- | length | <math>L = (2\pi^2\Omega^2)</math> | l | <math>unit = u^{-13}</math> |- | ampere | <math>A = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> | a | <math>unit = u^3</math> |} {| class="wikitable" |+Physical constants; calculated vs CODATA 2014 ! Constant ! Geometry ! Calculated from (R, c, μ₀, α) ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> |- | [[w:Planck constant | Planck constant]] | <math>{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 \alpha^{13} {R}^2},\; unit = u^{57}</math> | ''h^*'' = 6.626 069 134 e-34, unit = u¹⁹ | ''h'' = 6.626 070 040(81) e-34 |- | [[w:Gravitational constant | Gravitational constant]] | <math>{(G^*)}^5 = \frac{\pi^3 {\mu_0}}{2^{20} 3^6 \alpha^{11} {R}^2},\; unit = u^{30}</math> | ''G^*'' = 6.672 497 192 29 e11, unit = u⁶ | ''G'' = 6.674 08(31) e-11 |- | [[w:Elementary charge | Elementary charge]] | <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 {c}^4 \alpha^8 {R}},\; unit = u^{-81}</math> | ''e^*'' = 1.602 176 511 30 e-19, unit = u^-27 | ''e'' = 1.602 176 620 8(98) e-19 |- | [[w:Boltzmann constant | Boltzmann constant]] | <math>{(k_B^*)}^3 = \frac{\pi^5 {\mu_0}^3}{3^3 2 {c}^4 \alpha^5 {R}} ,\; unit = u^{87} </math> | ''k_B^*'' = 1.379 510 147 52 e-23, unit = u²⁹ | ''k_B'' = 1.380 648 52(79) e-23 |- | [[w:Electron mass | Electron mass]] | <math>{(m_e^*)}^3 = \frac{16 \pi^{10} {R} {\mu_0}^3}{3^6 {c}^8 \alpha^7},\; unit = u^{45} </math> | '' m_e^*'' = 9.109 382 312 56 e-31, unit = u¹⁵ | ''m_e'' = 9.109 383 56(11) e-31 |} {| class="wikitable" |+Dimensioned physical constants ! SI constant ! Natural constant ! unit |- | <math>\frac{k_B e c}{h} =</math> {{font color|blue|yellow|'''1.000 8254'''}} | <math>\frac{(k_B^*) (e^*) (c^*)}{(h^*)}</math> = {{font color|blue|yellow|'''1.0'''}} | <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1</math> |- | <math>\frac{h^3}{e^{13} c^{24}} =</math> {{font color|green|yellow| '''0.228 473 639... 10^-58'''}} | <math>\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} =</math> {{font color|green|yellow| '''0.228 473 759... 10^-58'''}} | <math>\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1</math> |- | <math>\frac{c^{35}}{\mu_0^9 R^7} =</math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}} | <math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} =</math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}} | <math>\frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1</math> |- | <math>\frac{c^9 e^4}{m_e^3} =</math> {{font color|blue|yellow| '''0.170 514 342... 10⁹²'''}} | <math>\frac{(c^*)^9 (e^*)^4}{(m_e^*)^3} =</math> {{font color|blue|yellow| '''0.170 514 368... 10⁹²'''}} | <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1</math> |- | <math>\frac{k_B}{e^2 m_e c^4} =</math> {{font color|green|yellow| '''73 095 507 858.'''}} | <math>\frac{(k_B^*)}{(e^*)^2 (m_e^*) (c^*)^4} =</math> {{font color|green|yellow| '''73 035 235 897.'''}} | <math>\frac{(u^{29})}{(u^{-27})^2 (u^{15}) (u^{17})^4} = 1</math> |- | <math>\frac{h c^2 e m_e}{G^2 k_B} =</math> {{font color|red|yellow| '''0.1413... 10^-21'''}} | <math>\frac{(h^*) (c^*)^2 (e^*) (m_e^*)}{(G^*)^2 (k_B^*)} =</math> {{font color|red|yellow| '''0.1415... 10^-21'''}} | <math>\frac{ (u^{19}) (u^{17})^2 (u^{-27}) (u^{15}) }{ (u^{6})^2 (u^{29}) } = 1</math> |- | <math>\frac{2 h}{\mu_0\; e^2 \;c} = \color{blue}\alpha \color{black}</math> | <math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = \color{blue}\alpha \color{black}</math> | <math> \frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1</math> |} {| class="wikitable" |+Dimensioned physical constants ! Constant ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! unit |- | [[w:Speed of light | Speed of light]] | ''c'' = 299 792 458 (exact) | <math>\frac{m}{s}</math> |- | [[w:Planck constant | Planck constant]] | ''h'' = 6.626 070 040(81) e-34 | <math>\frac{kg \;m^2}{s}</math> |- | [[w:Gravitational constant | Gravitational constant]] | ''G'' = 6.674 08(31) e-11 | <math>\frac{m^3}{kg \;s^2}</math> |- | [[w:Elementary charge | Elementary charge]] | ''e'' = 1.602 176 620 8(98) e-19 | <math>C = A s</math> |- | [[w:Boltzmann constant | Boltzmann constant]] | ''k_B'' = 1.380 648 52(79) e-23 | <math>\frac{kg \;m^2}{s^2 \;K}</math> |- | [[w:Vacuum permeability | Vacuum permeability]] | ''μ₀'' = 4π/10^7 (exact) | <math>\frac{kg \;m}{s^2 \;A^2}</math> |- | [[w:Electron mass | Electron mass]] | ''m_e'' = 9.109 383 56(11) e-31 | <math>kg</math> |} {| class="wikitable" |+Dimensioned physical constants ! SI constant ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! unit ! Natural constant* ! unit* |- | [[w:Speed of light | Speed of light]] | ''c'' = 299 792 458 (exact) | <math>\frac{m}{s}</math> | ''c*'' = V | 17 |- | [[w:Planck constant | Planck constant]] | ''h'' = 6.626 070 040(81) e-34 | <math>\frac{kg \;m^2}{s}</math> | <math>h^* = 2 \pi M V L</math> | 19 |- | [[w:Gravitational constant | Gravitational constant]] | ''G'' = 6.674 08(31) e-11 | <math>\frac{m^3}{kg \;s^2}</math> | <math>G^* = \frac{V^2 L}{M}</math> | 6 |- | [[w:Elementary charge | Elementary charge]] | ''e'' = 1.602 176 620 8(98) e-19 | <math>C = A s</math> | <math>e^* = A T</math> | -27 |- | [[w:Boltzmann constant | Boltzmann constant]] | ''k_B'' = 1.380 648 52(79) e-23 | <math>\frac{kg \;m^2}{s^2 \;K}</math> | <math>k_B^* = \frac{2 \pi V M}{A}</math> | 29 |- | [[w:Vacuum permeability | Vacuum permeability]] | ''μ₀'' = 4π/10^7 (exact) | <math>\frac{kg \;m}{s^2 \;A^2}</math> | <math>\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2}</math> | 56 |} {| class="wikitable" |+Dimensionless physical constants ! Constant ! value |- | [[w:Fine structure constant | Fine structure constant]] | ''α'' = 137.035 999 139(31) |- | [[v:Planck_units_(geometrical)#Omega | Omega]] | ''Ω'' = 2.007 134 9496 |} {| class="wikitable" |+Dimensioned physical constants ! Constant ! value ! unit |- | [[w:Planck mass | Planck mass]] | ''m_P'' = .217 647 0(51) e-7 | <math>kg</math> |- | [[w:Planck length | Planck length]] | ''l_p'' = .161 622 9(38) e-34 | <math>m</math> |- | [[w:Planck time | Planck time]] | ''t_p'' = 5.391 247(60) e-44 | <math>s</math> |} '''Natural Planck units as geometrical objects''' [[w:Planck units |Planck unit]] theories use basic units for mass, length, time and charge, and operate at the Planck scale. In a geometrical Planck theory, these basic units are assigned geometrical objects (''MLTA'') rather than numerical values, the advantage being that the geometries themselves can encode the function of the unit, for example the object for length (''L'') will encode the function of ''length'', the geometrical ''L'' is 1 unit of (Planck) length, such that, unlike numerical models, a dimensioned descriptive (i.e.: ''kg, m, s, A'', ... ) is not required. The ''MLTA'' geometrical objects are selected whereby they may interact with each other (the mass object for example is not independent of the length and the time objects). This permits a mathematical unit relationship linking the objects, and so a physical universe can be constructed [[w:Lego |Lego-style]] by combining the base (Planck unit) ''MLTA'' objects to form more complex objects such as electrons (i.e.: by embedding ''L'' and ''A'' into the geometry of the electron, the electron can have wavelength and charge). Furthermore, these objects can overlap and cancel in a particular ratio (according to that unit relationship), and this ratio occurs in the electron. And so, although the electron has physical parameters (wavelength, charge ...), '''the electron itself is a mathematical particle (units = 1)''', not a physical particle. === Geometrical objects === Base units for mass <math>M</math>, length <math>L</math>, time <math>T</math>, and ampere <math>A</math> can be constructed from the geometry of 2 [[w:dimensionless physical constant | dimensionless physical constants]], the (inverse) [[w:fine-structure constant | fine structure constant '''α''']] = 137.036 and [[v:Simulation_argument_(coding_Planck_units)#Omega | Omega]] '''Ω''' = 2.007 134 949 <ref>Macleod, M.J. {{Cite journal |title= Programming Planck units from a mathematical electron; a Simulation Hypothesis |journal=Eur. Phys. J. Plus |volume=113 |pages=278 |date=22 March 2018 | doi=10.1140/epjp/i2018-12094-x }}</ref>. Being independent of any numerical system and of any system of units, these MLTA units would qualify as "natural units"; {{bq|''...ihre Bedeutung für alle Zeiten und für alle, auch außerirdische und außermenschliche Kulturen notwendig behalten und welche daher als »natürliche Maßeinheiten« bezeichnet werden können...'' ...These necessarily retain their meaning for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as "natural units"... -Max Planck <ref>Planck (1899), p. 479.</ref><ref name="TOM">*Tomilin, K. A., 1999, "[http://www.ihst.ru/personal/tomilin/papers/tomil.pdf Natural Systems of Units: To the Centenary Anniversary of the Planck System]", 287–296.</ref>}} ==== Objects ==== Each object is assigned a specific geometry, embedded in this geometry is the object function (attribute) and a mathematical relationship between the objects defined using '''uⁿ''' where ''n'' is the unit number; {| class="wikitable" |+Geometrical units ! Attribute ! Geometrical object ! Unit relationship |- | mass | <math>M = (1)</math> | <math>unit = u^{15}</math> |- | time | <math>T = (\pi)</math> | <math>unit = u^{-30}</math> |- | [[v:Sqrt_Planck_momentum | sqrt(momentum)]] | <math>P = (\Omega)</math> | <math>unit = u^{16}</math> |- | velocity | <math>V = (2\pi\Omega^2)</math> | <math>unit = u^{17}</math> |- | length | <math>L = (2\pi^2\Omega^2)</math> | <math>unit = u^{-13}</math> |- | ampere | <math>A = \frac{16 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> | <math>unit = u^3</math> |} ==== Scalars ==== To translate from geometrical objects to a numerical system of units requires scalars ('''kltpva''') that can be assigned numerical values. For example, scalars for the SI units; :If we use '''k''' to convert '''M''' to the SI Planck mass <math>m_P</math> (M = 1k = <math>m_P</math>), then '''k''' = 0.2176728e-7kg and '''<math>u^{15}</math>''' will equate to '''kg'''. :To convert '''V''' = 2πΩ² = 25.3123819 to '''c''' requires scalar '''v''' = 11843707.905m/s ('''V''' ''v'' = 2πΩ²''v'' = 299792458m/s) with '''<math>u^{17}</math>''' equating to '''m/s'''. {| class="wikitable" |+Geometrical units ! Attribute ! Geometrical object ! Scalar ! Unit |- | mass | <math>M = (1)</math> | k | <math>unit = u^{15}</math> |- | time | <math>T = (\pi)</math> | t | <math>unit = u^{-30}</math> |- | [[v:Sqrt_Planck_momentum | sqrt(momentum)]] | <math>P = (\Omega)</math> | p | <math>unit = u^{16}</math> |- | velocity | <math>V = (2\pi\Omega^2)</math> | v | <math>unit = u^{17}</math> |- | length | <math>L = TV = (2\pi^2\Omega^2)</math> | l | <math>unit = u^{-13}</math> |- | ampere | <math>A = \frac{16 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})</math> | a | <math>unit = u^3</math> |} ===== Scalar relationships ===== The scalars follow the same unit relationship. This means that we can find ratios where the scalars cancel. Here the following ''uⁿ'' groups cancel ('''units = scalars = 1'''), as such '''only 2 numerical scalars are required''', for example, if we know '''a''' and '''l''' then we know '''t''' ('''t = a³l³'''), and from '''l''' and '''t''' we know '''k'''. :<math>\frac{u^{3*3} u^{-13*3}}{u^{-30}}\;(\frac{a^3 l^3}{t}) = \frac{u^{-13*15}}{u^{15*9} u^{-30*11}} \;(\frac{l^{15}}{k^9 t^{11}}) = \;...\; =1</math> And so we know any 2 constants, then we can solve the scalars for those constants, and from those 2 scalars we can solve the Planck units, and from these the dimensioned physical constants. This will apply to any set of units. In this example, to maintain integer exponents, a scalar ''p'' is defined in terms of a scalar ''r''. :<math>r = \sqrt{p} = \sqrt{\Omega},\; unit \;u^{16/2=8}</math> The SI Planck units are known with a low precision, conversely 2 of the CODATA 2014 physical constants have been assigned exact numerical values; ''c'' and permeability of vacuum ''μ₀''. Scalars ''r'' and ''v'' were chosen as they can be derived directly from ''V = c'' and ''μ₀'' = 4π/10^7 (see table [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |geometrical physical constants]] below). Using α = 137.035 999 139 (CODATA 2014), Ω = 2.007 134 949 636... :<math>v = \frac{c}{2 \pi \Omega^2}= 11 843 707.905 ...,\; units = m/s</math> :<math>r^7 = \frac{2^{11} \pi^5 \Omega^4 \mu_0}{\alpha};\; r = 0.712 562 514 304 ...,\; units = (\frac{kg.m}{s})^{1/4}</math> This gives scalars ''klta'' ([[v:Planck_units_(geometrical)#u_as_sqrt(velocity/mass)) |for derivation of units kg, m, s, A from r, v]]); :<math>k = \frac{r^4}{v}</math> = 0.217 672 817 580... ''x'' 10^-7kg, <math>\;\;\;u^{15} = \frac{(u^8)^4}{u^{17}}</math> :<math>l = \frac{r^9}{v^5}</math> = 0.203 220 869 487... ''x'' 10^-36m, <math>\;\;\;u^{-13} = \frac{(u^8)^9}{(u^{17})^5}</math> :<math>t = \frac{r^9}{v^6}</math> = 0.171 585 512 841... ''x'' 10^-43s, <math>\;\;\;u^{-30} = \frac{(u^8)^9}{(u^{17})^6}</math> :<math>a = \frac{v^3}{r^6}</math> = 0.126 918 588 592... ''x'' 10²³A, <math>\;\;\;u^{3} = \frac{(u^{17})^3}{(u^8)^6}</math> ===== Natural units MLTPA ===== Regardless of which system of units we use, alien or terrestrial, any combination of constants where '''scalars = 1''' (i.e.: the scalars overlap and cancel) will give the same numerical result, they will default to the MLTPA objects. This implies that these objects are Planck's 'natural' units, i.e.: that '''all possible systems of units''' are based on these objects, and so, given that these are geometrical objects, they can be construed as evidence of a mathematical universe. The following are examples of '''units = scalars = 1''' ratios using SI units <ref>Macleod, Malcolm J. {{Cite journal |title= Do the fundamental constants embed evidence of a mathematical universe at the Planck scale? |journal=RG | doi=10.13140/RG.2.2.15874.15041/1 }}</ref>. Note: the geometry <math>\color{red}(\Omega^{15})^n\color{black}</math> (integer n ≥ 0) is common to these ratios. ====== m_P, l_p, t_p ====== In this ratio, the MLT units and ''klt'' scalars both cancel; units = scalars = 1, reverting to the base MLT objects. Setting the scalars ''klt'' for SI Planck units; :k = 0.217 672 817 580... ''x'' 10^-7kg :l = 0.203 220 869 487... ''x'' 10^-36m :t = 0.171 585 512 841... ''x'' 10^-43s :<math>\frac{L^{15}}{M^{9} T^{11}} = \frac{(2\pi^2\Omega^2)^{15}}{(1)^{9} (\pi)^{11}} (\frac{l^{15}}{k^9 t^{11}}) = \frac{l_p^{15}}{m_P^{9} t_p^{11}} </math> (CODATA 2018 mean) The ''klt'' scalars cancel, leaving; :<math>\frac{L^{15}}{M^{9} T^{11}} = \frac{(2\pi^2\Omega^2)^{15}}{(1)^{9} (\pi)^{11}} (\frac{l^{15}}{k^9 t^{11}}) = 2^{15} \pi^{19} \color{red}(\Omega^{15})^2\color{black} = </math>{{font color|blue|yellow|'''0.109 293... 10²⁴ '''}}, <math>(\frac{l^{15}}{k^9 t^{11}}) = 1, \;\frac{u^{-13*15}}{u^{15*9} u^{-30*11}} = 1</math> Solving for the SI units; :<math>\frac{l_p^{15}}{m_P^{9} t_p^{11}} = \frac{(1.616255e-35)^{15}}{(2.176434e-8)^{9} (5.391247e-44)^{11}} = </math> {{font color|blue|yellow| '''0.109 485... 10²⁴'''}} ====== A, l_p, t_p ====== :a = 0.126 918 588 592... ''x'' 10²³A :<math>\frac{A^3 L^3}{T} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})^3 \frac{(2\pi^2\Omega^2)^3}{(\pi)} (\frac{a^3 l^3}{t}) = \frac{2^{24} \pi^{14} \color{red}(\Omega^{15})^1\color{black}}{\alpha^3} = </math> {{font color|green|yellow| '''0.205 571... 10¹³'''}}, <math>(\frac{a^3 l^3}{t}) = 1,\; \frac{u^{3*3} u^{-13*3}}{u^{-30}} = 1</math> :<math>\frac{(e / t_p)^3 l_p^3}{t_p} = \frac{(1.602176634e-19/5.391247e-44)^3 (1.616255e-35)^3}{(5.391247e-44)} = </math> {{font color|green|yellow| '''0.205 543... 10¹³'''}}, <math>units = \frac{(C/s)^3 m^3}{s} </math> The Planck units are known with low precision, and so by defining the 3 most accurately known dimensioned constants in [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |terms of these objects]] (c, R = Rydberg constant, <math>\mu_0</math>; CODATA 2014 mean values), we can test to greater precision; ====== c, μ₀, R ====== :<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = (2 \pi \Omega^2 v)^{35}/(\frac{\alpha r^7}{2^{11} \pi^5 \Omega^4})^9 .(\frac{v^5} {2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17} r^9})^7 = 2^{295} \pi^{157} 3^{21} \alpha^{26} \color{red}(\Omega^{15})^{15}\color{black} = </math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}}, <math>\frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1, \;(v^{35})/(r^7)^9 (\frac{v^5}{r^9})^7 = 1</math> :<math>\frac{c^{35}}{\mu_0^9 R^7} = \frac{(299792458)^{35}}{(4 \pi/10^7)^9 (10973731.568160)^7} = </math> {{font color|red|yellow| '''0.326 103 528 6170... 10³⁰¹'''}}, <math>units = \frac{m^{33}A^{18}}{s^{17}kg^9} == \frac{(u^{-13})^{33} (u^{3})^{18}}{(u^{-30})^{17} (u^{15})^9} = 1</math> The [[w:2019 redefinition of SI base units | 2019 SI unit revision]] assigned exact numerical values to 4 constants (c, e, k_B, h). {{see also |Planck units (geometrical)#2019 SI unit revision}} From the table [[v:Planck_units_(geometrical)#Physical_constants_(as_geometrical_formulas) |geometrical physical constants]], we get geometrical formulas and scalars for; :<math>h^* = 2 \pi MVL = 2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{15+17-13 = 19}</math> :<math>e^* = AT = \frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3},\; u^{3-30 = -27}</math> :<math>k_B^*= 2 \pi MV/A = \frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3},\; u^{17+15-3 = 29}</math> ====== c, e, k_B, h ====== :<math>\frac{(k_B^*) (e^*) (c^*)}{(h^*)} = (\frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3}) (\frac{2^7 \pi^4 \Omega^3}{\alpha} \frac{r^3}{v^3}) (2 \pi \Omega^2 v) / (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5}) </math> = {{font color|blue|yellow|'''1.0'''}}, <math>\frac{ (u^{29}) (u^{-27}) (u^{17}) }{ (u^{19}) } = 1,\; (\frac{r^{10}}{v^3}) (\frac{r^3}{v^3}) (v) / (\frac{r^{13}}{v^5}) = 1</math> :<math>\frac{k_B e c}{h} = </math> {{font color|blue|yellow|'''1.000 8254'''}}, <math>units = \frac{m C}{s^2 K} == \frac{(u^{-13}) (u^{-27})}{(u^{-30})^2 (u^{20})} = 1</math> ====== c, h, e ====== :<math>\frac{(h^*)^3}{(e^*)^{13} (c^*)^{24}} = (2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5})^3/(\frac{2^7 \pi^4 \Omega^3 r^3}{\alpha v^3})^7.(2\pi\Omega^2 v)^{24} = \frac{\alpha^{13}}{2^{106} \pi^{64} (\color{red}\Omega^{15})^5\color{black}} = </math> {{font color|green|yellow| '''0.228 473 759... 10^-58'''}}, <math>\frac{(u^{19})^{3}}{(u^{-27})^{13} (u^{17})^{24}} = 1, \;(\frac{r^{13}}{v^5})^3 / (\frac{r^3}{v^3})^{13} (v^{24}) = 1</math> :<math>\frac{h^3}{e^{13} c^{24}} = </math> {{font color|green|yellow| '''0.228 473 639... 10^-58'''}}, <math>units = \frac{kg^3 s^{21}}{m^{18} C^{13}} == \frac{(u^{15})^3 (u^{-30})^{21}}{(u^{-13})^{18} (u^{-27})^{13}} = 1</math> ====== m_e, λ_e ====== :<math>\sigma_{e} = \frac{3 \alpha^2 A L}{2\pi^2} = {2^7 3 \pi^3 \alpha \Omega^5}\frac{r^3}{v^2},\; u^{-10}</math> :<math>f_e = \frac{\sigma_{e}^3}{2 T} = 2^{20} 3^3 \pi^8 \alpha^3 (\color{red}\Omega^{15})\color{black},\; \frac{(u^{-10})^3}{u^{-30}} =1,\; (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1</math> :<math>(m_e^*) = \frac{M}{f_e} = \color{blue}9.109\;382\;3227 \;10^{-31}\color{black}\;u^{15}</math> :<math>(m_e^*) = \frac{2^3 \pi^5 (h^*)}{3^3 \alpha^6 (e^*)^3 (c^*)^5} = \frac{1}{2^{20} \pi^8 3^3 \alpha^3 (\color{red}\Omega^{15})\color{black}} \frac{r^4 u^{15}}{v} = \color{blue}9.109\;382\;3227 \;10^{-31}\color{black}\;u^{15}</math> :<math>m_e = \color{blue}9.109\;383\;7015... \;10^{-31}\color{black}\;kg</math> :<math>(\lambda_e^*) = 2 \pi L f_e = \color{purple}2.426\;310\;238\;667 \;10^{-12}\color{black}\;u^{-13}</math> :<math>\lambda_e = \frac{h}{m_e c} = \color{purple}2.426 \;310 \;238 \;67 \;10^{-12}\color{black}\;m</math> ====== c, e, m_e ====== :<math>(m_e^*)= \frac{M}{f_e}, \;f_e = 2^{20} 3^3 \pi^8 \alpha^3 (\color{red}\Omega^{15})^1\color{black} </math>, units = scalars = 1 ([[v:Planck_units_(geometrical)#Electron_formula |m_e formula]]) :<math>\frac{(c^*)^9 (e^*)^4}{(m_e^*)^3} = 2^{97} \pi^{49} 3^9 \alpha^5 (\color{red}\Omega^{15})^5\color{black} = </math> {{font color|red|yellow| '''0.170 514 368... 10⁹²'''}}, <math>\frac{(u^{17})^9 (u^{-27})^4}{(u^{15})^3} = 1,\; (v^9) (\frac{r^3}{v^3})^4 / (\frac{r^4}{v})^3 = 1</math> :<math>\frac{c^9 e^4}{m_e^3} = </math> {{font color|red|yellow| '''0.170 514 342... 10⁹²'''}}, <math>units = \frac{m^9 C^4}{s^9 kg^3} == \frac{(u^{-13})^9 (u^{-27})^4}{(u^{-30})^9 (u^{15})^3} = 1</math> ====== k_B, c, e, m_e ====== :<math>\frac{(k_B^*)}{(e^*)^2 (m_e^*) (c^*)^4} = \frac{3^3 \alpha^6}{2^3 \pi^5} = </math> {{font color|blue|yellow| '''73 035 235 897.'''}}, <math>\frac{(u^{29})}{(u^{-27})^2 (u^{15}) (u^{17})^4} = 1,\; (\frac{r^{10}}{v^3}) / (\frac{r^3}{v^3})^2 (\frac{r^4}{v}) (v)^4 = 1</math> :<math>\frac{k_B}{e^2 m_e c^4} = </math> {{font color|blue|yellow| '''73 095 507 858.'''}}, <math>units = \frac{s^2}{m^2 K C^2} == \frac{(u^{-30})^2}{(u^{-13})^2 (u^{20}) (u^{-27})^2} = 1</math> ====== m_P, t_p, ε₀ ====== These 3 constants, Planck mass, Planck time and the vacuum permittivity have no Omega term. :<math>\frac{M^4 (\epsilon_0^*)}{T} = (1) (\frac{2^9 \pi^3}{\alpha}) / (\pi) = \frac{2^9 \pi^2}{\alpha} = </math> {{font color|green|yellow| '''36.875'''}}, <math>\frac{(u^{15})^4 (u^{-90})}{(u^{-30})} = 1,\; (\frac{r^4}{v})^4 (\frac{1}{r^7 v^2}) / (\frac{r^9}{v^6}) = 1</math> :<math>\frac{m_p^4 (\epsilon_0)}{t_p} = </math> {{font color|green|yellow| '''36.850'''}}, <math>units = \frac{kg^4}{s} \frac{s^4 A^2}{m^3 kg} = \frac{kg^3 A^2 s^3}{m^3} == \frac{(u^{15})^3 (u^{3})^2 (u^{-30})^3}{(u^{-13})^3} = 1</math> ====== G, h, c, e, m_e, K_B ====== :<math>\frac{(h^*) (c^*)^2 (e^*) (m_e^*)}{(G^*)^2 (k_B^*)} = (m_e^*) (\frac{2^{11} \pi^3}{\alpha^2}) = </math> {{font color|red|yellow| '''0.1415... 10^-21'''}}, <math>\frac{ (u^{19}) (u^{17})^2 (u^{-27}) (u^{15}) }{ (u^{6})^2 (u^{29}) } = 1,\; (\frac{r^{13}}{v^5}) v^2 (\frac{r^{3}}{v^3})(\frac{r^{4}}{v^1}) / (\frac{r^5}{v^2})^2 (\frac{r^{10}}{v^3}) = 1</math> :<math>\frac{h c^2 e m_e}{G^2 k_B} = </math> {{font color|red|yellow| '''0.1413... 10^-21'''}}, <math>units = \frac{kg^3 s^3 C K}{m^4} == \frac{(u^{15})^3 (u^{-30})^3 (u^{-27}) (u^{20}) }{(u^{-13})^4} = 1</math> ====== α ====== :<math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{\alpha})^2(2 \pi \Omega^2) = \color{blue}\alpha \color{black},\; \frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1,\; (\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1</math> Note: The above will apply to any combinations of constants (alien or terrestrial) where '''scalars = 1'''. {| class="wikitable" |+Table of dimensioned physical constants ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> ! SI unit ! Natural constant* (scalars = 1) ! unit number* |- | ''c'' = 299 792 458 (exact) | <math>\frac{m}{s}</math> | ''c*'' = V = 25.3123819329 | 17 |- | ''h'' = 6.626 070 040(81) e-34 | <math>\frac{kg \;m^2}{s}</math> | ''h*'' = <math>2 \pi M V L</math> = 12647.240312 | 19 |- | ''G'' = 6.674 08(31) e-11 | <math>\frac{m^3}{kg \;s^2}</math> | ''G*'' = <math>\frac{V^2 L}{M}</math> = 50950.554778 | 6 |- | ''e'' = 1.602 176 620 8(98) e-19 | <math>C = A s</math> | ''e*'' = <math>A T</math> = 735.706358485 | -27 |- | ''k_B'' = 1.380 648 52(79) e-23 | <math>\frac{kg \;m^2}{s^2 \;K}</math> | ''k_B*'' = <math>\frac{2 \pi V M}{A}</math> = 0.679138336 | 29 |} ===== SI Planck unit scalars ===== :<math>M = m_P = (1)k;\; k = m_P = .217\;672\;817\;58... \;10^{-7},\; u^{15}\; (kg)</math> :<math>T = t_p = {\pi}t;\; t = \frac{t_p}{\pi} = .171\;585\;512\;84... 10^{-43},\; u^{-30}\; (s)</math> :<math>L = l_p = {2\pi^2\Omega^2}l;\; l = \frac{l_p}{2\pi^2\Omega^2} = .203\;220\;869\;48... 10^{-36},\; u^{-13}\; (m)</math> :<math>V = c = {2\pi\Omega^2}v;\; v = \frac{c}{2\pi\Omega^2} = 11\;843\;707.905... ,\; u^{17}\; (m/s)</math> :<math>A = e/t_p = (\frac{2^7 \pi^3 \Omega^3}{\alpha})a = .126\;918\;588\;59... 10^{23},\; u^{3}\; (A)</math> ====== MT to LPVA ====== In this example LPVA are derived from MT. The formulas for MT; :<math>M = (1)k,\; unit = u^{15}</math> :<math>T = (\pi) t,\; unit = u^{-30}</math> Replacing scalars ''pvla'' with ''kt'' :<math>P = (\Omega)\;\frac{k^{12/15}}{t^{2/15}},\; unit = u^{12/15*15-2/15*(-30)=16}</math> :<math>V = \frac{2 \pi P^2}{M} = (2 \pi \Omega^2)\; \frac{k^{9/15}}{t^{4/15}},\; unit = u^{9/15*15-4/15*(-30)=17} </math> :<math>L = T V = (2 \pi^2 \Omega^2) \; k^{9/15} t^{11/15},\; unit = u^{9/15*15+11/15*(-30)=-13}</math> :<math>A = \frac{2^4 V^3}{\alpha P^3} = \left(\frac{2^7 \pi^3 \Omega^3}{\alpha}\right)\; \frac{1}{k^{3/5} t^{2/5}},\; unit = u^{9/15*(-15)+6/15*30=3} </math> ====== PV to MTLA ====== In this example MLTA are derived from PV. The formulas for PV; :<math>P = (\Omega)p,\; unit = u^{16}</math> :<math>V = (2\pi\Omega^2)v,\; unit = u^{17}</math> Replacing scalars ''klta'' with ''pv'' :<math>M = \frac{2\pi P^2}{V} = (1)\frac{p^2}{v},\; unit = u^{16*2-17=15} </math> :<math>T = (\pi) \frac{p^{9/2}}{v^6},\; unit = u^{16*9/2-17*6=-30} </math> :<math>L = T V = (2\pi^2\Omega^2)\frac{p^{9/2}}{v^5},\; unit = u^{16*9/2-17*5=-13}</math> :<math>A = \frac{2^4 V^3}{\alpha P^3} = (\frac{2^7 \pi^3 \Omega^3}{\alpha})\frac{v^3}{p^3},\; unit = u^{17*3-16*3=3}</math> ==== Physical constants (as geometrical formulas) ==== note: <math>\color{red}(u^{15})^n\color{black}</math> constants have no Omega term. {| class="wikitable" |+Dimensioned constants; geometrical vs CODATA 2014 ! Constant ! In Planck units ! Geometrical object ! SI calculated (r, v, Ω, α^*) ! SI CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> |- | [[w:Speed of light | Speed of light]] | V | <math>c^* = (2\pi\Omega^2)v,\;u^{17} </math> | ''c^*'' = 299 792 458, unit = u¹⁷ | ''c'' = 299 792 458 (exact) |- | [[w:Fine structure constant | Fine structure constant]] | | | ''α^*'' = 137.035 999 139 (mean) | ''α'' = 137.035 999 139(31) |- | [[w:Rydberg constant | Rydberg constant]] | <math>R^* = (\frac{m_e}{4 \pi L \alpha^2 M})</math> | <math>R^* = \frac{1}{2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}}\frac{v^5}{r^9},\;u^{13} </math> | ''R^*'' = 10 973 731.568 508, unit = u¹³ | ''R'' = 10 973 731.568 508(65) |- | [[w:Vacuum permeability | Vacuum permeability]] | <math>\mu_0^* = \frac{4 \pi V^2 M}{\alpha L A^2}</math> | <math>\mu_0^* = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{17*2+15+13-6=7*8=56}</math> | ''μ₀^*'' = 4π/10^7, unit = u⁵⁶ | ''μ₀'' = 4π/10^7 (exact) |- | [[w:Vacuum permittivity | Vacuum permittivity]] | <math>\epsilon_0^* = \frac{1}{\mu_0^* (c^*)^2}</math> | <math>\epsilon_0^* = \frac{2^9 \pi^3}{\alpha}\frac{1}{r^7 v^2},\; \color{red}1/(u^{15})^6\color{black} = u^{-90}</math> | | |- | [[w:Planck constant | Planck constant]] | <math>h^* = 2 \pi M V L</math> | <math>h^* = 2^3 \pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{15+17-13 = 8*13-17*5 = 19}</math> | ''h^*'' = 6.626 069 134 e-34, unit = u¹⁹ | ''h'' = 6.626 070 040(81) e-34 |- | [[w:Gravitational constant | Gravitational constant]] | <math>G^* = \frac{V^2 L}{M}</math> | <math>G^* = 2^3 \pi^4 \Omega^6 \frac{r^5}{v^2},\; u^{34-13-15 = 8*5-17*2 = 6}</math> | ''G^*'' = 6.672 497 192 29 e11, unit = u⁶ | ''G'' = 6.674 08(31) e-11 |- | [[w:Elementary charge | Elementary charge]] | <math>e^* = A T</math> | <math>e^* = \frac{2^7 \pi^4 \Omega^3}{\alpha}\frac{r^3}{v^3},\; u^{3-30=3*8-17*3=-27}</math> | ''e^*'' = 1.602 176 511 30 e-19, unit = u^-27 | ''e'' = 1.602 176 620 8(98) e-19 |- | [[w:Boltzmann constant | Boltzmann constant]] | <math>k_B^* = \frac{2 \pi V M}{A}</math> | <math>k_B^* = \frac{\alpha}{2^5 \pi \Omega} \frac{r^{10}}{v^3},\; u^{17+15-3=10*8-17*3=29}</math> | ''k_B^*'' = 1.379 510 147 52 e-23, unit = u²⁹ | ''k_B'' = 1.380 648 52(79) e-23 |- | [[w:Electron mass | Electron mass]] | | <math>m_e^* = \frac{M}{f_e},\; u^{15}</math> | ''m_e^*'' = 9.109 382 312 56 e-31, unit = u¹⁵ | ''m_e'' = 9.109 383 56(11) e-31 |- | [[w:Classical electron radius | Classical electron radius]] | | <math>\lambda_e^* = 2\pi L f_e,\; u^{-13}</math> | ''λ_e^*'' = 2.426 310 2366 e-12, unit = u^-13 | ''λ_e'' = 2.426 310 236 7(11) e-12 |- | [[w:Planck temperature | Planck temperature]] | <math>T_p^* = \frac{A V}{\pi}</math> | <math>T_p^* = \frac{2^7 \pi^3 \Omega^5}{\alpha} \frac{v^4}{r^6} ,\; u^{3+17=17*4-6*8=20} </math> | ''T_p^*'' = 1.418 145 219 e32, unit = u²⁰ | ''T_p'' = 1.416 784(16) e32 |- | [[w:Planck mass | Planck mass]] | M | <math>m_P^* = (1)\frac{r^4}{v} ,\; \color{red}(u^{15})^1\color{black}</math> | ''m_P^*'' = .217 672 817 580 e-7, unit = u¹⁵ | ''m_P'' = .217 647 0(51) e-7 |- | [[w:Planck length | Planck length]] | L = TV | <math>l_p^* = (2\pi^2\Omega^2)\frac{r^9}{v^5},\;u^{-13} </math> | ''l_p^*'' = .161 603 660 096 e-34, unit = u^-13 | ''l_p'' = .161 622 9(38) e-34 |- | [[w:Planck time | Planck time]] | T | <math>t_p^* = (\pi)\frac{r^9}{v^6} ,\; \color{red}1/(u^{15})^2\color{black} </math> | ''t_p^*'' = 5.390 517 866 e-44, unit = u^-30 | ''t_p'' = 5.391 247(60) e-44 |- | [[w:Ampere | Ampere]] | <math>A = \frac{16 V^3}{\alpha P^3}</math> | <math>A^* = \frac{2^7\pi^3\Omega^3}{\alpha}\frac{v^3}{r^6} ,\; u^3 </math> | A^* = 0.297 221 e25, unit = u³ | ''e/t_p'' = 0.297 181 e25 |- | [[w:Quantum Hall effect | Von Klitzing constant ]] | <math>R_K^* = (\frac{h}{e^2})^*</math> | <math>R_K^* = \frac{\alpha^2}{2^{11} \pi^4 \Omega^2} r^7 v ,\; u^{73}</math> | ''R_K^*'' = 25812.807 455 59, unit = u⁷³ | ''R_K'' = 25812.807 455 5(59) |- | [[w:Gyromagnetic ratio | Gyromagnetic ratio]] | | <math>\gamma_e/2\pi = \frac{g l_p^* m_P^*}{2 k_B^* m_e^*},\; unit = u^{-42}</math> | ''γ_e/2π^*'' = 28024.953 55, unit = u^-42 | ''γ_e/2π'' = 28024.951 64(17) |} Note that ''r, v, Ω, α'' are dimensionless numbers, however when we replace ''u''ⁿ with the SI unit equivalents (''u''¹⁵ → kg, ''u''^-13 → m, ''u''^-30 → s, ...), the ''geometrical objects'' (i.e.: ''c^*'' = 2πΩ²v = 299792458, units = u¹⁷) become '''indistinguishable''' from their respective ''physical constants'' (i.e.: ''c'' = 299792458, units = m/s). If this mathematical relationship can therefore be identified within the SI units themselves, then we have an argument for a Planck scale mathematical universe <ref>[https://codingthecosmos.com/planck-scale.html Planck scale mathematical universe model]</ref>. ===== Electron formula ===== {{main|Electron (mathematical)}} Although the Planck units MLTA are embedded within the electron formula ''f_e'', this formula is both unit-less and non scalable (units = 1, scalars = 1). Furthermore it is the geometry of 2 dimensionless physical constants and so can also be defined as a dimensionless physical constant (if units = scalars = 1, then that constant will be independent of any numerical system and of any system of units, and so would qualify as a "natural unit"). :<math>f_e = 4\pi^2(2^6 3 \pi^2 \alpha \Omega^5)^3 = .23895453...x10^{23}</math> AL as an ampere-meter (ampere-length) are the units for a [[w:magnetic monopole | magnetic monopole]]. :<math>T = \pi \frac{r^9}{v^6},\; u^{-30}</math> :<math>\sigma_{e} = \frac{3 \alpha^2 A L}{2\pi^2} = {2^7 3 \pi^3 \alpha \Omega^5}\frac{r^3}{v^2},\; u^{-10}</math> :<math>f_e = \frac{\sigma_{e}^3}{2 T} = \frac{(2^7 3 \pi^3 \alpha \Omega^5)^3}{2\pi},\; units = \frac{(u^{-10})^3}{u^{-30}} = 1, scalars = (\frac{r^3}{v^2})^3 \frac{v^6}{r^9} = 1</math> The electron has dimensioned parameters, however the dimensions derive from the Planck units, ''f_e'' is a mathematical function that dictates how these Planck objects are applied, it does not have dimension units of its own, consequently there is no physical electron. [[w:electron mass | electron mass]] <math>m_e = \frac{M}{f_e}</math> (M = [[w:Planck mass | Planck mass]]) [[w:Compton wavelength | electron wavelength]] <math>\lambda_e = 2\pi L f_e</math> (L = [[w:Planck length | Planck length]]) [[w:elementary charge | elementary charge]] <math>e = A.T</math> ===== Fine structure constant ===== The Sommerfeld [[w:fine-structure constant | fine structure constant alpha]] is a dimensionless physical constant, the CODATA 2018 inverse alpha = 137.035999084. :<math>\frac{2 (h^*)}{(\mu_0^*) (e^*)^2 (c^*)} = 2({2^3 \pi^4 \Omega^4})/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})(\frac{2^{7} \pi^4 \Omega^3}{\alpha})^2(2 \pi \Omega^2) = \color{blue}\alpha \color{black},\; \frac{u^{19}}{u^{56} (u^{-27})^2 u^{17}} = 1,\; (\frac{r^{13}}{v^5})(\frac{1}{r^7})(\frac{v^6}{r^6})(\frac{1}{v}) = 1</math> ===== Omega ===== The most precise of the experimentally measured constants is the Rydberg ''R = 10973731.568508(65) 1/m''. Here ''c, μ₀, R'' are combined into a unit-less ratio; :<math>\frac{(c^*)^{35}}{(\mu_0^*)^9 (R^*)^7} = (2 \pi \Omega^2)^{35}/(\frac{\alpha}{2^{11} \pi^5 \Omega^4})^9 .(\frac{1} {2^{23} 3^3 \pi^{11} \alpha^5 \Omega^{17}})^7,\;units = \frac{(u^{17})^{35}}{(u^{56})^9 (u^{13})^7} = 1</math> We can now define ''Ω'' using the geometries for (''c^*, μ₀^*, R^*'') and then solve by replacing (''c^*, μ₀^*, R^*'') with the numerical (''c, μ₀, R''). :<math>\Omega^{225}=\frac{(c^*)^{35}}{2^{295} 3^{21} \pi^{157} (\mu_0^*)^9 (R^*)^7 \alpha^{26}}, \;units = 1</math> :<math>\Omega = 2.007\;134\;949\;636...,\; units = 1</math> (CODATA 2014 mean values) :<math>\Omega = 2.007\;134\;949\;687...,\; units = 1</math> (CODATA 2018 mean values) There is a close natural number for ''Ω'' that is a square root implying that ''Ω'' can have a plus or a minus solution and this agrees with theory. This solution would however re-classify Omega as a mathematical constant (as being derivable from other mathematical constants). :<math>\Omega = \sqrt{ \left(\frac{\pi^e}{e^{(e-1)}}\right)} = 2.007\;134\;9543... </math> ===== G, h, e, m_e, k_B ===== As geometrical objects, the physical constants (''G, h, e, m_e, k_B'') can also be defined using the geometrical formulas for (''c^*, μ₀^*, R^*'') and solved using the numerical (mean) values for (''c, μ₀, R, α''), i.e.:. :<math>{(h^*)}^3 = (2^3 \pi^4 \Omega^4 \frac{r^{13} u^{19}}{v^5})^3 = \frac{2\pi^{10} {(\mu_0^*)}^3} {3^6 {(c^*)}^5 \alpha^{13} {(R^*)}^2},\; unit = u^{57}</math> {| class="wikitable" |+Physical constants; calculated vs CODATA 2014 ! Constant ! Geometry ! Calculated from (R, c, μ₀, α) ! CODATA 2014 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2014)</ref> |- | [[w:Planck constant | Planck constant]] | <math>{(h^*)}^3 = \frac{2\pi^{10} {\mu_0}^3} {3^6 {c}^5 \alpha^{13} {R_\infty}^2},\; unit = u^{57}</math> | ''h^*'' = 6.626 069 134 e-34, unit = u¹⁹ | ''h'' = 6.626 070 040(81) e-34 |- | [[w:Gravitational constant | Gravitational constant]] | <math>{(G^*)}^5 = \frac{\pi^3 {\mu_0}}{2^{20} 3^6 \alpha^{11} {R_\infty}^2},\; unit = u^{30}</math> | ''G^*'' = 6.672 497 192 29 e11, unit = u⁶ | ''G'' = 6.674 08(31) e-11 |- | [[w:Elementary charge | Elementary charge]] | <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 {c}^4 \alpha^8 {R_\infty}},\; unit = u^{-81}</math> | ''e^*'' = 1.602 176 511 30 e-19, unit = u^-27 | ''e'' = 1.602 176 620 8(98) e-19 |- | [[w:Boltzmann constant | Boltzmann constant]] | <math>{(k_B^*)}^3 = \frac{\pi^5 {\mu_0}^3}{3^3 2 {c}^4 \alpha^5 {R_\infty}} ,\; unit = u^{87} </math> | ''k_B^*'' = 1.379 510 147 52 e-23, unit = u²⁹ | ''k_B'' = 1.380 648 52(79) e-23 |- | [[w:Electron mass | Electron mass]] | <math>{(m_e^*)}^3 = \frac{16 \pi^{10} {R_\infty} {\mu_0}^3}{3^6 {c}^8 \alpha^7},\; unit = u^{45} </math> | '' m_e^*'' = 9.109 382 312 56 e-31, unit = u¹⁵ | ''m_e'' = 9.109 383 56(11) e-31 |- | [[w:Planck mass | Planck mass]] | <math>(m_P^*)^{15} = \frac{2^{25}\pi^{13} \mu_0^6}{ 3^6 c^5 \alpha^{16} R_\infty^2},\; unit = (u^{15})^{15}</math> | ''m_P^*'' = .217 672 817 580 e-7, unit = u¹⁵ | ''m_P'' = .217 647 0(51) e-7 |- | [[w:Planck length | Planck length]] | <math>(l_p^*)^{15} = \frac{\pi^{22} \mu_0^9}{2^{35} 3^{24} \alpha^{49} c^{35} R_\infty^8},\; unit = (u^{-13})^{15} </math> | ''l_p^*'' = .161 603 660 096 e-34, unit = u^-13 | ''l_p'' = .161 622 9(38) e-34 |- | [[w:Gyromagnetic ratio | Gyromagnetic ratio]] | <math>(\gamma_e/2\pi)^3 = \frac{g^3 3^3 c^4}{2^8 \pi^8 \alpha \mu_0^3 R_\infty^2},\; unit = u^{-126}</math> | ''γ_e/2π^*'' = 28024.953 55, unit = u^-42 | ''γ_e/2π'' = 28024.951 64(17) |} ==== 2019 SI unit revision ==== Following the 26th General Conference on Weights and Measures ([[w:2019 redefinition of SI base units|2019 redefinition of SI base units]]) are fixed the numerical values of the 4 physical constants (''h, c, e, k_B''). In the context of this model however only 2 base units may be assigned by committee as the rest are then numerically fixed by default and so the revision may lead to unintended consequences. For example, the von Klitzing constant Rk = h/e2, yet to derive h and e using Rk suggests an (inverse) alpha = 137.0359952 <ref>[https://codingthecosmos.com/physical-constants-calc.html Physical constants calculator]</ref>. {| class="wikitable" |+Physical constants ! Constant ! CODATA 2018 <ref>[http://www.codata.org/] | CODATA, The Committee on Data for Science and Technology | (2018)</ref> |- | [[w:Speed of light | Speed of light]] | ''c'' = 299 792 458 (exact) |- | [[w:Planck constant | Planck constant]] | ''h'' = 6.626 070 15 e-34 (exact) |- | [[w:Elementary charge | Elementary charge]] | ''e'' = 1.602 176 634 e-19 (exact) |- | [[w:Boltzmann constant | Boltzmann constant]] | ''k_B'' = 1.380 649 e-23 (exact) |- | [[w:Fine structure constant | Fine structure constant]] | ''α'' = 137.035 999 084(21) |- | [[w:Rydberg constant | Rydberg constant]] | ''R'' = 10973 731.568 160(21) |- | [[w:Electron mass | Electron mass]] | ''m_e'' = 9.109 383 7015(28) e-31 |- | [[w:Vacuum permeability | Vacuum permeability]] | ''μ₀'' = 1.256 637 062 12(19) e-6 |- | [[w:Quantum_Hall_effect#Applications | Von Klitzing constant]] | ''R_K'' = 25812.807 45 (exact) |} For example, if we solve using the above formulas; <math>R^* = \frac{4 \pi^5}{3^3 c^4 \alpha^8 e^3} = 10973\;729.082\;465</math> <math>{(m_e^*)}^3 = \frac{2^4 \pi^{10} R \mu_0^3}{3^6 c^8 \alpha^7},\;m_e^* = 9.109\;382\;3259 \;10^{-31}</math> <math>{(\mu_0^*)}^3 = \frac{3^6 h^3 c^5 \alpha^{13} R^2}{2 \pi^{10}},\;\mu_0^* = 1.256\;637\;251\;88\;10^{-6}</math> <math>{(h^*)}^3 = \frac{2 \pi^{10} \mu_0^3}{3^6 c^5 \alpha^{13} R^2},\;h^* = 6.626\;069\;149\;10^{-34}</math> <math>{(e^*)}^3 = \frac{4 \pi^5}{3^3 c^4 \alpha^8 R},\; e^* = 1.602\;176\;513\;10^{-19}</math> ==== u as sqrt(velocity/mass) ==== We find there is a single base unit '''u''' from which the other units and numerical values can be derived. This base unit incorporates [[v:Sqrt_Planck_momentum |MLT as square roots]]. =====''u = √{L/M.T}''===== :<math>u,\; units = \sqrt{\frac{L}{M T}} = \sqrt{u^{-13-15+30=2}} = u^1</math> Setting: :<math>x,\;units = \sqrt{\frac{M^9 T^{11}}{L^{15}}} = u^0, units = 1, scalars = 1</math> :<math>y,\;units = M^2T = u^0, units = 1, scalars <> 1</math> Gives; :<math>u^3 = \frac{L^{3/2}}{M^{3/2} T^{3/2}} = A,\; (ampere)</math> :<math>u^6 (y) = \frac{L^3}{T^2 M},\; (G)</math> :<math>u^{13} (xy) = \frac{1}{L},\; (1/l_p)</math> :<math>u^{15} (xy^2) = M,\; (m_P)</math> :<math>u^{17} (xy^2) = V,\; (c)</math> :<math>u^{19} (xy^3) = \frac{ML^2}{T},\; (h)</math> :<math>u^{20} (xy^2) = \frac{L^{5/2}}{M^{3/2} T^{5/2}} = AV,\;(T_P)</math> :<math>u^{27} (x^2y^3) = \frac{M^{3/2}\sqrt{T}}{L^{3/2}} = 1/AT,\; (1/e)</math> :<math>u^{29} (x^2y^4) = \frac{M^{5/2}\sqrt{T}}{\sqrt{L}} = ML/AT,\; (k_B)</math> :<math>u^{30} (x^2 y^3) = \frac{1}{T},\; (1/t_p)</math> :<math>u^{56} (x^4 y^7) = \frac{M^4 T}{L^2},\;(\mu_0)</math> :<math>u^{90} (x^6 y^{11}) = \frac{M^4}{T} </math> ===== ''β'' (unit = ''u'') ===== ''i'' (from ''x'') and ''j'' (from ''y''). :<math>R = \sqrt{P} = \sqrt{\Omega} r,\; units = u^8</math> :<math>\beta = \frac{V}{R^2} = \frac{2\pi R^2}{M} = \frac{A^{1/3} \alpha^{1/3}}{2} \;..., \; unit = u</math> :<math>i = \frac{1}{2\pi {(2\pi \Omega)}^{15}},\; units = 1, scalars = 1</math> :<math>j = \frac{r^{17}}{v^8} = k^2t = \frac{k^8}{r^{15}} ...,\; unit = \frac{u^{17*8}}{u^{8*17}} = u^{15*2}u^{-30} ... = 1,\; units = 1, scalars <> 1</math> For example; the constants solved in terms of (''r, v'') :<math>\beta = \frac{V}{R^2} = \frac{2\pi \Omega^2 v}{\Omega r^2} = \frac{2\pi \Omega v}{r^2},\; u^1 = u</math> :<math>A = \beta^3 (\frac{2^4}{\alpha}) = \frac{2^7 \pi^3 \Omega^3}{\alpha}\frac{v^3}{r^6},\; u^3</math> :<math>G = \frac{\beta^6}{2^3 \pi^2} (j) = 2^3 \pi^4 \Omega^6 \frac{r^5}{v^2},\; u^6</math> :<math>L^{-1} = 4\pi \beta^{13} (ij) = \frac{1}{2\pi^2 \Omega^2} \frac{v^5}{r^9},\; u^{13}</math> :<math>M = 2\pi \beta^{15} (ij^2) = \frac{r^4}{v},\; u^{15}</math> :<math>P = \beta^{16} (ij^2) = \Omega r^2,\; u^{16}</math> :<math>V = \beta^{17} (ij^2) = 2\pi \Omega^2 v,\; u^{17}</math> :<math>h = \pi \beta^{19} (ij^3) = 8\pi^4 \Omega^4 \frac{r^{13}}{v^5},\; u^{19}</math> :<math>T_P^* =\frac{2^3 \beta^{20}}{\pi \alpha} (ij^2) = \frac{2^7 \pi^3 \Omega^5}{\alpha} \frac{v^4}{r^6} ,\; u^{20} </math> :<math>e^{-1} = \frac{\alpha \pi \beta^{27} (i^2j^3)}{4} = \frac{\alpha}{128\pi^4 \Omega^3} \frac{v^3}{r^{3}},\; u^{27} </math> :<math>k_B = \frac{\alpha \pi^2 \beta^{29}(i^2j^4)}{4} = \frac{\alpha}{32 \pi \Omega} \frac{r^{10}}{v^3},\; u^{29}</math> :<math>T^{-1} = 4\pi \beta^{30} (i^2 j^3) = \frac{1}{\pi}\frac{v^6}{r^9},\; u^{30}</math> :<math>\mu_0^* = \frac{\pi^3 \alpha \beta^{56}}{2^3} (i^4 j^7) = \frac{\alpha}{2^{11} \pi^5 \Omega^4} r^7,\; u^{56}</math> :<math>\epsilon_0^{*-1} = \mu_0^* (c^*)^2 = \frac{\pi^3 \alpha \beta^{90}}{2^3} (i^6 j^{11}) = \frac{\alpha}{2^9 \pi^3} v^2 r^7,\; u^{90} </math> ===== limit ''j'' ===== The SI values for ''j'' suggest a limit (numerical boundary) to the values the SI constants can have. :<math>j = \frac{r^{17}}{v^8} = k^2 t = \frac{k^{17/4}}{v^{15/4}} = ... </math> gives a range from 0.812997... ''x''10^-59 to 0.123... ''x''10⁶⁰ In SI terms unit ''β'' can be derived via these ratio; :<math>a^{1/3} = \frac{v}{r^2} = \frac{1}{t^{2/15}k^{1/5}} = \frac{\sqrt{v}}{\sqrt{k}} ... = 23326079.1...; unit = u</math> ===== Rydberg formula ===== The [[w:Rydberg_formula |Rydberg formula]] can now be re-written in terms of amperes <math>A^2</math> :<math>\frac{hc}{2\pi \alpha^2} = \frac{j^2 A^2}{2^8 2\pi t_p}</math> ==== External links ==== * [[v:electron_(mathematical) | Mathematical electron]] * [[v:Relativity_(Planck) | Programming relativity at the Planck scale]] * [[v:Quantum_gravity_(Planck) | Programming gravity at the Planck scale]] * [[v:Black-hole_(Planck) | Programming the cosmic microwave background at the Planck scale]] * [[v:Sqrt_Planck_momentum | The sqrt of Planck momentum]] * [[v:God_(programmer) | The Programmer God]] * [[w:Simulation_hypothesis | The Simulation hypothesis]] * [https://codingthecosmos.com/ Programming at the Planck scale using geometrical objects] -Malcolm Macleod's website * [http://www.simulation-argument.com/ Simulation Argument] -Nick Bostrom's website * [https://www.amazon.com/Our-Mathematical-Universe-Ultimate-Reality/dp/0307599809 Our Mathematical Universe: My Quest for the Ultimate Nature of Reality] -Max Tegmark * [https://dx.doi.org/10.2139/ssrn.2531429 The mathematical electron model in a Planck scale universe] -(article) * [https://link.springer.com/article/10.1134/S0202289308020011/ Dirac-Kerr-Newman black-hole electron] -Alexander Burinskii (article) ==== References ==== {{Reflist}} [[Category: Physics]] [[Category: Philosophy of science]] duhqs6fmmzq90qspfsobvquwu3cprd7 0 276123 2416350 2416011 2022-08-18T23:08:12Z Dave Braunschweig 426084 Reverted edits by [[Special:Contributions/137.92.214.39|137.92.214.39]] ([[User_talk:137.92.214.39|talk]]) to last version by [[User:Spurzem|Spurzem]] using [[Wikiversity:Rollback|rollback]] wikitext text/x-wiki {{title|Cognitive dissonance and motivation:<br> What is the effect of cognitive dissonance on motivation?}} {{MECR3|1=https://youtu.be/_j-aHMRBi3A}} __TOC__ ==Overview== {{RoundBoxTop|theme=3}} '''Introductory case study:''' Jill loves animals. One day, she finds a documentary on Netflix about the treatment of cattle in feedlots. She is horrified at how inhumane the conditions are. Later that afternoon, Jill is purchasing pot plants from a hardware store and smells the delicious aroma of the sausage sizzle. She decides to buy a sausage sandwich. {{RoundBoxBottom}} [[wikipedia:Cognitive_dissonance|Cognitive dissonance]] is the aversive [[w: Mental state|psychological state]] created by simultaneously holding inconsistent [[wikipedia:Knowledge|knowledge]], [[wikipedia:Opinion|opinions]] or [[wikipedia:Belief|beliefs]]. This aversive state is a precursor to action (Festinger, 1957, p. 3). As such, [[w: Leon Festinger|Leon Festinger's]] theory of cognitive dissonance provides an explanation for a common motivational experience (McGrath, 2017). As outlined by Festinger (1957, p. 3), the [[w: Cognition|cognitions]] considered by the theory are constituted by knowledge, opinions and beliefs. Despite a number of revisions to Festinger's theory, a synthesis of these revisions provides a complete view of cognitive dissonance. The experimental paradigms highlight the application of the theory to different research contexts. Furthermore, research indicates cognitive dissonance may have a basis in [[w: Physiology|physiological]] processes. In summary, cognitive dissonance is itself a source of motivation. {{RoundBoxTop|theme=6}} '''Learning outcomes''' # Understand Festinger's theory of cognitive dissonance and subsequent revisions. # Understand how cognitive dissonance is a fundamentally motivational state. # Understand how cognitive dissonance is applied to different experimental paradigms. # Understand the physiological mechanisms involved in cognitive dissonance. {{RoundBoxBottom}} ==Festinger's theory== [[File:Noun Conflict 1325494.svg|thumb|''Figure 1''. Festinger's theory of cognitive dissonance describes the experience of holding inconsistent cognitions.|225x225px]] The theory of cognitive dissonance was proposed by [[wikipedia:Leon_Festinger|Leon Festinger]] (1957). He considered that people strive for the state of mental consistency or consonance, and dissonance describes the opposite state of inconsistency. In particular, Festinger proposed two [[w: Hypothesis|hypotheses]]: # the existence of dissonance is a motivator for achieving cognitive consonance #where dissonance exists, people will try to reduce the dissonance and actively avoid circumstances which increase it. Regarding motivation, Festinger (1957, p. 3) considered cognitive dissonance as a 'motivating factor in its own right'. Implicit in a discussion of the theory of cognitive dissonance is its effects on motivation. Whilst Festinger's theory may be considered simplistic (Mills, 2019) and has been subject to revisions, it does provide a broad conceptualisation of cognitive dissonance. == Reduction == Where cognitive dissonance is the aversive motivational state, reduction describes the mechanisms used to remove the dissonance. As proposed by Leon Festinger (1957), and supported in all subsequent revisions of the theory (Harmon-Jones & Harmon-Jones, 2019), the existence of cognitive dissonance influences action due to the creation of a negative psychological state (McGrath, 2017). === Reduction and motivation === Cognitive dissonance is a motivational state of psychological discomfort. Elliot and Devine (1994) describe it in terms of a "fundamentally motivational state" due to this discomfort. Festinger (1957, p. 18) outlined that the "presence of dissonance leads to action to reduce it". Reduction is the response to dissonance. As such, cognitive dissonance is a source of motivation which drives behaviours which facilitate a reduction in dissonance (McGrath, 2017). As initially proposed by Festinger (1957, p. 16), the magnitude of the dissonance correlates with the motivation to reduce it. This magnitude may be in relation to either the number of inconsistent cognitions, or the importance of the inconsistent cognitions. === Methods for reducing dissonance === Although reduction is the overarching strategy to remove cognitive dissonance, it may be divided into individual methods. Festinger (1957) originally proposed three different methods which are used to reduce or remove dissonance: # changing a behavioural cognitive element # changing an environmental cognitive element # adding new cognitive elements Subsequent literature has clarified these methods further into seven categories (McGrath, 2017; see Table 1). Table 1. ''Reduction strategies from McGrath (2017).'' {| class="wikitable" |+ !Method for reducing dissonance !Example from the introductory case study |- |Changing attitudes |Jill decides that feedlots are necessary. |- |[[w: Distraction|Distraction]] or [[w: Forgetting|forgetting]] |Jill distracts herself reading the instructions for her new plants. |- |[[w: Minimisation (psychology)|Trivialisation]] |Jill tells herself that the use of feedlots is probably not that widespread. |- |Denying responsibility |Jill says that she can't change the situation herself. |- |Adding consonant cognitions |Jill reads about how feedlots are humane. |- |Behavioural changes |Jill decides not to buy a sausage. |- |[[w: Rationalization (psychology)|Act rationalisation]] |Jill tells herself that because there are not other food options available, it's okay. |} Whilst studies of cognitive dissonance have typically given particular attention to the use of one of these strategies to reduce cognitive dissonance, research has highlighted this is not necessarily clear cut. In an experiment by Gosling et al. (2006) participants who were exposed to conditions which aroused cognitive dissonance were seen to utilise the first method of reduction made available to them. There is a distinct lack of literature surrounding the methods people use to reduce dissonance in everyday life, as this depends on a number of factors (McGrath, 2017). Nevertheless, particular examples do exist such as act rationalisation for smoking (Foruhi et al., 2013). {{RoundBoxTop|theme=3}} '''Act rationalisation case study:''' Alex is a smoker. Despite knowing the negative health effects, he tells himself that he is going to die of something, so he may as well have a smoke and enjoy life. {{RoundBoxBottom}} == Revisions of cognitive dissonance theory == Despite cognitive dissonance being a persistent theory, it has been subject to a number of revisions. A notable area of dispute amongst theorists regards the underlying motivation or rationale behind dissonance effects (Mills, 2019). Indeed, the four revisions presented below represent different conceptualisations of the underlying cognitions that cause cognitive dissonance. === Action-based model === The action-based model of cognitive dissonance considers cognitions with regard to connected action. This means the negative psychological state created by cognitive dissonance arises specifically in relation to conflicting cognitions with action implications (Harmon-Jones et al., 2015). The action-based model considers cognitions in relation to connected action, where cognitive dissonance arises in relation to difficult decisions and commitments to act in a certain way. Cognitive dissonance may be seen to hinder effective action (Harmon-Jones & Harmon-Jones, 2019).{{vague}}{{example}} === Self-consistency === [[File:Gabriel Cornelius Ritter von Max - The Mirror.jpg|thumb|''Figure 2.'' The self-consistency, self-affirmation and new look revisions of cognitive dissonance consider self-reflection at the centre of cognitive dissonance.|left|276x276px]] The self-consistency model considers that cognitive dissonance is caused by an inconsistency between the [[w: Self-concept|self-concept]] or personal standards, and a course of behaviour (Thibodeau & Aronson, 1992). As proposed by Aronson (1968), the self-consistency revision considers that cognitive dissonance arises specifically in relation to how inconsistency impacts on a positive self-concept, such as where people act in a way that makes them feel immoral, incompetent or guilty. In order to reduce cognitive dissonance, reduction strategies seek to rescue the self-image through justification of the action (Stone & Cooper, 2000). === Self-affirmation === Similarly to the self-consistency revision of cognitive dissonance, the self-affirmation revision considers cognitive dissonance to arise specifically in relation to cognitions about the self. The self-affirmation revision considers that people seek to maintain a perception of [[w: Morality|moral]] integrity (Steele, 1988). As such, the revision considers dissonance to be "the distress of a threatened sense of [overall] self-integrity" (Steele et al., 1993). The self-affirmation theory considers that the goal of a reduction strategy is to restore overall self-integrity through focussing on positive cognitions of the self (Stone & Cooper, 2000). === The new look === The new look perception of cognitive dissonance considers that people experience cognitive dissonance in response to something they are personally responsible for. Cooper and Fazio (1984) propose that people experience cognitive dissonance for consequences that they personally cause, and then seek to use reduction strategies to justify this course of action. However, this revision has been subject to criticism, where not all the experimental paradigms presented below support this revision (Harmon-Jones & Harmon-Jones, 2007). === Synthesis of the different revisions === Despite conceptual disagreement amongst the different revisions of cognitive dissonance, they may be incorporated into the original theory of cognitive dissonance. However, Harmon-Jones and Mills (2019) argue that while subsequent research has shown that revisions to the theory of cognitive dissonance are largely unnecessary, they do indeed provide a viewpoint of the different ways cognitive dissonance may arise. Despite the differences, they contribute to the overall picture of cognitive dissonance. Thus, a synthesis of the different revisions should consider these as distinct, but interconnected perspectives of cognitive dissonance theory. {{RoundBoxTop|theme=2}}<quiz display="simple"> {Which revision of Festinger's cognitive dissonance theory considers dissonance to arise from an inconsistency between a person's self-concept and actions? |type="()"} - The new look model + Self-consistency model - Self-affirmation model - Action-based model </quiz> {{RoundBoxBottom}} ==Experimental paradigms== Experimental paradigms illustrate the specific components of cognitive dissonance. These may be conceptualised as the different research observations of cognitive dissonance. As such, they provide a greater view of the specific effects of the theory on motivation. === Free-choice paradigm === [[File:Rayon hygiene dentaire Supermarché Cameroun 02.jpg|thumb|''Figure 3.'' The free-choice paradigm may be implicated in the choices made in everyday contexts such as the supermarket.|247x247px]] The free-choice paradigm describes the cognitive dissonance associated with making a choice between competing options (see Figure 3). After making a choice, the negative aspects of the choice, and the positive aspects of the alternative are dissonant with the actual choice.{{example}} A seminal study by Brehm (1956) tested the free-choice paradigm amongst a range of household items such as an automatic coffee-maker, a sandwich grill and a fluorescent desk lamp. Participants were asked to rate the desirability of each item on a scale. After rating the items, the participants were given a choice between two items, with the high dissonance condition created by asking participants to choose between similarly rated items. After the choice, there was a second rating conducted and results showed a reduction in the desirability of the forgone choice in the high dissonance condition. This change indicates that participants were reducing dissonance through reducing the desirability of the alternative (Brehm, 1956). The free-choice paradigm continues to be applied in studies of cognitive dissonance. Harmon-Jones et al. (2008) observed increased left [[w: Frontal lobe|frontal cortical]] activity amongst participants engaging in a reduction strategy through the free-choice paradigm. Similarly, a study by Eagan et al. (2007) observed dissonance reduction from a variation of the free-choice paradigm in both monkeys and young children, using stickers and coloured M&Ms{{explain}}. Nevertheless, the methods behind the free-choice paradigm may be criticised due to the effects of existing preferences (Chen & Risen, 2010). Chen and Risen (2010) found that a reduction in the desirability of an alternative could occur even when it could not have been in response to a previously made choice. Whilst the free-choice paradigm should be interpreted cautiously, it does present an everyday example of cognitive dissonance. {{RoundBoxTop|theme=3}} '''Free-choice case study:''' Mark needs new toothpaste and walks down the personal care aisle at the supermarket. He picks up a box of Col-white whitening toothpaste. But he looks at the other options, and sees Sensotime gum care. The positive aspects of the alternative such as superior gum care, and the negative aspects of his choice such as its abrasiveness are dissonant with his choice. {{RoundBoxBottom}} === Effort-justification paradigm === The [[w: Effort justification|effort-justification paradigm]] describes the cognitive dissonance associated with undertaking an unpleasant task for a positive result. As such, cognition about the unpleasantness of the task is dissonant with engaging in the task. A study by Aronson and Mills (1959) tested this effect through participants engaging in initiation activities to join particular groups. Participants were 63 college women who were split into a [[w: scientific control|control condition]], a mild initiation condition and a severe initiation condition, which involved an embarrassing task. After the initiation activity, participants listened to a fake conversation of the group and were asked how desirable the discussion and group members were. Participants in the severe initiation condition rated the group more highly compared to those in the control condition. This indicates that cognitive dissonance may be reduced by exaggerating the desirability of the result (Aronson & Mills, 1959). [[File:Interior of the IKEA Băneasa 29.jpg|thumb|''Figure 4.'' The IKEA effect may be considered as a particular subset of the effort-justification paradigm.|324x324px]] The [[w: IKEA effect|IKEA effect]] may be considered as a particular subset of the effort-justification paradigm in which people exhibit an increased value for self-made products (Norton et al., 2011). In a series of three experiments, participants were asked to assemble an [[w: IKEA|IKEA]] box (see ''Figure 4''), [[w: Origami|origami]] and [[w: Lego|Lego]]. [[w: Willingness to pay|Willingness-to-pay]] was used to measure the value participants placed in the products. Overall, Norton et al. (2011) concluded that successful assembly of the items led to an over-valuing of these items in comparison to participants who merely observed the items. Further, participants who assembled the products believed that their products rivalled those made by experts. === Induced-compliance paradigm === Also known as the insufficient justification or [[w: Forced compliance theory|forced compliance paradigm]], induced-compliance describes the cognitive dissonance a person experiences when doing something inconsistent with a held belief. Originally coined as the forced compliance paradigm (Festinger & Carlsmith, 1959), external sources of motivation are consonant with the behaviour, whilst prior held beliefs are dissonant. The experienced dissonance is greater where the degree of external motivation such as a reward or punishment is less. Festinger and [[w: Merrill Carlsmith|Carlsmith]] (1959) observed the existence of the induced-compliance paradigm in their seminal study. Participants were 71 male psychology students who completed a boring task for an hour. Subsequently, participants were asked to convey to the next participant how enjoyable the tasks were. Participants were divided into a control condition with no reward, a condition where the participant was paid $1 and a condition where the participant was paid $20. Afterwards, participants rated how interesting the activity was. The results showed that participants paid $1 rated the tasks as the most enjoyable. The study demonstrates that people with high choice have few consonant cognitions for engaging in the activity against their held belief, and therefore needed to create these consonant cognitions to reduce the cognitive dissonance. In other words, students in the condition that was paid $1 had the greatest free choice because the reward wasn't high, but came to believe the lie that the task was enjoyable. Where the participants were paid $20, the external motivation meant that participants experienced less dissonance. [[File:Feuerwehrauto von Günthermann - tin toy (um 1950).jpg|left|thumb|278x278px|''Figure 5.'' As a subset of the induced-compliance paradigm, the forbidden toy experiment used toys such as a toy fire truck to create dissonance.]] As a subset of the induced-compliance paradigm, the forbidden toy paradigm involves the threat of punishment rather than a positive reward incentive. In a study by Aronson and Carlsmith (1963) young children rated the desirability of different toys (see Figure 5). Afterwards, they played with the toys, with one toy having a punishment associated with it. Participants were split into a mild punishment condition and a severe punishment condition. After playing with the toys, the children re-evaluated the desirability of the toys. Participants in the mild punishment condition rated the forbidden toy less desirable than those in the severe punishment condition, indicating that they experienced dissonance because the external punishment was less. {{RoundBoxTop|theme=3}} '''Induced compliance case study:''' Lisa sells dietary supplements as part of Herb-Alive, a multi-level marketing scheme. Despite working very hard to sell the supplements, she is not making money, and believes that she can't just from sales. She reads in the pamphlet that if she recruits others to join, she will receive a bonus. She convinces her friend Michael to join, telling him that he will make lots of money from the business. To reduce dissonance, she internalises the narrative that she is actually helping out Michael by making him rich. {{RoundBoxBottom}} === Belief-disconfirmation paradigm === The belief-disconfirmation paradigm describes the cognitive dissonance associated with exposure to information that is inconsistent with existing beliefs. In response to the cognitive dissonance, people are motivated to misinterpret the conflicting information, reject the information, seek support from those who agree with the belief, and attempt to persuade others to agree with the belief. In the study by Festinger et al. (1956), the researchers observed a [[w:Cults|cult]] group called the [[w: The Seekers (rapturists)|Seekers]] through [[w: participant observation|participant observation]]. The group predicted the end of the world through [[w: Flood|flooding]] which would occur on the 21st of December 1954. On the day before the flooding was due to occur, the group was told that they would be taken away in a [[w: Flying saucer|flying saucer]]. When this did not occur, the group's beliefs were disconfirmed. However, after the disconfirmation, the group continued to engage in [[w: Proselytism|proselytising]] to encourage others to join. This indicates that members were removing the cognitive dissonance between the previously held belief and the exposure to the reality by adding cognitions consonant with their own beliefs (Festinger et al., 1956). {{RoundBoxTop|theme=2}} <quiz display="simple"> {Betty purchases a new phone. She was deciding between an e-phone and a Suesung which are similar. After making her choice, what paradigm of cognitive dissonance would she likely experience? |type="()"} + Free-choice paradigm - Effort-justification paradigm - Induced-compliance paradigm - Belief-disconfirmation paradigm </quiz> {{RoundBoxBottom}} == Physiological basis == [[File:190603 Functional magnetic resonance imaging at the Imperial Centre for Psychedelic Research.jpg|thumb|''Figure 6.'' Functional magnetic resonance imaging (fMRI) has allowed researchers to investigate the specific areas of the brain associated with cognitive dissonance.|258x258px]] Different areas of the brain may be implicated in cognitive dissonance. Through the use of neuroimaging methods such as [[w: Functional magnetic resonance imaging|functional magnetic resonance imaging]] (fMRI) (see Figure 6) and [[w: Electroencephalogram|electroencephalogram]] (EEG), researchers have a greater understanding of the cognitive processes underlying cognitive dissonance. In particular, increased activity in the posterior medial frontal cortex (pMFC) may be observed with experiencing cognitive dissonance. In a study by Van Veen et al. (2009) the induced compliance experimental paradigm was adapted to allow for participants to be monitored by fMRI. The results of the fMRI showed increased activity in the pMFC when participants experienced dissonance. Similarly, using the free-choice paradigm, Izuma et al. (2010) found that the pMFC was associated with a choice-making condition designed to create dissonance. Activity in the dorsal [[w: Anterior cingulate cortex|anterior cingulate cortex]] (dACC) and dorsomedial [[w: Prefrontal cortex|prefrontal cortex]] (dmPFC) tracked these situations (Izuma et al., 2010). Other neural regions have also been observed in studies on cognitive dissonance. The [[w: Anterior insula|anterior insula]] has been observed to be active in both the induced compliance and free-choice paradigms (van Veen et al., 2009; Izuma et al., 2010). Nevertheless, in each of these studies, it is important to note that observations are made through reverse inference (Izuma & Murayama, 2019). ==Conclusion== Cognitive dissonance is an important topic of psychological science. Understanding why people make choices to justify or engage in certain courses of action provides insight into their motivation. Cognitive dissonance is fundamentally a motivational state, which causes people to engage in behaviours to reduce its effects. Further, its presence in everyday life makes understanding its effects beneficial on a personal level. The theory and its experimental paradigms aptly describe behaviours which are frequently experienced. Cognitive dissonance remains a pertinent theory in attributing behaviours to motivational states. In conclusion, cognitive dissonance is fundamentally motivational. ==See also== * [[w:Cognitive dissonance|Cognitive dissonance]] (Wikipedia) * [[Motivation and emotion/Book/2018/Cognitive dissonance and emotion|Cognitive dissonance and emotion]] (Book chapter, 2018 * [[Motivation and emotion/Book/2020/Motivated reasoning|Motivated reasoning]] (Book chapter, 2020) * [[Motivation and emotion/Book/2018/Self-perception theory and motivation for positive change|Self-perception theory and motivation for positive change]] (Book chapter, 2018) ==References== {{Hanging indent|1= Aronson, E., & Carlsmith, J. M. (1963). Effect of the severity of threat on the devaluation of forbidden behavior. ''The Journal of Abnormal and Social Psychology'', ''66''(6), 584–588. https://doi.org/10.1037/h0039901 Aronson, E., & Mettee, D. R. (1968). Dishonest behavior as a function of differential levels of induced self-esteem. ''Journal of Personality and Social Psychology'', ''9''(2, Pt.1), 121–127. https://doi.org/10.1037/h0025853 Aronson, E., & Mills, J. (1959). The effect of severity of initiation on liking for a group. ''The Journal of Abnormal and Social Psychology'', ''59''(2), 177–181. https://doi.org/10.1037/h0047195 Brehm, J. W. (1956). Postdecision changes in the desirability of alternatives. ''The Journal of Abnormal and Social Psychology'', ''52''(3), 384–389. https://doi.org/10.1037/h0041006 Chen, M. K., & Risen, J. L. (2010). How choice affects and reflects preferences: Revisiting the free-choice paradigm. ''Journal of Personality and Social Psychology'', ''99''(4), 573–594. https://doi.org/10.1037/a0020217 Cooper, J., & Fazio, R. H. (1984). A new look at dissonance theory. In ''Advances in Experimental Social Psychology'' (Vol. 17, pp. 229–266). Elsevier. https://doi.org/10.1016/S0065-2601(08)60121-5 Egan, L. C., Santos, L. R., & Bloom, P. (2007). The origins of cognitive dissonance: Evidence from children and monkeys. "Psychological Science", "18"(11), 978–983. https://doi.org/10.1111/j.1467-9280.2007.02012.x Festinger, L. (1957). ''A theory of cognitive dissonance''. Stanford University Press. Festinger, L., & Carlsmith, J. M. (1959). Cognitive consequences of forced compliance. ''The Journal of Abnormal and Social Psychology'', ''58''(2), 203–210. https://doi.org/10.1037/h0041593 Festinger, L., Riecken, H., & Schachter, S. (1956). ''When Prophecy Fails''. University of Minnesota Press. Fotuhi, O., Fong, G. T., Zanna, M. P., Borland, R., Yong, H.-H., & Cummings, K. M. (2013). Patterns of cognitive dissonance-reducing beliefs among smokers: A longitudinal analysis from the International Tobacco Control (ITC) Four Country Survey. ''Tobacco Control'', ''22''(1), 52–58. https://doi.org/10.1136/tobaccocontrol-2011-050139 Gosling, P., Denizeau, M., & Oberlé, D. (2006). Denial of responsibility: A new mode of dissonance reduction. ''Journal of Personality and Social Psychology'', ''90''(5), 722–733. https://doi.org/10.1037/0022-3514.90.5.722 Harmon-Jones, E., Gerdjikov, T., & Harmon-Jones, C. (2008). The effect of induced compliance on relative left frontal cortical activity: A test of the action-based model of dissonance. ''European Journal of Social Psychology'', ''38''(1), 35–45. https://doi.org/10.1002/ejsp.399 Harmon-Jones, E., & Harmon-Jones, C. (2007). Cognitive dissonance theory after 50 years of development. ''Zeitschrift für Sozialpsychologie'', ''38''(1), 7–16. https://doi.org/10.1024/0044-3514.38.1.7 Harmon-Jones, E., & Harmon-Jones, C. (2019). Understanding the motivation underlying dissonance effects: The action-based model. In E. Harmon-Jones (Ed.), ''Cognitive dissonance: Reexamining a pivotal theory in psychology (2nd ed.).'' (pp. 63–89). American Psychological Association. https://doi.org/10.1037/0000135-004 Harmon-Jones, E., Harmon-Jones, C., Fearn, M., Sigelman, J. D., & Johnson, P. (2008). Left frontal cortical activation and spreading of alternatives: Tests of the action-based model of dissonance. ''Journal of Personality and Social Psychology'', ''94''(1), 1–15. https://doi.org/10.1037/0022-3514.94.1.1 Harmon-Jones, E., Harmon-Jones, C., & Levy, N. (2015). An action-based model of cognitive-dissonance processes. ''Current Directions in Psychological Science'', ''24''(3), 184–189. https://doi.org/10.1177/0963721414566449 Izuma, K., Matsumoto, M., Murayama, K., Samejima, K., Sadato, N., & Matsumoto, K. (2010). Neural correlates of cognitive dissonance and choice-induced preference change. ''Proceedings of the National Academy of Sciences'', ''107''(51), 22014–22019. https://doi.org/10.1073/pnas.1011879108 Izuma, K., & Murayama, K. (2019). Neural basis of cognitive dissonance. In E. Harmon-Jones (Ed.), ''Cognitive dissonance: Reexamining a pivotal theory in psychology (2nd ed.).'' (pp. 227–245). American Psychological Association. https://doi.org/10.1037/0000135-011 McGrath, A. (2017). Dealing with dissonance: A review of cognitive dissonance reduction. ''Social and Personality Psychology Compass'', ''11''(12), e12362. https://doi.org/10.1111/spc3.12362 Mills, J. (2019). Improving the 1957 version of dissonance theory. In E. Harmon-Jones (Ed.), ''Cognitive dissonance: Reexamining a pivotal theory in psychology (2nd ed.).'' (pp. 27–39). American Psychological Association. https://doi.org/10.1037/0000135-002 Norton, M. I., Mochon, D., & Ariely, D. (2012). The IKEA effect: When labor leads to love. ''Journal of Consumer Psychology'', ''22''(3), 453–460. https://doi.org/10.1016/j.jcps.2011.08.002 Steele, C. M. (1988). The psychology of self-affirmation: Sustaining the inteegrity of the self. In ''Advances in Experimental Social Psychology'' (Vol. 21, pp. 261–302). Elsevier. https://doi.org/10.1016/S0065-2601(08)60229-4 Steele, C. M., Spencer, S. J., & Lynch, M. (1993). Self-image resilience and dissonance: The role of affirmational resources. ''Journal of Personality and Social Psychology'', ''64''(6), 885–896. https://doi.org/10.1037/0022-3514.64.6.885 Stone, J., & Cooper, J. (2001). A self-standards model of cognitive dissonance. ''Journal of Experimental Social Psychology'','' 37''(3), 228–243. https://doi.org/10.1006/jesp.2000.1446 Thibodeau, R., & Aronson, E. (1992). Taking a closer look: Reasserting the role of the self-concept in dissonance theory. ''Personality and Social Psychology Bulletin'', ''18''(5), 591–602. https://doi.org/10.1177/0146167292185010 van Veen, V., Krug, M. K., Schooler, J. W., & Carter, C. S. (2009). Neural activity predicts attitude change in cognitive dissonance. ''Nature Neuroscience'', ''12''(11), 1469–1474. https://doi.org/10.1038/nn.2413}} == External links == * [https://www.khanacademy.org/test-prep/mcat/processing-the-environment/cognition/v/cognitive-dissonance Cognitive dissonance] (Khan Academy) * [https://www.youtube.com/watch?v=NqONzcNbzh8&t=19s Ash Donaldson - Cognitive dissonance] (TEDxCanberra) *[https://www.simplypsychology.org/cognitive-dissonance.html Cognitive dissonance] (Simply Psychology) *[https://www.youtube.com/watch?v=9Y17YaZRRvY&ab_channel=AndyLuttrell Cognitive dissonance] (Opinion Science Podcast)) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Cognitive dissonance]] rlsz4l5njpkmo44xjc5hul503s5ox7x 0 277132 2416450 2413908 2022-08-19T06:56:16Z StormSar 2947766 wikitext text/x-wiki {{METE}} {{title|Task Initiation. What are the challenges with task initiation and how to get get started? }} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }}nt with those tasks until its completion even if they encounter challenges. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} ==Task initiation== === '''What is task initiation?''' === === '''How does task initiation develop in adults?''' === === '''How does task initiation develop in children?''' === How you are going to structure the chapter? Aim for three to six main headings between the Overview and Conclusion. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} == Task initiation in Autism == === How does Task initiation differ in those with Autism spectrum disorder (ASD)? === === What is the neurological basis of task initiation in ASD? === == Task initiation in ADHD == === How does task initiation differ in those with ADHD? === === What is the neurological basis of task initiation in ADHD? === ==Interactive learning features== What brings an online book chapter to life, compared to an essay, are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Feature boxes=== Feature boxes can be used to highlight content, but don't overuse them. There are many different ways of creating feature boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some example 3 x 3 tables which could be adapted: Table 1. ''Example of a Table with an APA Style Caption'' {| align=center border=1 cellspacing=0 cellpadding=5 ! '''Children''' ! '''Gather''' ! '''Round''' |- | Mary | had | a |- | little | lamb | it's |- | fleece | was | white |} Table 2. ''Another Example of a Table with an APA Style Caption'' <div align="center"> {| class="wikitable" ! '''Nursery''' ! '''Rhyme''' ! '''Time''' |- | Incy | Wincy | spider |- | climbed | up | the |- | water | spout | down |} </div> Table 3. ''Example of a Sortable Table with an APA Style Caption'' <div align="center"> {| class="wikitable sortable" |- ! Fruit ! Price/kg ! Popularity |- | Tomatoes | style="text-align:right;" | $6.00 | 1st |- | Bananas | style="text-align:right;" | $5.00 | 2nd |- | Watermelon | style="text-align:right;" | $2.99 | 3rd |- | Oranges | style="text-align:right;" | $3.85 | 4th |- | Apples | style="text-align:right;" | $4.95 | 5th |- | Grapes | style="text-align:right;" | $9.50 | 6th |- | Mangoes | style="text-align:right;" | $12.00 | 7th |- | Avocados | style="text-align:right;" | $12.00 | 8th |} </div> ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Working notes== It could be useful to have a temporary section for working notes during the topic development and chapter drafting. This section will be ignored when the topic development is marked, but remove it before finalising the book chapter. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read only the Overview and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== In this section, provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== In this section, list the cited references in [[w:APA style|APA style]] (7th ed.). For example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by a en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== In this section, provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant external resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] fixfa8y0sotzckq5kc3bbo1aadao8x8 0 277657 2416336 2415945 2022-08-18T22:56:26Z Jtneill 10242 [/Neuroimaging and mood disorders/]] - How can neuroimaging assist in diagnosing and treating mood disorders? - [[User:Jdebear|Jdebear]] wikitext text/x-wiki {{/Banner}} ==Motivation == # [[/Academic help-seeking/]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:Ibm4444|Ibm4444]] # [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[User:U3216563|U3216563]] # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:Teermeej Hossain|Teermeej Hossain]] # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]] # [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:U3222363|U3222363]] # [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]] # [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - [[User:U3141987|U3141987]] # [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - [[User:SoSilverLibby|SoSilverLibby]] # [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:CaitlinEmc|CaitlinEmc]] # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]] # [[/Choice overload/]] - What is choice overload? What is the optimal amount of choice? - [[UserGeorgiaFairweather|GeorgiaFairweather]] # [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:GiovanniBartlett|GiovanniBartlett]] # [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:JimmyOC1985|JimmyOC1985]] # [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:U3193000|U3193000]] # [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:U3215103|U3215103]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:U3203936|U3203936]] # [[/Comprehensive action determination model/]] - What is the CADM and how can it be applied to understanding human motivation? - [[User:MyUserName|MyUserName]] # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - [[User:KingMob221|KingMob221]] # [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:Hanarose123|Hanarose123]] # [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]] # [[/Discounts and consumer purchase behaviour/]] - What role do discounts play in consumer purchase behaviour? - [[User:MyUserName|MyUserName]] # [[/Drugs-violence nexus and motivation/|Drugs-violence nexus and motivation]] - What is the role of motivation in the drugs-violence nexus? - [[Atu3202070|Atu3202070]] # [[/Domestic energy conservation motivation/]] - How can domestic energy conservation and reduced consumption be motivated and behaviour changed? - [[User:MyUserName|MyUserName]] # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]] # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:U3246310|U3246310]] # [[/Equity theory/]] - What is equity theory and how can it be applied? - [[EKS2001|EKS2001]] # [[/ERG theory/]] - What is Alderfer's ERG theory? - [[User:Wuser1307|Wuser1307]] # [[/Environmental volunteering motivation/]] - What motivates environmental volunteering? - [[User:MyUserName|MyUserName]] # [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]] # [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[Cedevlin9|Cedevlin9]] # [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [[User:Sebastian Armstrong|Sebastian Armstrong]] # [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117|U3214117]] # [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:Btarmstrong24|Btarmstrong24]] # [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[User:U3211125|U3211125]] # [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:U3161584|U3161584]] # [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:Earthxangel|Earthxangel]] # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[User:Dee320|Dee320]] # [[/Help-seeking among boys/]] - What are the barriers to help-seeking for boys and what motivates them to seek help? - [[User:BradMcGrath|BradMcGrath]] # [[/Hidden costs of reward/]] - What are the hidden costs of motivating by reward? - [[User:SLoCE|SLoCE]] # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292|U3218292]] # [[/Honesty motivation/]] - What motivates honesty? - [[User:U3200859|U3200859]] # [[/Humour, leadership, and work/]] - What role does humour play in effective leadership in the workplace? - [[User:U3210264|U3210264]] # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[User:U3216963|U3216963]] # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]] # [[/Kindness motivation/]] - What motivates kindness? - [[User:U3205429|U3205429]] # [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:U3183466|U3183466]] # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation and reduced consumption be motivated and behaviour changed? - [[User:MyUserName|MyUserName]] # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:U3213250|U3213250]] # [[/Perfectionism/]] - What motivates perfectionism? Is perfectionism good or bad? How can it be managed? - [[User:AEMOR|AEMOR]] # [[/Physiological needs/]] - How do human's physiological needs affect motivation? - [[User:U3203655|U3203655]] # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:U3200956|U3200956]] # [[/Psychological trauma and subsequent drug use/]] - How does psychological trauma motivate drug use? What are the strategies for treating trauma-induced drug use? - [[User:U3210431|U3210431]] # [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:U3191574 (PHP)|U3191574 (PHP)]] # [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:Will-U3214082|Will-U3214082]] # [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:U3216654|U3216654]] # [[/Self-efficacy and academic achievement/]] - What role does self-efficacy play in academic achievement? - [[User:U943292|U943292]] # [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:U3216513mt|U3216513mt]] # [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:U3037979|U3037979]] # [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]] # [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:U3215976|U3215976]] # [[/Staff retention motivation/]] - How can organisations and managers help to motivate long-term retention of employees? - [[User:MyUserName|MyUserName]] # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]] # [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:Emily.desilva|Emily.desilva]] # [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:U3148161|U3148161]] # [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:StormSar|StormSar]] # [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]] # [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:Lturner2311|Lturner2311]] # [[/Time management/]] - How can one's time be managed effectively? - [[User:CNK.20|CNK.20]] # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:U3207458|U3207458]] # [[/Treatment motivation in juvenile delinquency/]] - What is the role of treatment motivation for juvenile delinquency and how can it be enhanced? - [[User:MyUserName|MyUserName]] # [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:Franklin Brightt|Franklin Brightt]] # [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - [[User:U3055143|U3055143]] # [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]] # [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:U962051|U962051]] # [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:U3201643|U3201643]] # [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:U3215603|U3215603]] # [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:U3213441|U3213441]] ==Emotion== # [[/Animal emotion/]] - What is the emotional experience of animals? - [[User:U3216502|u3203545]] # [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]] # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:U3186959|U3186959]] # [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - [[User:FulaAjeo22|FulaAjeo22]] # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:SunandaUC|SunandaUC]] # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:U3202788|U3202788]] # [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:Tatjurate|Tatjurate]] # [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:Micabaker1|Micabaker1]] # [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - [[User:U3203545|U3203545]] # [[/Compassion fatigue in mental health professionals/]] - What causes compassion fatigue in MH professionals and how can it be prevented? - [[User:MyUserName|MyUserName]] # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]] # [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:U3202788|U3219905]] # [[/Core emotions/]] - What are the core emotions and what is their function? U3203140 # [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[User:SashaBrooksby|SashaBrooksby]] # [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:Tamika Afeaki|Tamika Afeaki]] # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]] # [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:U3158968|U3158968]] # [[/Disappointment/]] - What is disappointment, what causes it, and how can it be managed? - [[User:U3216256|U3216256]] # [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[User:DenniseSoleymani|DenniseSoleymani]] # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? -]] [[User:MyUserName|MyUserName]] # [[/Ecological grief/]] - What is ecological grief and what can be done about it? - [[User:Brewerjr|Brewerjr]] # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress? - [[User:MyUserName|MyUserName]] # [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - [[User:U3190353|U3190353]] # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:Eimilerous22|Eimilerous22]] # [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:GabbieUC|GabbieUC]] # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - [[User:U3230861|U3230861]] # [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:RWilliams12|Rwilliams12]] # [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:Gabrielle Eagling|Gabrielle Eagling]] # [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]] # [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:U3131472|U3131472]] # [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:U3246554|U3246554]] # [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:Lewis.Kusk|Lewis.Kusk]] # [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:Icantchooseone|Icantchooseone]] # [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:U3216963|U3216963]] # [[/Flourishing in the elderly/]] - How can psychological flourishing be supported in the elderly? - [[User:MyUserName|MyUserName]] # [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]] # [[/Gratitude and subjective wellbeing/]] - What is the relationship between gratitude and subjective wellbeing? - [[User:MyUserName|MyUserName]] # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:Lyndel Lemon|Lyndel Lemon]] # [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[User:MyUserName|MyUserName]] # [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:Alec.cortez|Alec.cortez]] # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:U3189442 - K.Ryan|U3189442 - K.Ryan]] # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]] # [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:U3230861|U3227354]] # [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:U3190094|U3190094]] # [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:U3203265|U3203265]] # [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:U3183521|U3183521]] # [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:U3119310|U3119310]] # [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:U3217109|U3217109]] # [[/Mental toughness in the workplace/]] - What can mental toughness be useful in the workplace? How can it be developed? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[User:CaityDcr1603|U3217109]] # [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:Clairelogan|Clairelogan]] # [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:U3210490|U3210490]] # [[/Mudita/]] - What is mudita and how can it be developed? -[[User:Inandonit365|Inandonit365]] # [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? -[[User:U3148366_Chris|U3148366_Chris]] # [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:Ana028|Ana028]] # [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:A Super Villain|A Super Villain]] # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]] # [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:U3166273|U3166273]] # [[/Neuroimaging and mood disorders/]] - How can neuroimaging assist in diagnosing and treating mood disorders? - [[User:Jdebear|Jdebear]] # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? - [[User:MyUserName|MyUserName]] # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|JorjaFive]] # [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:U3190773|U3190773]] # [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:MyUserName|MyUserName]] # [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:U3083720|U3083720]] # [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]] # [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:Noah O'Brien|Noah O'Brien]] # [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:U3216389|U3216389]] # [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:BenjiD'Ange|BenjiD'Ange]] # [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:Jingru shao 0906|Jingru shao0906]] # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - [[User:U3200902|U3200902]] # [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[JaimeTegan|JaimeTegan]] # [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - [[User:Isaacem13|Isaacem13]] # [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:Brookewin|Brookewin]] # [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]] # [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:RSPMeredith|RSPMeredith]] # [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:Bill.miosge|Bill.miosge]] # [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:U3216958 - Tiarna.Wilson-Ginn|U3216958 - Tiarna.Wilson-Ginn]] # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|MyUserName]] # [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[User:Jamieepiper|Jamieepiper]] # [[/Window of tolerance/]] - What is the window of tolerance and how it be applied? - [[User:U3223109|U3223109]] # [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[User:ArtOfHappiness|ArtOfHappiness]] # [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:u3211603|U3211603]] ==Motivation and emotion== # [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[User:U3217287|U3217287]] # [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]] # [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:Molzaroid|Molzaroid]] # [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]] # [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]] # [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:U3100193|U3100193]] # [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:U3162201|U3162201]] [[Category:Motivation and emotion/Book/2022]] j6ymekk6en2p1dm65hxuskoodpxaecf 2416469 2416336 2022-08-19T10:59:50Z U3195233 2948122 /* Emotion */ Topic selection wikitext text/x-wiki {{/Banner}} ==Motivation == # [[/Academic help-seeking/]] - What are the barriers and enablers of AHS and how can AHS be fostered? - [[User:Ibm4444|Ibm4444]] # [[/Academic self-regulation/]] - What is academic self-regulation, why does it matter, and how can it be fostered? - [[User:U3216563|U3216563]] # [[/Actively open-minded thinking/]] - How can AOT be used to improve human performance? - [[User:Teermeej Hossain|Teermeej Hossain]] # [[/Active transport motivation/]] - What motivates use of active transport and how can people be encouraged to use it? - [[User:MyUserName|MyUserName]] # [[/Antidepressants and motivation/]] - What are the effects of popular antidepressants on motivation? - [[User:U3222363|U3222363]] # [[/Approach motivation/]] - What is approach motivation and how does it lead to behaviour? - [[User:U3189370|U3189370]] # [[/Behavioural economics and motivation/]] - What aspects of motivation theory are useful in behavioural economics? - [[User:U3141987|U3141987]] # [[/Behavioural model of health services/]] - What is the BMHS and how can it be used? - [[User:SoSilverLibby|SoSilverLibby]] # [[/Beneficence as a psychological need/]] - What is beneficence and what are its implications as a psychological need? - [[User:CaitlinEmc|CaitlinEmc]] # [[/Brief motivational interviewing as a health intervention/]] - How can brief motivational interviewing be used as a health intervention? - [[User:MyUserName|MyUserName]] # [[/Choice overload/]] - What is choice overload? What is the optimal amount of choice? - [[UserGeorgiaFairweather|GeorgiaFairweather]] # [[/Chunking and goal pursuit/]] - How does chunking affect goal pursuit? - [[User:GiovanniBartlett|GiovanniBartlett]] # [[/Cognitive entrenchment/]] - What is cognitive entrenchment and how can it be avoided? - [[User:JimmyOC1985|JimmyOC1985]] # [[/Climate change helplessness/]] - How does learned helpless impact motivation to engage in behaviours to limit climate change? - [[User:U3193000|U3193000]] # [[/Closeness communication bias/]] - What is the CCB, why does it occur, and how can it be overcome? - [[User:U3215103|U3215103]] # [[/Commitment bias/]] - What motivates escalation of commitment even it does not lead to desirably outcomes? - [[User:U3203936|U3203936]] # [[/Comprehensive action determination model/]] - What is the CADM and how can it be applied to understanding human motivation? - [[User:MyUserName|MyUserName]] # [[/Conspiracy theory motivation/]] - What motivates people to believe in conspiracy theories? - [[User:KingMob221|KingMob221]] # [[/Construal level theory/]] - What is construal level theory and how can it be applied? - [[User:MyUserName|MyUserName]] # [[/Courage motivation/]] - What is courage, what motivates courage, and how can courage be enhanced? -[[User:Hanarose123|Hanarose123]] # [[/Death drive/]] - What is the death drive and how can it be negotiated? - [[User:U3086459|U3086459]] # [[/Discounts and consumer purchase behaviour/]] - What role do discounts play in consumer purchase behaviour? - [[User:MyUserName|MyUserName]] # [[/Drugs-violence nexus and motivation/|Drugs-violence nexus and motivation]] - What is the role of motivation in the drugs-violence nexus? - [[Atu3202070|Atu3202070]] # [[/Domestic energy conservation motivation/]] - How can domestic energy conservation and reduced consumption be motivated and behaviour changed? - [[User:MyUserName|MyUserName]] # [[/Episodic future thinking and delay discounting/]] - What is the relationship between between EFT and DD? - [[User:MyUserName|MyUserName]] # [[/Episodic memory and planning/]] - What role does episodic memory play in planning? - [[User:U3246310|U3246310]] # [[/Equity theory/]] - What is equity theory and how can it be applied? - [[EKS2001|EKS2001]] # [[/ERG theory/]] - What is Alderfer's ERG theory? - [[User:Wuser1307|Wuser1307]] # [[/Environmental volunteering motivation/]] - What motivates environmental volunteering? - [[User:MyUserName|MyUserName]] # [[/Frame of reference and motivation/]] - How does frame of reference affect motivation? - [[User:MyUserName|MyUserName]] # [[/Freedom and motivation/]] - What is the effect of freedom on motivation? - [[Cedevlin9|Cedevlin9]] # [[/Fully functioning person/]] - What is a FFP and how can full functioning be developed? - [[User:Sebastian Armstrong|Sebastian Armstrong]] # [[/Functional fixedness/]] - What is functional fixedness and how can it be overcome? - [[User:U3214117|U3214117]] # [[/Functional imagery training/]] - What is FIT and how can it be applied? - [[User:Btarmstrong24|Btarmstrong24]] # [[/Gamification and work motivation/]] - How can gamification enhance work motivation? - [[User:U3211125|U3211125]] # [[/Giving up goals/]] - When should we give up goals and when should we persist? - [[User:U3161584|U3161584]] # [[/Green prescription motivation/]] - What motivates green prescription compliance? - [[User:Earthxangel|Earthxangel]] # [[/Health belief model/]] - What is the HBM and how can it be used to enhance motivation for health-promoting behaviour? - [[User:Dee320|Dee320]] # [[/Help-seeking among boys/]] - What are the barriers to help-seeking for boys and what motivates them to seek help? - [[User:BradMcGrath|BradMcGrath]] # [[/Hidden costs of reward/]] - What are the hidden costs of motivating by reward? - [[User:SLoCE|SLoCE]] # [[/Hijack hypothesis of drug addiction/]] - What is the hijack hypothesis, what is the evidence, and how does it help to understand drug addiction? - [[U3218292|U3218292]] # [[/Honesty motivation/]] - What motivates honesty? - [[User:U3200859|U3200859]] # [[/Humour, leadership, and work/]] - What role does humour play in effective leadership in the workplace? - [[User:U3210264|U3210264]] # [[/IKEA effect/]] - What is the IKEA effect and how can it be applied? - [[User:U3216963|U3216963]] # [[/Intertemporal choice/]] - What are intertemporal choices and how can they be effectively negotiated? - [[User:MyUserName|MyUserName]] # [[/Kindness motivation/]] - What motivates kindness? - [[User:U3205429|U3205429]] # [[/Motivational music and exercise/]] - How can music be used to help motivate exercise? - [[User:U3183466|U3183466]] # [[/Non-residential energy conservation motivation/]] - How can non-residential building energy conservation and reduced consumption be motivated and behaviour changed? - [[User:MyUserName|MyUserName]] # [[/Novelty-variety as a psychological need/]] - What is novelty-variety and what are its implications as a psychological need? - [[User:MyUserName|MyUserName]] # [[/Nucleus accumbens and motivation/]] - What role does the nucleus accumbens play in motivation? - [[User:U3213250|U3213250]] # [[/Perfectionism/]] - What motivates perfectionism? Is perfectionism good or bad? How can it be managed? - [[User:AEMOR|AEMOR]] # [[/Physiological needs/]] - How do human's physiological needs affect motivation? - [[User:U3203655|U3203655]] # [[/Protection motivation theory and COVID-19/]] - How does PMT apply to managing COVID-19? - [[User:U3200956|U3200956]] # [[/Psychological trauma and subsequent drug use/]] - How does psychological trauma motivate drug use? What are the strategies for treating trauma-induced drug use? - [[User:U3210431|U3210431]] # [[/Relative deprivation and motivation/]] - What is the effect of relative deprivation on motivation? - [[User:U3191574 (PHP)|U3191574 (PHP)]] # [[/Retrospective regret/]] - What is the motivational role of retrospective regret? - [[User:Will-U3214082|Will-U3214082]] # [[/Revenge motivation/]] - What motivates revenge and how does it affect us? - [[User:U3216654|U3216654]] # [[/Self-efficacy and academic achievement/]] - What role does self-efficacy play in academic achievement? - [[User:U943292|U943292]] # [[/Self-efficacy and achievement/]] - What role does self-efficacy play in achievement outcomes? - [[User:U3216513mt|U3216513mt]] # [[/Sexual harassment at work motivation/]] - What motivates sexual harassment at work and what can be done about it? - [[User:U3037979|U3037979]] # [[/Signature strengths/]] - What are signature strengths and how can they be applied? - [[User:MyUserName|MyUserName]] # [[/Social cure/]] - What is the social cure and how can it be applied? - [[User:U3215976|U3215976]] # [[/Staff retention motivation/]] - How can organisations and managers help to motivate long-term retention of employees? - [[User:MyUserName|MyUserName]] # [[/System justification theory/]] - What is SJT, how does it affect our lives, and what can be done about it? - [[User:MyUserName|MyUserName]] # [[/Stretch goals/]] - What are stretch goals? Do they work? - [[User:MyUserName|MyUserName]] # [[/Sublimation/]] - What is sublimation and how can it be fostered? - [[User:Emily.desilva|Emily.desilva]] # [[/Survival needs and motivation/]] - What are survival needs and how do they influence motivation? - [[User:U3148161|U3148161]] # [[/Task initiation/]] - What are the challenges with task initiation and how to get get started? - [[User:StormSar|StormSar]] # [[/Theoretical domains framework/]] - What is the TDF and how can be used to guide behaviour change? - [[User:MyUserName|MyUserName]] # [[/Time and motivation/]] - What is the effect of time on motivation? - [[User:Lturner2311|Lturner2311]] # [[/Time management/]] - How can one's time be managed effectively? - [[User:CNK.20|CNK.20]] # [[/To-do lists/]] - Are to-do lists a good idea? What are their pros and cons? How can they be used effectively? - [[User:U3207458|U3207458]] # [[/Treatment motivation in juvenile delinquency/]] - What is the role of treatment motivation for juvenile delinquency and how can it be enhanced? - [[User:MyUserName|MyUserName]] # [[/Uncertainty avoidance/]] - What is uncertainty avoidance, why does it occur, and what are its consequences? - [[User:Franklin Brightt|Franklin Brightt]] # [[/Urgency bias and productivity/]] - What is the impact of urgency bias on productivity and what can be done about it? - [[User:U3055143|U3055143]] # [[/Vocational identity/]] - What is vocational identity and how does it develop? - [[User:MyUserName|MyUserName]] # [[/Volunteer tourism motivation/]] - What motivates volunteer tourism? - [[User:U962051|U962051]] # [[/Wanting and liking/]] - What are the similarities and differences between wanting and liking, and what are the implications? - [[User:U3201643|U3201643]] # [[/Work breaks, well-being, and productivity/]] - How do work breaks affect well-being and productivity? - [[User:U3215603|U3215603]] # [[/Work and flow/]] - What characteristics of work can produce flow and how can flow at work be fostered? - [[User:U3213441|U3213441]] ==Emotion== # [[/Animal emotion/]] - What is the emotional experience of animals? - [[User:U3216502|u3203545]] # [[/Attributions and emotion/]] - How do attributions affect emotion? - [[User:MyUserName|MyUserName]] # [[/Autonomous sensory meridian response and emotion/]] - What emotions are involved in ASMR experiences and why do they occur? - [[User:U3186959|U3186959]] # [[/Benzodiazepines and emotion/]] - What are the effects of benzodiazepines on emotion? - [[User:FulaAjeo22|FulaAjeo22]] # [[/Bewilderment/]] - What is bewilderment and how can it be dealt with? - [[User:SunandaUC|SunandaUC]] # [[/Burnout/]] - What is burnout and how can be it be managed and prevented? - [[User:U3202788|U3202788]] # [[/Cognitive dissonance reduction/]] - What strategies do people use to reduce cognitive dissonance and how effective are they? - [[User:Tatjurate|Tatjurate]] # [[/Colonisation and emotion in Australia/]] - What are the emotional responses to colonisation in Australia? - [[User:Micabaker1|Micabaker1]] # [[/Compassion/]] - What is compassion, what are its pros and cons, and how can it be fostered? - [[User:U3203545|U3203545]] # [[/Compassion fatigue in mental health professionals/]] - What causes compassion fatigue in MH professionals and how can it be prevented? - [[User:MyUserName|MyUserName]] # [[/Connection to country and well-being/]] - What is the relationship between connection to country and well-being? - [[User:MyUserName|MyUserName]] # [[/Contempt/]] - What is contempt, what causes it, and how can it be managed? - [[User:U3202788|U3219905]] # [[/Core emotions/]] - What are the core emotions and what is their function? U3203140 # [[/Creative arts and trauma/]] - How can creative arts help in dealing with trauma? - [[User:SashaBrooksby|SashaBrooksby]] # [[/Cultural influences on shame, guilt, and pride/]] - How does culture influence shame, guilt, and pride? - [[User:Tamika Afeaki|Tamika Afeaki]] # [[/Default mode network and the self/]] - What is the relationship between the DMN and the self? - [[User:MyUserName|MyUserName]] # [[/Difficult conversations and emotion/]] - What communication and emotional skills are needed to successfully negotiate difficult conversations? - [[User:U3158968|U3158968]] # [[/Disappointment/]] - What is disappointment, what causes it, and how can it be managed? - [[User:U3216256|U3216256]] # [[/DMT and spirituality/]] - How can DMT facilitate spiritual experiences? - [[User:DenniseSoleymani|DenniseSoleymani]] # [[/Durability bias in affective forecasting/]] - What role does durability bias play in affective forecasting? -]] [[User:MyUserName|MyUserName]] # [[/Ecological grief/]] - What is ecological grief and what can be done about it? - [[User:Brewerjr|Brewerjr]] # [[/Ecopsychology and stress/]] - How can ecopsychology help to explain and deal with stress? - [[User:MyUserName|MyUserName]] # [[/Embarrassment/]] - What is embarrassment, what causes it, and how can it be managed? - [[User:U3190353|U3190353]] # [[/Emotional intelligence training/]] - How can emotional intelligence be trained? - [[User:Eimilerous22|Eimilerous22]] # [[/Emotion knowledge/]] - What is emotion knowledge and how can it be developed? - [[User:GabbieUC|GabbieUC]] # [[/Emotion across the lifespan/]] - How does emotion develop across the lifespan? - [[User:U3230861|U3230861]] # [[/Endocannabinoid system and emotion/]] - What is the role of the endocannabinoid system in emotion? - [[User:RWilliams12|Rwilliams12]] # [[/Environmental grief/]] - What is eco-grief, its causes and consequences, and what can be done? - [[User:Gabrielle Eagling|Gabrielle Eagling]] # [[/Exercise and endocannabinoids/]] - What is the relationship between exercise and the endocannabinoid system? - [[User:MyUserName|MyUserName]] # [[/Expressive suppression and emotion regulation/]] - What is the role of expressive suppression in emotion regulation? - [[User:U3131472|U3131472]] # [[/Fairness and emotion/]] - What is the relation between fairness and emotion? - [[User:U3246554|U3246554]] # [[/Fatigue and emotion/]] - What is the effect of fatigue on emotion and what can be done about it? - [[User:Lewis.Kusk|Lewis.Kusk]] # [[/Fear/]] - What is fear, what causes it, and how can it be managed? - [[User:Icantchooseone|Icantchooseone]] # [[/Fear of working out/]] - What is FOWO and how can it be overcome? - [[User:U3216963|U3216963]] # [[/Flourishing in the elderly/]] - How can psychological flourishing be supported in the elderly? - [[User:MyUserName|MyUserName]] # [[/Fundamental attribution error and emotion/]] - What is the relationship between the FAE and emotion? - [[User:MyUserName|MyUserName]] # [[/Gratitude and subjective wellbeing/]] - What is the relationship between gratitude and subjective wellbeing? - [[User:MyUserName|MyUserName]] # [[/Gloatrage/]] - What is gloatrage, what causes it, and what are its consequences? - [[User:MyUserName|MyUserName]] # [[/Heart rate variability and emotion regulation/]] - What is the relationship between HRV and emotion regulation? - [[User:MyUserName|MyUserName]] # [[/Hedonic adaptation prevention model/]] - What is the HAP model and how can it be applied? - [[User:Lyndel Lemon|Lyndel Lemon]] # [[/Humility/]] - What is humility, what causes it, and is it desirable? - [[U3195233]] # [[/Hypomania and emotion/]] - What are the emotional characteristics of hypomania? - [[User:Alec.cortez|Alec.cortez]] # [[/Impact bias/]] - What is impact bias, what causes it, what are its consequences, and how can it be avoided? - [[User:MyUserName|MyUserName]] # [[Indigenous Australian emotionality]] - In what ways is emotionality experienced by Indigenous Australian people? - [[User:U3189442 - K.Ryan|U3189442 - K.Ryan]] # [[/Indigenous Australian mindfulness/]] - How has Indigenous Australian culture traditionally conceived of, and practiced, mindfulness? - [[User:MyUserName|MyUserName]] # [[/Inspiration/]] - What is inspiration, what causes it, what are its consequences, and how can it be fostered? - [[User:U3230861|U3227354]] # [[/Insular cortex and emotion/]] - What role does the insular cortex play in emotion? - [[User:U3190094|U3190094]] # [[/Interoception and emotion/]] - What is the relationship between interoception and emotion? - [[User:U3203265|U3203265]] # [[/Kama muta/]] - What is kama muta, what are its effects, and how can it be fostered? - [[User:U3183521|U3183521]] # [[/Linguistic relativism and emotion/]] - What is the role of linguistic relativism in emotion? - [[User:U3119310|U3119310]] # [[/Menstrual cycle mood disorders/]] - What causes menstrual cycle mood disorders and how can they be managed? - [[User:U3217109|U3217109]] # [[/Mental toughness in the workplace/]] - What can mental toughness be useful in the workplace? How can it be developed? - [[User:MyUserName|MyUserName]] # [[/Mindfulness and creativity/]] - How can mindfulness enhance creativity? - [[User:CaityDcr1603|U3217109]] # [[/Mindful self-care/]] - What is mindful self-care, why does it matter, and how can it be developed? - [[User:Clairelogan|Clairelogan]] # [[/Mixed emotions/]] - What are mixed emotions, what causes them, and how can they be managed? - [[User:U3210490|U3210490]] # [[/Mudita/]] - What is mudita and how can it be developed? -[[User:Inandonit365|Inandonit365]] # [[/Natural disasters and emotion/]] - How do people respond emotionally to natural disasters and how can they be supported? -[[User:U3148366_Chris|U3148366_Chris]] # [[/Nature therapy/]] - What is nature therapy and how can it be applied? - [[User:Ana028|Ana028]] # [[/Narcissism and emotion/]] - What is the relationship between narcissism and emotion? - [[User:A Super Villain|A Super Villain]] # [[/Narrative therapy and emotion/]] - What is the role of emotion in narrative therapy? - [[User:MyUserName|MyUserName]] # [[/Needle fear/]] - How does needle fear develop, what are its consequences, and what can be done about it? - [[User:U3166273|U3166273]] # [[/Neuroimaging and mood disorders/]] - How can neuroimaging assist in diagnosing and treating mood disorders? - [[User:Jdebear|Jdebear]] # [[/Occupational violence, emotion, and coping/]] - What are the emotional impacts of occupational violence and how can employees cope? - [[User:MyUserName|MyUserName]] # [[/Positivity ratio/]] - What is the positivity ratio and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Post-traumatic stress disorder and emotion/]] - What is the effect of PTSD on emotion? - [[User:JorjaFive|JorjaFive]] # [[/Psychological distress/]] - What is PD, what are the main types, and how can they be managed? - [[User:U3190773|U3190773]] # [[/Psychological trauma/]] - What causes psychological trauma, what are the consequences, and how can people recover from psychological trauma? - [[User:MyUserName|MyUserName]] # [[/Psilocybin assisted psychotherapy/]] - How can psilocybin be used to assist psychotherapy? - [[User:U3083720|U3083720]] # [[/Rational compassion/]] - What is rational compassion and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Reflected glory/]] - What is reflected glory and what are its pros and cons? - [[User:MyUserName|MyUserName]] # [[/Religiosity and coping/]] - What is the relationship between religiosity and coping? - [[User:Noah O'Brien|Noah O'Brien]] # [[/Resentment/]] - What is resentment, what causes it, and what are its consequences? - [[User:U3216389|U3216389]] # [[/Risk-as-feelings/]] - What is the emotional experience of risk and how does it influence decision-making and behaviour? - [[User:BenjiD'Ange|BenjiD'Ange]] # [[/Self-esteem and culture/]] - What are the cultural influences on self-esteem? - [[User:Jingru shao 0906|Jingru shao0906]] # [[/Smiling and emotion/]] - What is the relationship between smiling and emotion? - [[User:U3200902|U3200902]] # [[/Social media and suicide prevention/]] - How can social media be used to help prevent suicide? - [[JaimeTegan|JaimeTegan]] # [[/Sorry business/]] - What is sorry business and what role does it play in Indigenous communities in Australia? - [[User:Isaacem13|Isaacem13]] # [[/Stress control mindset/]] - What is a SCM, why does it matter, and how can it be cultivated? - [[User:MyUserName|MyUserName]] # [[/Suffering as emotion/]] - What is the emotional experience of suffering and how can people cope with suffering? - [[User:Brookewin|Brookewin]] # [[/Telemental health/]] - What are the pros and cons of TMH and what are the key ingredients for effective TMH practices? - [[User:MyUserName|MyUserName]] # [[/Topophilia/]] - What is topophilia, how does it develop, and what are the psychological impacts? - [[User:RSPMeredith|RSPMeredith]] # [[/Triumph/]] - What is triumph, what causes it, and how can it be managed? - [[User:Bill.miosge|Bill.miosge]] # [[/Unemployment and mental health/]]: What is the relationship between unemployment and mental health? - [[User:U3216958 - Tiarna.Wilson-Ginn|U3216958 - Tiarna.Wilson-Ginn]] # [[/Viewing natural scenes and emotion/]] - What is the effect of viewing natural scenes on emotion and how can this be applied? - [[User:MyUserName|MyUserName]] # [[/Wave metaphor for emotion/]] - In what respects is an ocean wave a helpful metaphor for understanding human emotions? - [[User:Jamieepiper|Jamieepiper]] # [[/Window of tolerance/]] - What is the window of tolerance and how it be applied? - [[User:U3223109|U3223109]] # [[/Workplace mental health training/]] - What is WMHT, what techniques are used, and what are the impacts? - [[User:ArtOfHappiness|ArtOfHappiness]] # [[/Zoom fatigue/]] - What is Zoom fatigue, what causes it, what are its consequences, and what can be done about it? - [[User:u3211603|U3211603]] ==Motivation and emotion== # [[/Financial investing, motivation, and emotion/]] - What role does motivation and emotion play in financial investing? - [[User:U3217287|U3217287]] # [[/Hostage negotiation, motivation, and emotion/]] - What role does motivation and emotion play in hostage negotiation? - [[User:U3213549|U3213549]] # [[/Money priming, motivation, and emotion/]] - What is the effect of money priming on motivation and emotion? - [[User:Molzaroid|Molzaroid]] # [[/Motivational dimensional model of affect/]] - What is the motivational dimensional model of affect and what are its implications? - [[User:MyUserName|MyUserName]] # [[/Napping, motivation, and emotion/]] - What are the motivational and emotional effects of napping? - [[User:MyUserName|MyUserName]] # [[/Overchoice, emotion, and motivation/]] - What are the emotional and motivational effects of overchoice? - [[User:MyUserName|MyUserName]] # [[/Patience and impatience/]] - What are the psychological causes and consequences of patience and impatience? - [[User:U3100193|U3100193]] # [[/Reward system, motivation, and emotion/]] - What role does the reward system play in motivation and emotion? - [[User:U3162201|U3162201]] [[Category:Motivation and emotion/Book/2022]] 4hib71bdom76myjsczd5qmiocqutmq3 1 277716 2416434 2411052 2022-08-19T04:53:10Z Greg at Higher Math Help 2911873 /* Test */ new section wikitext text/x-wiki {{Eventmath tabs}} Talk page for Eventmath. The top of this page is still under construction, but you're welcome to post questions, suggestions, or other discussion topics below. Note: We may want to have a "seeking collaborators" section, or else a link to a dedicated section of the Tasks page for that, or maybe a whole page dedicated to that. == Contrast / Accessibility == The contrast for the tabs appears to be inaccessible. See [https://webaim.org/resources/contrastchecker/ WebAIM: Contrast Checker]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 01:09, 19 September 2021 (UTC) :{{At|Dave Braunschweig}} Thanks for taking a look, and for sharing the contrast checker tool. The contrast absolutely needs to be fixed. Right now, the Eventmath pages are very much under construction, and we're still sorting out how to change the link color. We will likely choose a style similar to that of the [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Project_impact Project-impact table in our grant proposal], although we may change the colors some for increased contrast. Our [[Template:Eventmath tabs|current template]] is based on a [https://en.wikipedia.org/wiki/Template:Start_tab tab template]; that template has parameters for some CSS styles, but it has no such parameter for the link color. Perhaps we can use the freeform-n parameter with a span element and a style attribute (although maybe a separate style sheet would be preferable), or maybe we can pursue an update to the underlying template. We'll look into it, but if you happen to have an idea that would speed up the process, or if you know of any general pages about setting up custom CSS on Wikiversity, please let us know! A quick search revealed this [[Help:CSS at Wikiversity|Wikiversity help page on CSS]], and this [https://en.wikipedia.org/wiki/Help:Cascading_Style_Sheets Wikipedia help page on CSS], but I haven't had a chance to read through those yet. --[[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 01:23, 20 September 2021 (UTC) ::It's not as simple as you might like. Link colors are controlled by the [[mw:Manual:Skins|skin]] selected by the user. There are four different skins available and four possible [https://www.w3schools.com/css/css_link.asp link states]. In addition, users can customize their own skin options to enhance contrast, etc. So, it's not a matter of simply specifying a link color, as whatever you select is likely to conflict with another setting somewhere that you can't and/or shouldn't try to control. I suppose it would be possible to try and work with some of the UX designers to customize this, but we just have a copy of the Wikipedia tab templates. With 4,266 uses on Wikipedia, no one has felt the need to customize this so far. Rather than spending valuable time trying to customize the user interface in ways it is not designed for, I encourage you to select an accessible background tab color and focus on creating meaningful content instead. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 16:34, 20 September 2021 (UTC) :::Thanks for the information. --[[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 02:23, 21 September 2021 (UTC) :::'''Update: It's fixed :)''' --[[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 09:06, 21 February 2022 (UTC) {{Eventmath end tab}} [[Category:Eventmath project pages]] == Test == This is a test. It's probably better to use a sandbox, but a quick live test will be helpful in this case. [[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 04:53, 19 August 2022 (UTC) 4ibamhmjixbh1sois6qfqn3626tw83t 2416436 2416434 2022-08-19T04:54:04Z Greg at Higher Math Help 2911873 Undo revision 2416434 by [[Special:Contributions/Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|talk]]) wikitext text/x-wiki {{Eventmath tabs}} Talk page for Eventmath. The top of this page is still under construction, but you're welcome to post questions, suggestions, or other discussion topics below. Note: We may want to have a "seeking collaborators" section, or else a link to a dedicated section of the Tasks page for that, or maybe a whole page dedicated to that. == Contrast / Accessibility == The contrast for the tabs appears to be inaccessible. See [https://webaim.org/resources/contrastchecker/ WebAIM: Contrast Checker]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 01:09, 19 September 2021 (UTC) :{{At|Dave Braunschweig}} Thanks for taking a look, and for sharing the contrast checker tool. The contrast absolutely needs to be fixed. Right now, the Eventmath pages are very much under construction, and we're still sorting out how to change the link color. We will likely choose a style similar to that of the [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Project_impact Project-impact table in our grant proposal], although we may change the colors some for increased contrast. Our [[Template:Eventmath tabs|current template]] is based on a [https://en.wikipedia.org/wiki/Template:Start_tab tab template]; that template has parameters for some CSS styles, but it has no such parameter for the link color. Perhaps we can use the freeform-n parameter with a span element and a style attribute (although maybe a separate style sheet would be preferable), or maybe we can pursue an update to the underlying template. We'll look into it, but if you happen to have an idea that would speed up the process, or if you know of any general pages about setting up custom CSS on Wikiversity, please let us know! A quick search revealed this [[Help:CSS at Wikiversity|Wikiversity help page on CSS]], and this [https://en.wikipedia.org/wiki/Help:Cascading_Style_Sheets Wikipedia help page on CSS], but I haven't had a chance to read through those yet. --[[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 01:23, 20 September 2021 (UTC) ::It's not as simple as you might like. Link colors are controlled by the [[mw:Manual:Skins|skin]] selected by the user. There are four different skins available and four possible [https://www.w3schools.com/css/css_link.asp link states]. In addition, users can customize their own skin options to enhance contrast, etc. So, it's not a matter of simply specifying a link color, as whatever you select is likely to conflict with another setting somewhere that you can't and/or shouldn't try to control. I suppose it would be possible to try and work with some of the UX designers to customize this, but we just have a copy of the Wikipedia tab templates. With 4,266 uses on Wikipedia, no one has felt the need to customize this so far. Rather than spending valuable time trying to customize the user interface in ways it is not designed for, I encourage you to select an accessible background tab color and focus on creating meaningful content instead. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 16:34, 20 September 2021 (UTC) :::Thanks for the information. --[[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 02:23, 21 September 2021 (UTC) :::'''Update: It's fixed :)''' --[[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 09:06, 21 February 2022 (UTC) {{Eventmath end tab}} [[Category:Eventmath project pages]] 3pw4dnk85ftqzgyrf6qatp6pdd1x7u0 0 279839 2416376 2348987 2022-08-19T00:48:10Z Cedevlin9 2947652 wikitext text/x-wiki {{title|Chapter title:<br>Subtitle?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] {{METP}} ==See also== * [[Motivation and emotion/Book/2021/Freedom and motivation|Freedom and motivation]] (Book chapter, 2021) 1qw3cnaks3s6sw6ld5f8vagme1no7bf 0 279851 2416369 2416107 2022-08-19T00:15:14Z U3216389 2947531 got rid of the template wikitext text/x-wiki {{METP}} ==See also== * [[Motivation and emotion/Book/2021/Resentment|Resentment]] (Book chapter, 2021) ==External links== * [https://www.abc.net.au/radionational/programs/allinthemind/overcoming-resentment-with-gratitude/13660930 Overcoming resentment with gratitude] (All in the Mind, 2021) == Link to User Page (U3216389) == [[User:U3216389|My User Page]] [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[Category:Motivation and emotion/Book/Resentment]] {{title|Resentment<br>What is Resentment, what causes it, and what are it's consequences?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== An overview of Resentment and the topics that I will potentially discuss throughout this book chapter. [[File:Emotions - 3.png|thumb|Figure 1. A diagram of the primary emotions that we all experience, with resentment sitting under anger. ]] ==Resentment == * What is Resentment? * Where does the emotion derive from? == What Causes Resentment and What are its Consequences? == == Resentment and Morality == == Ressentiment == ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] 3myiyw309l530tpqsayuyavu7tu66w3 0 280327 2416441 2360767 2022-08-19T05:05:43Z U3216256 2942574 wikitext text/x-wiki {{METP}} ==See also== * [[Motivation and emotion/Book/2021/Creative arts and trauma|Creative arts and trauma]] (Book chapter, 2021) {{title|Chapter title:<br>Subtitle?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[User:U3216256|U3216256]] ([[User talk:U3216256|discuss]] • [[Special:Contributions/U3216256|contribs]]) 05:05, 19 August 2022 (UTC) j37vbtn1ge6gyg4k23ege73najasli1 0 281095 2416464 2413766 2022-08-19T09:24:51Z Hamish84 1362807 /* Basic Principles. */Adding text wikitext text/x-wiki = <ref>'''William Shakespeare''' (bapt. 26 April 1564 – 23 April 1616</ref>A message. = What humanity needs is not any individual approach but a governance powerful body of excellence that has modern technology, knowledge, and freedom they can use to disseminate clear information. Clear information about a new language structure of absolutes beneficial to an International forum, and eventually to reach a Universal status.Its benefits reach toward conceptual language on a planet that speaks more than 7,000 languages. No matter the language spoken the concepts of: Air - Food - Water are recognised. The overall development of conceptual language can only be beneficial and will be as appropriate to Absolutes definitions. Reaching for the Stars might show us the way. Universal Language of Absolutes. "A very grand title but it took many years to explain its value" = Our history. = Born in Scotland in 1927 left school at 14 years of age. Married at 21 years of age and we had two children. We emigrated to New Zealand in 1953 and lived there for approximately thirty years. During our stay there I did a Liberal Studies Course at Canterbury University Christchurch and graduated. We have since had ten books published through Google books on the subject matter at hand and my wife Jean Caldwell McMillan is the co-author to most of the work presented here. My wife Jean was an avid reader of many works on philosophy and psychology. She was influenced by the works of Erich Fromm. Jean died 9th of January 2011. To refresh the original purpose of  our  earlier writings my wife and I  went on an odyssey  looking for any data, ancient or otherwise, on human consciousness, specifically related to Alzheimer’s disease. Now at 95 years of age (well past my used by date) it may well be that I am a candidate with a focus on my own pending dementia. If so, then the theory and the method I now write about is holding it at bay. To address the health of my mind in this way could be the catalyst that retains its own functional activity. The creation of a  semantic template is well documented below. No semantic definition of absolutes or principles can be ill-defined. They are always interconnected, interdependent and infinite. Each configuration constructed by anyone has meaning particular to them, although its value is universal. That  is why it is never personal property! [[File:JimNJean.png|center|thumb|228x228px|Jim and Jean]] = The Beginning. = "The road's half traveled when you know the way" [[File:Tree lined path - geograph.org.uk - 2269906.jpg|center|thumb]] Oxford dictionary definitions: Principle: "A fundamental truth used as a basis of reasoning". Absolute: " Complete - Entire - Perfect - Pure. These particular dictionary given definitions offers us guidelines to ‘existing conditions’ necessary for complementary understanding, and experience. We can only examine that which is real, basic evidence, that is fundamentally true, and we must ‘use’ it, to establish that which is reasonable. The general consensus is that there are no Absolutes. The following material is set out to show the very reverse is true and that everything that is is Absolute. Establish that there are no dichotomies that will leave the primary terms alone to create a semantic template. There are no dichotomies. Mythical dichotomies distort Reality. Everything is: The computer you use today has always existed, it is the arrangement of particles that have materialized it. The subject matter "Universal Language of Absolutes' is promoted to provide a new understanding of spoken language. This understanding was initially constructed by the cognitive experiences of both my wife and myself many years ago. Just like the principle of a jigsaw puzzle, meaning lies significantly in the fact that all pieces of the puzzle are interdependent and interconnected. When completed they provide a picture of the whole. We have endeavoured to produce a picture of the evolutionary process of language in human history because the evolution of language prefixes all modes of thought in human culture. The material directs the reader towards a new view that all that evolves is in a vertical direction, not the linear direction commonly understood. Human consciousness is of itself the phenomenon of evolution and to recognize its existence is part of the process. Shakespeare expressed this succinctly through the voice of Juliet who proclaimed, “a rose by any other name- would smell as sweet.” [[File:Comestible rose in the Laquenexy orchard garden, Moselle, France (01).jpg|thumb|center|237x237px]] === Conceptual language. === My wife and I recognized how profound the extension of this observation would mean conceptually. Of all the languages spoken on this planet, it would be fair to say that all of them would contain the properties of, air, food, and water conceptually, etc. This is a form of consciousness equality that is available to us all. It points to the reality of our constant relation to each other and our existence. We can never exist in a world of individuality, but only in relation to the consciousness of one another. Consequently, that exceptional experience can only be shared superficially. We cannot ‘know’ any other life experiences other than our own introspection. <gallery> File:Einstein 1921 by F Schmutzer - restoration.jpg </gallery>" ''Albert Einstein 1921. We experience our thoughts and feelings as something separate from the rest. A kind of optical delusion of consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us."'' ''Albert Einstein, in One Home, One Family, One Future,p.99'' Einstein came very close. In reality, every human being has a backpack from the day they were conceived. In the backpack every experience in their mother’s womb is experienced. At birth and throughout their lives, everything that happens to them in life is registered and creates their personality. That life with all its experiences can never be known to anyone else, consequently, we can never “know” another person. It creates equality of consciousness that we must understand. We can know details about a person, but that is all. That life is sacrosanct. Who we are really goes beyond normal human experience and into the realm of the Absolute. Werner Karl Heisenberg (physicist). The Heisenberg Uncertainty Principle. As a young layman with no knowledge of Heisenberg but interested in principles it seemed to me that the Uncertainty Principle was just a contradiction in terms. In later years I found that Heraclitus describes life as being in a state of flux a replica of the Uncertainty Principle which in fact can be defined as an absolute state. Within the context of knowing who we are and the backpack we carry our life in, we can never know each individual life as that life experience is singularly their own and sacrosanct. It now seems that the Heisenberg Uncertainty Principle can fall into the category of being an Absolute. Evolution proceeds in advance of our need to evolve. In our pure active state, we are.There is no static end (an abomination) - only beginning. As we cannot know what tomorrow will bring, living with expectations is rather futile. Nature has its own agenda. Zen Koan recorded 1228: 'An instant realization sees endless time. Endless time is as one moment. When one comprehends the endless moment He, or she, realizes the person who is seeing it.' We do not own Space. We do not own Time We do not own Energy We do not own Matter [[File:Universum.jpg|center|thumb]] "What we call the beginning is often the end And to make an end is to make a beginning. The end is where we start from We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the first time" T.S Eliot = Everything is in scale. = The present moment is the point in which Eternity has placed us – we all live in that moment, and whether we like it or not, we exist in it, experience it, have knowledge of it, and we all share it, measure by measure. There are no dichotomies. Illusion is a measure of Reality, as Stupidity is a measure of Intelligence If one keeps measuring illusion it is an attempted downward spiral to nothingness.There is no opposite to Reality – that illusion is a measure of Reality. There is no such thing as ‘nothing’ in the elemental construction of Homo sapiens. All the innate ‘something’s’ are the fundamentals of our being human and all our experiences. The correct use, and understanding of who we are, is an extension therefrom. Does it require any interpretation on anyones part to say ‘we are? Any attempted denial of that statement would be perverse use of the language, and delusional. === Try saying ‘we are not” === ‘We are’ is the foundation of all affirmation, and within that spectrum, we can know, and be. ‘We are’ is self-evident Truth. We can neither know, nor experience what isn’t. Eternity is the here and now, that is why it is possible to explain the experience of Eternity. Nothing is ever lost in Time. We are located in a vast Universe. ‘twas a moment’s pause,- All that took place within me came and went As in a moment; yet with Time it dwells, And grateful memory, as a thing divine. Wordsworth Prelude, Book V111 We already know – the ''basic'' condition that must exist for us to re-cognise. It is at that moment of pure affirmation, when all that is, is manifest. Unless there was mutual identity we could not know anything. It is why we are urged to evacuate the Platonic cave. Sadly most prefer the shadows rather than confront who they are. We already know – the basic condition that must exist for us to recognise. It is at that moment of pure affirmation, when all that is, is manifest. == Lost Shadows == [[File:People Shadow.JPG|left|thumb]] The shadows move Lost in confusion Lost in despair Imagination shrouds the real Looking back Looking forward Is this the Centre? James Brines. = Basic Principles. = We are all in the business of living and attempting to understand the principles involved in that human process up to the end of life. The implicit principles necessary for life eventually disappear and all measurable criteria pronounce a body to be devoid of life. Throughout historical agreement we know what that means, and we act accordingly on *common knowledge*. We know that dying is a necessary factor of life. It is a Natural law that if we live - we also die. Natural law is Universal, for us to *know* that a body lives; we also *know* that a body’s life will end. Albeit that reports tell us that today millions of people die of disease, starvation, wars, we of necessity accept that as the ongoing reality because again we are universally connected and know the results of such carnage. Because it is in our realm of common knowledge we have graveyards, crematoriums, undertakers, doctors who pronounce bodies to have died. We understand the consequences of leaving such bodies unburied, the diseases that would prevail. Again, please explore the definition of principle (Universal principle) and try to go to the limitations of the definition without using mythical dichotomies. [[File:( A great picture of outer space ).jpg|center|thumb]] Principle: A fundamental truth used as a basis of reasoning. All of these questions are based on singularity (the Cartesian dogma) If Descartes had only introduced inclusion into his musings (they were taken as conservative singularity) he may have realized the difficulty of addressing thought as reality. He then may have quite easily concluded that universally - *We are!!* That pronouncement is inclusive, and conclusive in every way, *we are - and we know!!* Because my knowledge is not a private, personal piece of property concerning principles, but Universal (Archimedes et al), then that innate knowledge has completeness we can share. Whatever identical resonance we may be able to share (concerning completeness), that can only be accomplished by understanding the principles involved and their constant relationship to each of us. Clearly the plethora of present and past discoveries establishes the existence of that which is fundamentally true, and the foundation for law. The principles are established, and always have been, we are in the business of making them transparent and complete. However tenuous the link we are all connected through communication, the air we breathe, the ground we walk on, the universe we live in, the common principles we live by. We all must have sustenance to survive, or we will not live. (See above) <nowiki>*</nowiki>There is nothing else to experience.* When we actively explore the reality of anything, all principles involved in that exploration are complementary, and honest, and we understand the wasteful divisive mythology that people attempt to attach to them. We cannot */partially know/* the truth, it must be complete. Dichotomies attempt to deny the existence of truth, and are misleading. How do we more reasonably completely know anything? The complexity of language systems with contemporary usage requires new and creative structures to provide clear information. Internal and external reasoning capacities can only develop in concert with Man’s recognition of the principles that essentially form our lives. The inevitability of human consciousness rising beyond its historical beginnings posits a future outside our normal perceptions, and  a factual reality that points to the existence of new perceptions that are infinite.   It is natural to know when we are no longer trapped in any mythical ideologies that gives credence to dichotomies that stifle the recognition of simple principles. There was no cause for knowledge to be established - it has always existed.* Ask yourself - ‘how do you know to ask any questions at all?’ There is no hidden dimension or mystical world. The only philosophical reality is ‘that which is’. To access that we need a new structure to explore ‘knowledge’, a new transformational language. Real knowledge is not amassed information, nor is it the establishment of dogmas, isms, or mythical belief systems. [[File:FAIR data principles.jpg|center|thumb]] Basic principles are the source and foundation of all /*knowledge*/. Until that is recognized, extension from mythical sources only leads to a denial of one’s own senses. Trying to conceive of a contemporary world without principles is to posit a world without reason, or intelligence. The principle of pure knowledge (Truth) could be said to exist in another dimension given the present state of human understanding. To progress that mistaken belief there would be strong support in the need for a comparative reality. Curiously it seems that philosophy (the seeking of knowledge) constantly discounts any knowledge that does not come within the sphere of established philosophy, and the comparative reality dictum. That consequence profoundly distils the purity of any experience and alienates the observer in their confrontation of that which is real. The measure of that ‘comparative reality’ knowledge bounded by dichotomies is so restrictive that it lies in a mythical dimension where denial of its very purpose is the order of the day. Evidently it will not allow doubt to undermine its own denials. A basic principle of Nature is /*knowledge*/ and it constantly communicates innately in every living structure. How to grow, develop, and disseminate. Knowledge is reciprocal truth that depends on our relationship and the recognition of principles operating. Knowledge (unlike information) is not stored in an individual box; it is ubiquitously manifest in everything we do. Knowledge is the experience of a positive reality, and its true construction is a dependable source of secure information (not to be misused). We daily have the opportunity of witnessing ‘knowledge’ in action as expressed by the ‘the principles of knowledge’, namely the principles themselves. The questions lie in a continuing mythical belief in a mystical unknown (the Cartesian stance) which because of its non-existence can never be known. It is a belief that is detrimental (because it attempts to deny all existing factual knowledge) to dealing with Nature and Reality and the fundamental necessity of our relationship with them. Making that relationship transparent is our basic obligation and the ongoing evolutionary activity. Although there is an obvious avoidance to address the definition of principle itself, it is a factor that must be paid attention to, to realize that ‘common knowledge’ is the only reality that exists. Amazingly although there is avoidance of principles - truth - reason etc, there is acceptance of the Cartesian dogma,. Paradoxically this determined acceptance of Descartes supports the reality of innate knowledge existing (I think - therefore I am) which establishes for him innate knowledge; however mistaken he is concerning the interpretation of his experience. = Leverage. = Long before I read of Archimedes and his various principle discoveries I was using the principle of leverage in a variety of ways, prying lids off boxes, moving articles with a lever well beyond my physical strength to do so without said lever, and I knew how to do it. Transferring that knowledge to a student or apprentice is relatively easy because innately they also /*know*/ how to do it. Every aspect of human industry uses the principle in a myriad of ways because it is our obligation to constantly progress the principle and confirm the constant utilization of knowledge. We wholeheartedly adopted Archimedes principles (et al) because we recognized their fundamental utility. It is preposterous to question the widespread /*factual knowledge*/ of all principles, more especially so when we cannot escape their ubiquitous daily existence in all our lives - Nature and Reality do not lie. One may abstain from admitting their existence. To do so is simply attempting the impossible, and is devoid of all reason. Real knowledge has been put into the realm of the mystical unknowable even to the point that knowledge practitioners go to the outer extremes and deny the gift of their natural senses. With their adherence to what they consider is knowledge they become captive automatons to any prescription for life that is expressed in that ’knowledge medium’, which then becomes the authority. When ’knowledge’ is addressed as having a collective source in Universal principles then we have the potential to experience its complete reality (microcosm - macrocosm) without any imaginary, or divisive comparative content. There is then a re-orientation process toward our true being and recognition of our own reality in relation to the natural processes we share. Real knowledge is elementary and Natural. We know, because that which is knowable is constantly expressed by the principles involved. We all share those principles and can correctly infer the most simple and obvious truths. All social life functions by our adherence to the implicit laws operating within them. Seeking experimental contradiction to a fact of life offers us nothing but proof which is the establishing of ‘complete knowledge’ however ‘more reasonably’ one wishes to extend the exercise. The construction of new philosophies must seek a mandate to fully explore the relationship between experience and innate knowledge as the foundation for pure knowledge to emerge. To repeat, knowledge or truth do not reside in any individual domain, nor are they the private possession of any human being. We know, because ’knowledge’ is an innate natural possession that we constantly share - otherwise civilization could not exist. == Knowledge: Evident facts about mutual standards that provide us with security. == We do not become human beings because we can ‘think’; we become more human because we learn to conform to the implicit principles in Nature and Reality. Denying them - denies our being. It is natural to know. There was no cause for knowledge to be established - it has always existed. Knowledge is an evolutionary process. Human beings developed from primitive innate instinctual knowledge to contemporary cultures. Some know more than others through experience, and make that knowledge transparent. Insistence on how we can ’completely’ know is an ephemeral philosophical question that attempts to deny that we can have ’knowledge’ at all, as you understand it. Knowing that we ’know’ the inherent completeness of everything through the existence of principles, is the natural catalyst to make ’that which is’ transparent. Heraclitus: "No man steps in the same river twice " He believed in the "Unity of Opposites (Absolutes). He cried for the needless unconsciousness of mankind.. “Exploration of a mythical dichotomy below for the purpose of establishing principles. Principles that are not a 'mind' construct, but the very essence of our being. Independence, is the curious and dangerous malady where humanity has lionized negative mythology in denial of its own reality. The human fundamental reality has at its base the simple natural law that we are dependent beings. From conception, the human embryo is entirely dependent on the health and well-being of its mother to provide it adequate healthy sustenance to enable its entry into the world. That form of innate dependability the human species carries with it throughout its spectrum of life. Every aspect of human activity is premised on the availability of air, food, and water without which the organism cannot survive (this would be an incontrovertible 'more reasonable' truism or an Absolute). In a perverse way, that which is our natural state has become the target for what appears to be open defiance of the laws that govern our behavior. When a basic premise is either used mistakenly, or deliberately, its consequences can be socially far reaching, for any deviation however far it is extended is a distortion of the truth, and a denial of who we are. The erroneous conclusion through exercising responsibility that we can confer independence to our actions has gained a distorting and ubiquitous influence which paradoxically undermines the very responsibility practiced. Within the context of being dependent we can correctly be responsible for our own actions but with the surety of knowledge that that responsibility is contingent on the measure of life giving forces available that we are dependent on. The mythical dichotomy 'independence' connotes with the myth of separateness, division, alienation, and the force of these particular myths is expressed in wars, genocide, criminality. Alarmingly the mythological term has become a residual in our lexicon and is used more widely with acclaim than its true counterpart. To uphold delusional 'independence' as a value to strive for erodes our human heritage by diverting useless energy toward a dubious goal, and consequences that leave us questioning our means of arrival. Sadly it is a loss of being with the paradoxical view that the energy expended will deliver up a personal reality. The cult of independent individuality with its mythical ideology based on personal intimacy is now taken for granted, which then passes into the acceptance of the spurious dichotomy as a tangible reality. This in turn disposes the adherents to discredit the very essence of their being, and in the process dehumanizes many cultures. The presumption of independent individuality leading hopefully to a superior future is in fact counter-productive to the purpose, and leads eventually to corrupt power, and subsequently the invention of immoral policies that continue the negative spiral, which in the end has no ethical base to extend from. The alienation from our substantial being creates inevitable tension anxiety, and the need to somehow relieve that anxiety with any artificial means available. = Responsibility. = Being responsible for any social activity would best be enacted with regard to the effect it will have (directly or indirectly) on the lives of those who are dependent on a beneficial outcome. To recognize with some significance the basic structure of our being in turn significantly increases the measure of our understanding of human relationships. Being dependent is not a mental construct choice - it is a state of being, and there can be no being-ness without at least one other being, there is then the possibility that the principle of true egalitarianism becomes the manifest reality. Being-ness can only be identified and expressed in relationship. [[File:Togetherness - Wiki Conference India - CGC - Mohali 2016-08-06 7666.JPG|center|thumb]] This is why the cult of 'independence' is eventually so socially destructive, as it creates that alienation which attempts to deny each real human need, and leads to a depravation of honesty. Human relationships between children and adults where independence is the accepted norm is severely undermined when the educational process predominantly teaches an unnatural form of living (either intentionally or unintentionally). The educational process is then reduced to the adoption of a fiction, which in turn puts at risk any educational program. The effect of interpreting mythical dichotomies as described here is symptomatically ascribed to the existence of all other principles that govern life. The construction of any ethological debate should not be premised on comparative perspectives, based on human thought, but rather on the issues that we can recognize as being universally compatible, therein lies the common denominator point of extension. The focus of attention on comparative perspectives denies justice to 'what is'. To contemplate the 'right or wrong' of any circumstance is a deviation from the truth. The correctness of any debate (however minute the finding), is the justifiable extension, and the only true trajectory we are morally obliged to travel. Truth is not defined, nor experienced by comparison, but by 'what we are'. An orange is to an orange, what an orange is to an orange. To define correctly there should be careful and disciplined action toward establishing 'true factors' that we constantly use to promote reasonable standards. = Time. = In that moment of time, we have the potential to merge on an equal basis with the reality that exists, and to know what true interaction is. That is the point of 'direct experience'. It is then that we know the truth about ourselves and the beauty of this Universe which also reveals to us the folly of our present conditioning. In that experience, it becomes very clear that all so called cerebral activity has nothing to do with reality. The fundamental repository of our knowledge and relationship with life is our being-ness, which is not located inside a mental box to be analyzed, accepted, or discarded at will, but the very privileged natural gift of being. What some scholars deal with is the appearance of life prescribed by the illusion of comparative perspective which functions on the basis of dichotomous ideology. It is in effect a denial of our humanity to conclude that all things that exist [from our perspective] exist only in the mind, that is, they are purely notional. It compounds the denial of 'what is', and an extraordinary refusal to observe transparent life. It should be noted that there is ample contemporary exposure to the Cartesian doctrine, and in this regard, I would refer you to the works of Professor Gilbert Ryle, notably his publication The Concept of Mind.<ref>{{Cite journal|date=1977|title=Professor Gilbert Ryle|url=http://dx.doi.org/10.1093/mind/lxxxvi.341.1-s|journal=Mind|volume=LXXXVI|issue=341|pages=1–s-1|doi=10.1093/mind/lxxxvi.341.1-s|issn=0026-4423}}</ref> The heuristic principle applies throughout when establishing our connectedness with reality. It is only through our contact with reality that we can discover, and equate with the mutual structure of the principles that govern all existence. Have already noted that it is also a peculiar form of arrogance that presumes that life is only a notional existence beyond the boundaries of the 'mind in a box' assertion. It would be foolish of anyone to assert that ice cream has a cold smooth satisfying texture and taste on a warm summers day unless they had actually experienced it, preferably on more than one occasion. For anyone who has never enjoyed that experience, it would be foolish of them to discount the very numerous accounts of such an experience that is available just because they had not been party to that event. From a logical point of view, given the avowed experiences of ice cream eaters, we could reasonably ascertain the validity of each experience by documenting their separate opinions. Each participant would have 'direct experience' in the consumption of ice cream, which at that point in time has the potential for that participant to experience the full measure of that factual reality. We have the natural capacity to experience coldness, smoothness, which equates with the reality that exists, and the potential for those realities to unify. It is not a question of how to get outside of our minds (mind in a box position); we are constantly outside our so-called minds performing acts of transparency throughout our entire existence. The belief that our constant engagement with reality can never be based on a rational acceptance of 'what is', is at the least, very sad. The Platonic Cave shadows are a metaphor for the (mind in a box) syndrome. [[File:An Illustration of The Allegory of the Cave, from Plato’s Republic.jpg|center|thumb]] The need to reach simple, and obvious conclusions and accept them for the reality they are provides the opportunity to engage the complete reality of the moment. It is indeed going too far beyond the reality of the moment searching for philosophical profoundness which does not exist, that fails to establish the constant principles that always operate. Pure principles are not amorphous shadowy ideals; they are represented in everything that exists. The only way we can equate our inner knowledge of reality is through direct experience of its truth. [[File:JUL Soul Iris.png|center|thumb]] Within that context then, life cannot take on a notional existence but is an existence that is very real, and that we continually share through our innate knowledge. That our so-called minds are defined by comparison - incompleteness - dualism would have extreme difficulty in pursuing the proposition that we are defined by our direct relationship with reality which is expressed in our innate ability to directly interact with 'what is'. The reality of interconnection, and interaction, are not idealistic concepts of a notional nature, but actual and consistent transparent realities. We do not live in a shadowy world that is hidden from our direct experience, but we are constantly engaged in the process of life, and we do not have the right, nor the choice, to deny it. The man whose book is filled with quotations has been said to creep along the shore of authors as if he were afraid to trust himself to the free compass of reasoning. I would rather defend such authors by a different allusion and ask whether honey is the worse for being gathered from many flowers. Anonymous, quoted in Tryon Edwards (1853) The World’s Laconics: Or, The Best Thoughts of the Best Authors. p. 232 == Create your own semantic template. == That will consist of an alphabetical list of Absolutes that are all interdependent and interconnected. Their unifying construction creates a ‘new’ consciousness meaning. That ‘meaning’ is yours specifically. The greatest knowledge you can ever have is your own! [[File:Knowledge belongs to all of us. 2030 Wikimedia.jpg|center|thumb]] That meaning also creates its own moral construction that cannot be misused . The semantic template is available to everyone, and its dissemination is our responsibility. “Consider that the language structure, concepts, and definitions now in use no longer always deliver, accurate, reasonable, and responsible information. Indeed at times, they can be quite ambiguous. The statement ‘mutual agreement’, and its physical manifestation in whatever form, is its own dialectic, and will carry within it all other principles necessary for the activity to proceed. Given the Socratic assertion that if something is true then it cannot lead to false consequences no matter how circular any argument may be. Then extrapolating the statement into extended definitions must only lead to a better understanding of the inherent truths available. This can promote recognition of the underlying essence of all things, which can become more real than our conventional understanding of Reality. There is a contemporary need to find new definitions, new paradigms to explore the concepts that govern our existence. Examples of principles and how to define them without dichotomies, just add conjunctions to create any sentence. Communication. Truth. Standard. Proof. Express. Contribute. Mutual. Direction. Advance. Comfort. Organize. Certain. Immediate. Interest. Improve. Present. Constructive. Gain. Trust. Progress. Source. Knowledge. Basic. Original Reality. Awareness.Freedom. Purpose. Connect. Understand. Support. Peace. Cause. Unity. Ability. Rights. Honest. Discover. Positive. Energy. Balance. Good. Courage. Willing.  Control. Use. Association.  Observe. Reason. Easy. Wealth. Simple. Law. Increase. Order. Flow.Co-operation. Exact. Quality. Accuracy. Strength. Responsible. Operating. Creative. Measure. Recognition. Accept. Constant. Obligation. Include. Dependence. Relationship. Value. Success. Principle. Equality. Stable. Share. Love. Sustenance. Action. Identity. Intelligence. Education. Secure. Facts. Agreement. Information. For. Rules.Clear. Yield. Example: Success = Securing facts through responsible co-operation and using constructive knowledge to develop your success. There are no dichotomies! Any principle is correctly defined by any two other principles. You create a new language. Using conjunctions you can write your own book. Where a circular argument is based on an untruth, then it cannot lead to a truth. The reverse of that is that when the truth is established, it cannot be denied. Establishing ‘mutual agreement’ as a center from which we can reach out for extended knowledge in its ever-evolving radius, is not a limitation, or a stop, it is only a beginning! When any concept is truly established the superficial exemplification ceases to dominate, and we can truly experience the apparent essence of ‘what is’. Paradoxically to resource innate knowledge, we must recognize our profound ignorance of Nature, and Reality. Completeness does not lie in individuality. This is an extreme form of monastic expectation. There can be no individuality (or completeness) unless there is at least one other individual. This is the true foundation of completeness. Whenever we are privileged to experience that instantaneous essence of one other, then we know in that moment that we experience ourselves. It is complete complementation with the knowledge paradoxically that it is an endless process. There are many paradoxes we live within that strain our conventional views of what is ‘more reasonable’. Any true relationship experience is not based on a causality premise, but on an experience that is necessarily complementation. Individuality in terms of completeness is a fundamental circular argument back to one, which in its form of denial excludes any form of reasonable argument to the contrary. It is a non sequitur, which denies the pressure of facts that are in abundance, despite the evidence of their reality. To observe ‘mutual agreement’ is looking at things as they are. True observation of ‘mutual agreement’ in action is observing essence transparency – it is knowing ‘who we are’. That form of recognition is essence duplication. The proposition that we can observe the Truth may well be the highest attainment of Realities properties, for Truth is knowledge. Consider the hypothesis of a human entity (an individual) being born in a black space with no other form of life in that environment. How could there be Agreement? How could there be Intelligence? How could there be Understanding? How could there be Recognition? How could there be Love? How could there be Law? How could there be Reason? [[File:Knowledge-Reid-Highsmith.jpeg|center|thumb]] All of the above principles are the transparent manifestation of Nature and Realities properties that are constantly evolving. They are ethical imperatives, and we have developed the positive properties of language to establish them for our use. We can only be defined through relationship principles for they offer us the best hope to recognize the factors that lead to complementation. There is a fundamental need to grasp simple common-sense essentials. The Here and Now is not a temporary transitional time phase that we move in and out of. It is a constant certainty that is essential to recognize, so that our focus of attention has a foundation. Centrism can imply a fixation, which also implies vulnerability, which can be perfectly true if it does not lead to extension. To understand who we are, it is essential that we recognize and become aware of the very principles that we operate from. They encapsulate all the measure of any human societies ethics, morals, and laws, which is a continuous evolutionary educational process within which the realization of its total essence is always available. To use the doctrine that reason is a reliable tool to discover Truth – therefore ‘mutual agreement’ in the context ‘correct information’ translates to the Truth to reason! Evolution is a constant dynamic process. The human phenomena of ‘who we are’ is only understood in our union with each other, and ‘what is’. The paradox again is that there never is any separation. Separation is a mythical non-existent. The principles that are our necessities have continuous expansion properties that as humans we are privileged to assist their propagation. [[File:Making Scholarly Articles Findable- Towards Ensuring F of FAIR Data Principles, John Samuel, WikidatCon 2021.pdf|center|thumb]] The human constellation in its evolutionary march must use these fundamental principles to ensure continuity. To maintain coherence and consistency our source is centered in the principles and factors that we have interpreted from our association with Nature, and Reality. Whatever we write that is of any consequence, or at any other time, is written with the hope that stronger interpreters than us overtake what we present. To ‘see’ Reality as we have seen, and be intoxicated by it, as we have been, will ensure its progression.” == Discovery == The consciousness whole is the sum of all its parts and experiences. As we are all on an evolutionary path, our life and knowledge hopefully develop in the right direction. The exploration took us through a plethora of data and opinions about reality from authorities on science, religion, philosophy and metaphysics. Nowhere could we find a definitive conclusive argument, or agreement, that met our needs. For us, the question came down to “Is there anything at all that provides some form of construction, and certainty?” Something that has its own inherent ethical standards. The alternative proposition to that is a nihilistic “nothingness”. A pathological proposition that makes no sense. Heraclitus’s “unity of opposites” seemed the most promising. Our understanding of that now made dichotomies a semantic illusion. If achieved in a mindful way it is the act of uniting them, providing a conscious correct experience of ‘what is’. We live our lives with secondary knowledge that everything that is – is always interconnected and interdependent. Yet our illusory experience belies that form of knowledge. It is here that we understood Heraclitus and his “unity of opposites”. Mentally uniting opposites replaces the existing illusion of their existence – there are no dichotomies! [[File:Heraclitus, Johannes Moreelse.jpg|center|thumb]] Once the illusion is gone a new solution manifests that is peculiar to the mindfulness operator, and belongs to a higher form of consciousness. Heraclitus was known as “The Weeping Philosopher”. He wept for the needless unconsciousness of mankind when the ‘unity of opposites” was always available. He was also known as Heraclitus The Obscure. A title we suspect that came about because the successful conclusion to uniting opposites and replacing the illusion, opened a door to a new dimension. Semantic description at this time may not have been available. This brings us to the ancient Yin and Yang symbol of the ‘unity of opposites’. As separate entities in Chinese philosophy, they are complementary, and in fusion they represent the whole. So as dichotomies they don’t exist. [[File:Ying and Yang Street Art - panoramio.jpg|center|thumb]] The whole is the elemental answer to any fusion of opposites, whatever that may be. Symbolize a line as being infinite in the sense that any line can be categorised as being infinite. Apply an infinite number of points in any line. Intersect any line through any point by another line then we have a specific identifiable point at the intersection, which at that point in time has an infinite quality, yet constant and complete. Any such point has Matter, Energy, Space, and Time, the epitome of the microcosm. We may locate a Reality point that establishes the Truth. Conventional mechanical ‘thought processes’ deal with dichotomies that are based on a comparative perspective ideology, and consequently, skew any real experience of that which is real. We must use correct ‘measuring sticks’ to secure proper standards, but from the point of view that there is a belief in dichotomies, it will always be a compelling argument that aspects of reality can be contradictory. The element of denial within human historical memory accumulates to establishing dichotomies as being real. We are defined not by how ‘different’ we are, but by our commonality of existence. When we locate that Reality point we will then know that the definition in itself has a whole, and complete explanation of ‘reason” in all possible senses. All the reality we can deal with is here, and now. There is no possibility that ‘infinite regress’ (an imagined reality) is any part of our immediate experience. Infinite regress through thought processes, deals with questionable imponderables. It is a descending spiral, which further removes one from reality, which only produces illusion, and correct meanings are always deferred. It is making a holy virtue out of complexity. The epitome of completeness is the active realization of the operational principle. [[File:Brick house 09.jpg|center|thumb]] Conclusion: A brick – a house. Each complete in themselves. A house is not composed of one single brick, but each brick in its composition is complete, and whole in itself in that it has matter, energy, space, and time. In that context, it is a microscopic whole which has implicit within it the macrocosmic whole, a house. We cannot define that which isn’t. We constantly use negative dichotomous terms in language, which are in essence factually indefinable, and therefore non-existent, but they are used as though we can support a view as to their existence. At this time we constantly use mythical concepts as though they had real substance. That erroneous belief in turn diminishes that which is real and compounds the problem of recognition of Reality. The flat earth society no doubt had to be persuaded of the mythical nature of their beliefs. This dictates that we must research ‘that which is” to achieve an understanding that supports that reality. Separation is the mythical measure we use in an attempt to justify the real identity of either ‘relationship’ or ‘completion’, but it has no substance in fact.That we are connected, that we are related, that we are communicating, that we agree that ‘mutual agreement’ exists, all of these factors fall into the category of ‘that which is. There is nothing that is real that is not immediately available to us, there is ‘mutual agreement’. Attempting to view true relationships as having a necessary separation link, or dichotomy is a clear misunderstanding of the nature and completeness of all that we are related to. == Connectedness. == A relationship is defined as we are by the measure of contact (especially homo sapiens) that is apparent. It would be true to say that I have a measure of relationship with everyone who reads this material. No matter how tenuous the link we have a measure of relationship with all life – we are related! Depending on the strength of that relationship defines ‘who we are’. ‘Who we are’ is not defined by any spurious separation from life, quite simply because we cannot be separated from it, we are engaged in it at every moment in time. Any attempt to establish ‘separation’ as a reality is an attempt to deny ‘who we are’, and another exercise in futility! Again artificial interioralisation of concepts or principles leads only to a denial of their external reality. We are all connected by the very simple fact that we all exist on this planet. It is a very simple axiom that all life on this planet is supported by the conditional properties this planet provides. It is also a very simple, and more reasonable axiom to conclude that no matter how tenuous the link that all life in this regard has very concrete and definitive forms of relationship. We all must breathe, we all must eat, and we all must drink, and if you need any further certainty of ‘completion’ relationships, we certainly, all must die! [[File:Universal Theories.jpg|center|thumb]] To set in qualifications from the premise that there is a ‘separate mind’ (a kind of Platonic cave) to get outside of. This premise precludes either in part or in whole the evidence and experience of Nature, and Reality, within which our beingness is located. It would all be beyond our grasp if indeed our conventional concepts of consciousness was adhered to, which in effect attempts to deny us that direct ‘relationship’ to ‘what is’, and the completeness of that experience. Knowing or being and solipsistic theories advise that we cannot truly know each others experience, nor can we penetrate others experience, but in the democratic recognition that we know and have our being in relationship, and the mutual, and natural convergence of everything there is. Homo sapiens (race, color, or creed aside) necessarily conduct themselves in ways that extend recognition, and understanding at every level, without the constant need of ‘completion’ recognition that is inherent in all our interrelated actions. The flat earth society eventually moved on to a realization that their visionary scope was shrunken, and severely limited. They were deprived of a planetary (never mind a universal) relationship that one can only imagine severely curtailed the very expansion of consciousness necessary for humanity to progress. We have evolved some little way because of our understanding of the natural relationship. The centered in the mind condition - which connotes with the separation ideology - screams to be released from that mythology, and engage itself directly in real relationship with everything that is. = Relationship. = Separation is the mythical tool we attempt to use to maintain a false continuity of an imaginary individuality that does not exist. The taking for granted conditional mythologies (the flat earth society) engage the victims in what can best be described as serious problems in recognizing the very limitations that restrict their development. We must learn to view wholes, which equates with viewing ‘completeness’. The whole is greater than the sum of its parts, but the parts are not necessarily separate conceptual parts. We can ‘see’ the whole when we are able to identify the factors that constitute their existence as a complete reality. That which is complete in Law = Agreements that produce secure and dependent outcomes. We know in essence the concept of ‘completeness’, and we demonstrate the evidence in myriad ways. Each act is a microcosm of the whole – view from the other end of the telescope! In the traffic analogy the driver, and all other drivers, conform to the law by driving off when the light turns green. There is an implicit agreement about the value of traffic laws, and traffic lights that control the flow of traffic. At that moment there is a complete relationship understanding of those values. The ‘complete’ or ‘wholesome’ activity of motorists waiting at traffic lights for the green signal to go, and they then move off, validates all the factors implicit within the properties of ‘mutual agreement’. [[File:ParkStreetTraffic2.jpg|center|thumb]] Whatever we communicate for the benefit of future generations should not be based on mythical assumptions, but should be based on necessary factors. It is ordinary life that portrays all the dignity, honor, and the complementary wonder of the human species. We are here – we are present – we are communicating. We have an obligation (which we necessarily fulfill) to make transparent the basic principles that govern our existence. That, which is factual, provides us with a correct motive for behavior, and we do a disservice to Reality when we attempt to deny it. We exist and live in a world where acts of ‘completeness’ expressed in one form as ‘mutual agreement’, are continually enacted. It is the form of expression, and continuance of processes that we constantly use to arrive at reasonable solutions, and we employ factors that are necessary to provide us with a clear, and unambiguous understanding. They motivate reasonable behavior toward activity that we can accept as being a logical process. == Natural Experience: == No form of life can exist in and of itself, it is brought into existence through a relationship with its own environment, or its species. The obvious egotistical monistic nature of oneness (if there was such an entity) could not leave any room for the realization of anything that might disturb its comfort zone. There is no real knowledge where any concentration is on the “I am” syndrome. "No man is an island, entire of itself; Every man is a piece of the continent, a part of the main." - John Donne, 'No Man Is An Island, Meditation XVII - Devotion Upon Emergent Occasion. “We are’ is an inclusive affirmation that deals with “what is’. There is no constructive dialogue, no real understanding, without a relationship. Based on personal experience, we are a distinct, and unique species born of Nature and Reality that has combined to provide us with the innate ability to recognize the very properties that created us, and utilize them through an evolutionary process toward ever-increasing transparency. That transparency can only become available through a matching process between innate knowledge, and the reality we share, a reality that is our heritage. We like all other forms of life are the product of Nature, and subject to its laws, and principles. Necessity created a language that evaluated Reality, and provided us with guidelines to emulate its constant properties. The measure, and quality of knowledge is dependent on the realization of ‘what is’. The crux of correct knowledge is to know the base that we function from. The principles that are implicit within Nature, and Reality we have now translated through the evolved language systems with symbols and definitions that we now use to share the experience. When principles are fundamentally true and recognized for what they are, misguided belief systems will evaporate. To ask what is the source of the principles we present is ipso facto to ask what is the source of Nature, and Reality, and we repeat, that is an exercise in futility, but that does not mean that we cannot recognize that which is natural to us, and express it, as best we can through language. The experience of ‘who we are’ is the Ultimate transparency that transcends all doubt, or denial. We can know with an understanding that is pure and indisputable, that is the motivational drive for evolutionary continuance. To understand who we are we must address Nature, and Reality, and ask ‘what is’ Here, and Now, with an understanding of the dichotomies that exist in language. The ‘Eureka’ moments, epiphanies, enlightenments, etc, are all evolutionary evidence of who we are, and when we can translate them into principles, and concepts, then the assertions of an Archimedes (and many others) are recognized, and properly used. Through Nature, and Reality we can establish what Truth is! Is it true to say that most people conform to the rules of the road? It would be more reasonable to assert that the answer is yes! Consequently, we can say that we have ‘Mutual agreement’, and ‘Co-operative Understanding’ as observable realities. There is no conceptual source through Time, or history where there is an end. There is only ‘beginning’. Here and Now is always the ‘beginning’, and a more reasonable platform to explore than any exploration into the past concerning our true identity. Contemporary terms like Absolute, Complete, End, we use to match our conditional understanding of ‘what is’. When we use contemporary conditional language to address concepts like Truth, Knowledge, Understanding, they are limited by the measure of our progression, but we use them all as stepping-stones. Language is a constantly evolving process. When we agree that there is ‘some certainty’ and ‘limited knowledge’ you have agreed to the concepts of certainty and knowledge as factors that are part of our natural reality. All of us function within the framework of certainty, and knowledge, to some degree. Given that we agree to their existence, these are the factors that can lead us to the experience of ‘who we are’. They are a part of us that can lead us to recognition of ‘what is’, and make a transparent reality of the very things we do on a daily basis. We do not need absolute and certain knowledge to perform everyday tasks, but those performances are structured contemporarily because of our advanced understanding of the things we do, based on our own innate reality. To honestly perceive the consistency of ‘what is’ (to be interdependent, and mutually connected) in interaction, can and does promote the visible reality of ‘who we are’. That visible evidence translates into the knowledge of our complete presence. We know with certainty that our beingness is of pure essence, and from that experience, we are obliged to formulate, as best we can, the structures that are responsible for making that transparent (witness the explosion of human progression, without the necessity in Time of experiencing fully ‘who we are’). To take a more reasonable stand please observe the multiplicity of human action where interdependence, and connection naturally proliferate. To realize that we are ‘interdependent, and mutually connected’ is the realization of a consistent fundamental truth – ‘what is’. Through identifying ‘what is’ as an internal reality we can make transparent the factors that are our natural construct. It is only through sharing this reality that we recognize it!! These factors are not based on any ideology, belief systems, or opinions; they are composed of the Reality that is available to us all. We are unique in that we have the opportunity to be able to use their value in the manner that is implicit within their structure. That use is evidence of our understanding of Reality. What could have happened without the assertion that traffic lights are a safe way to control crossroads, or the assertion that the rules of the road are beneficial for our safety? Any true experience, epiphany, enlightenment, etc, of ‘who we are’ provides fundamental, and indisputable knowledge of that Reality. Applying the recognized principles provided by Nature, and Reality consistently advances the evolutionary process, hence we have Science, Philosophy, Religion, Education, Art, and Law. When there is a Pure Realization of who ‘we are’ through relationship recognition, it is unquestionably the recognition of the encapsulated, and innate principles we all share, and there is no place for the dissolving of another Real identity through that recognition. Indeed it is a privileged insight into the epitome of purity. Nature and Reality can give us direction and guidance to our human existence, and we repeat, it is an exercise in futility to seek any cause to their beginnings. We have proffered the concept of two as a basis from which human reality can be ‘experienced’. To recognize through, and equate with the true substance, and essence of one other is to automatically experience the totality of ‘who we are’ in full measure. This does not mean that the terms ‘totality’ and ‘full measure’ convey an ‘end’ to ‘what is’. We have consistently offered recognizable facts (not assertions) that are part of our natural human activity, and give correct direction and meaning to our basic essence. We do function within the structure of ‘mutual agreement’, and we do communicate and ‘make known’ – basic obligations. These are evident simple examples of innate knowledge, and our understanding of ‘what is’ made transparent. To repeat we could not recognize anything without innate knowledge. All knowledge is a continual matching process 1 + 1 = 2. toward the realization of ‘who we are’ the development of language structures that correctly establish basic reality as it is, provides continuous knowledge that makes transparent the very nature of that reality. It is vitally important to recognize that we have active communal agreements concerning the existence of basic principles and concepts that form the very foundation of our lives. Constant change and movement in Space-Time - Energy - Matter are applicable absolutes to be recognised, which equates to evolution. There never will be a static property involved in the evolution advance. Evolution encompasses its own absolute properties to provide cognitive connection confirmation. The evolution of conceptual language exists to provide natural equality and to promote cognition between language states. No matter the languages spoken the concepts of air - food- water is the same and can provide a gateway to explore the future dynamics of human relationships. === Knowing. === Knowing what all the truth is is not some miraculous state of perception. Nor is it a high academic achievement of amassed information. It is simply an objective common-sense view of ‘what is” and in reality what must be. It is what must be for life to function within the principles that exist that are its natural foundation. We are always of necessity the living expression of a reality that must be experienced in the whole. Our recognition of the same principles operating universally is also our recognition of who we are. That proliferate ‘sameness’ is an evident easily recognizable identity. When a child is afraid of an imaginary monster in the dark, we generally do not accuse them of being absurd, or that they are lying. Appeasement comes with an explanation of reality at that level which is truthful. An explanation, which the child can grow up with, and find comfort. It is simply introducing a child to a level of truth that is more real to them. In every instance, the only reality that ever exists is truth. However, distorted it may be expressed. One of the major distortions as the result of thought processes is to consider that we can manufacture something other than the reality that exists. The ‘fact’ that you recognize contradictory or absurd statements is that behind them there is a measure of truth. The habitual liar lives in a world in which he or she imagines that truth is something they can manufacture. Where human ‘thought’ constructs its reality in terms of dichotomies it can never deal with the truth because it continually makes those comparative perception judgments. Those judgments are always in question because again they cannot deal with reality as it is. No matter how absurd or contradictory any statement is, that is the measure of truth expressed. Ergo whatever it is that is expressed, or made manifest, is the truth to some degree. Ergo everything that is, is the truth. It is our responsibility to recognize it for what it is. It may be appropriate to review previous observations on dichotomies and gradient scale. Consistently we have contended that there are no dichotomies, which then properly puts each principle into the category of an absolute. To identify ‘truth’ as an absolute in that category then everything that is must have a measure of truth. It is a very simple and sensible approach to establish ‘what is’. It is the means of identifying a reality that must have truth as a base – whatever it is, and however nonsensical it may appear. All principles have an elementary gradient scale that we must use to identify knowledge that is honest. That gradation scale knows no dichotomies. Dichotomies are always the imaginary properties of pseudo subjective reasoning. Necessary factors establish that gradient scale where only objective realities exist to furnish a healthy subjective reason with truth, and so we learn to apply the conjunction to address reality for what it is. Truth comes in an abundant variety of ways in its commonality – and therein lays its overall ‘complete’ power, despite any denial to the contrary. Embedded knowledge as we see it is neither experience nor knowledge without principle content. A person may be well educated in all aspects of the geography of a beautiful South Sea island, but have no practical experience at all of its beauty. Being clever about a subject does not necessarily equate to an understanding of the subject. Nor should it lend itself to posing as adjudicators on a proposition preset we imagine by the same adjudicators, or essentially the same school of embedded thought processes. Long before human evolution, the principle of leverage has always existed in all Nature (as have all other principles, wherever there is space – time – energy – matter). Our adaptation to the existence of principles has added to the sum total of ‘knowledge’ as we know it, including the concept of knowledge itself. That form of ‘knowledge’, and our ‘knowing’ is natural and not any personal or esoteric acquisition. Just as a fish knows what its natural habitat is, or a bird to fly in the air, the human species uses all available principles it recognizes to add to its knowledge (already said). Any valid theory of knowledge must have as its base constructive definitive principles to support it, and it is evident that our accumulated common knowledge equates to our common experience. No matter how erudite or convoluted any argument may be, if in the end it is reduced to inane observations that have no factual basis in principle, then it is time to abandon them. Do try to consider the sort of ‘mind’ processes that offers us up a world that knows nothing but separation. How can we possibly evaluate what ‘wholesomeness’ really is? How can we possibly evaluate ‘who we are’? How can there be any theory of knowledge without addressing Nature or our innate and biological relationship with it? Any attempt to debate ‘who we are’ and the completeness of that concept must have some sense of reality on the real meaning of ‘completeness’, and some understanding of the principles that are the nucleus of human society. To wrap any argument around a non-existent concept that can never be realized is apropos to attaching oneself to a system of belief in things that do not exist. One can make ponderous and convoluted statements about those beliefs but in reality, they are morally and ethically misguided. The ability to correlate correct definitions to the reality of life offers up that direct link to the truths that are common to us all. It corrects the presumptive notion that there can be ‘different’ perspectives on the same reality. There can be ‘differences’ but there can not be ‘different’ measuring sticks for the same reality. No matter the multiplicity of perspectives, they can never alter the core principle of ‘what is’. Historically evolutionary progress can best be measured by the adoption of recognized principles. Reality at whatever level we find it can only be understood by addressing ‘what is’. Nothing can be understood by attempting to relegate it to a non-entity through questionable theories of ‘knowledge’, which in essence negate the very content of knowledge itself. The perpetuation of any theory of knowledge, which cannot recognize the principles that are its foundation, can only be a shadow of its own reality. Construct the ‘necessary factors’ around the skeleton and a body will take form. If any particular theory of knowledge cannot identify simple truths, how can we possibly question how anyone ‘knows’? A dichotomy is the human attempt to deny the existence of a whole reality of a principle. We have the principle of leverage and its necessary gradient scale. Mutual agreements of a consistent reality, at a communal level, are a passive form of the Eureka moment, which recognizes fundamental principles that relates to truth. In every social structure, there are varying degrees of recognition, which determine social use. The mosaics of differences, which make up the rich pattern of life, are a testament to human creativity. Principles offer up a form of predictability in which our brain forms knowledge through the process of interaction. The experience accumulated through each moment, forms exponentially in use, or becomes transparent immediately in a Eureka moment, in which we know. Real knowledge is through the constant interaction with natural principles, much more than the transference of divided embedded information. The problem we face is that ‘knowing’ or ‘how we know” is never a personal possession. Any theory of knowledge no matter how in that respect, is true interaction. Peeling a potato and ‘knowing’ it, is rewarding enough! All principles are the repository of pure erudite knowledge. We recognize Universal principles in play at all times in Nature and its by-product – human societies. The correlation between determined interdependence to objective reality requires our intellect to ‘honestly observe ‘what is’, and assimilate that subjectively. Then the equality of the external and the internal becomes a reality and we ‘know’. Knowledge is the process of natural action, reaction, and interaction. It is nonsensical to ask how do we ‘know’.Every moment in time is complete because it must contain all the principles that form its nucleus. It can only be like that to facilitate the immediate experience of Eternity, or the wholesomeness of any of its principles. On the gradient scale of experience, we all exist somewhere on that scale. It is called life. === Relativism: === Relativism can be consistent with interconnectedness and a gradient scale of knowledge. What it cannot do is confuse the relationship that correct gradient scales of principles have in reality. Hot and cold would be on a temperature scale – no dichotomy! Leverage could only be measured by its own scale (say a child’s sea-saw to a high-rise industrial crane) – no dichotomy! The human being is a human being whether it is a child or an adult – no dichotomy! The domestic cat is the same animal species as a wild lion – no dichotomy! The domestic dog is the same animal species as a wolf – no dichotomy! How can we manage to classify these as dichotomies? Gradient scales are the natural human mechanisms used to recognize constant principles. They ensure the human perspective is aligned correctly to identify ‘what is’. The distorted human perspective is the result of human thought processes unable to establish constants that must exist in each moment of time. The problem with embedded information is that it becomes stultified and it can stifle healthy reaction. The injection of recognizable principles invigorates and brings new life that offers countless avenues for human energy to be released. More importantly, those energies are used to enhance the evolutionary process. We are collectively gifted with the potential to elevate life itself. We can correctly use such information by transforming its content so that its inherent truth is made recognizable. It would be impossible for life to function if it was composed of ‘different’ opposite realities. All theories of knowledge are in essence interconnected and can only contain validity when the principles that are the coalescent mechanisms are recognized. The unification of the truth that must exist in any theory needs to be harvested and used to offer up a body of ‘knowledge’ that has commonality of meaning. The identification of principles, truth, knowledge, and their subsequent establishment can only be achieved through direct interaction with Nature and life. Gifted with life we have an obligation to demonstrate its capacity to use every resource to sustain and nourish its own environment. We all know through the constant natural process of action – re-action – interaction. Depending on the quality of that process, knowledge will take its appropriate place on the gradient scale. That we ‘know’ is natural. It is not some extraordinary esoteric attainment, posited by a body of theories that, by their very nature, look for difficulties where none exists. Universal belief systems based on mythologies can have an entrenched view of good principles being established because of their beliefs. Indeed the perpetuation of the beliefs throughout history offers a dynamic that is counterproductive to the ‘realization’ of principles that are necessarily true. Principles used in this approximate way, paradoxically hold no real meaning, and in fact, impose unhealthy dysfunctionality. When there is a critical change toward establishing correct principles, it is axiomatic that the diffusion of mythologies becomes an automatic process. True interaction lies in the knowledge that correct action is its own reward Any other interpretation is less than tangible. The accuracy in interpreting basic principles, and the alternate knowledge implicit in the interpretation, will always establish the primary principle sought. Archimedes et al. Truth can be found in the oddest places. (Archimedes bath image here please) Archimedes cognition on how to weigh metals in water through displacement. === Truth. === Language is the construct of human action and the word “truth” seems to hold pride of place by the power of its usage and the meanings it evokes. It is preferable if we could turn our attention to the unity of principles (including truth) that are the construct of every language we use. By uniting the principal terms we can elevate the meanings we desire. Reasonable constructs and the correct duality of established principles always lead toward meaning. It is the only form of meaning that leads to its own extension eg. how to weigh metals – how else could it be? All principles have reciprocal value one to the other. No foundation principle can stand alone. They can only exist in a union, one with the other, the source of reciprocity. All absolutes are universal. There is no hierarchy beyond the meanings they evoke in their joint construction. The binary connotations, however, one may express them, provide a constant reality beyond conventional consciousness. That experience is the immediate reward through disciplined application of their use. That discipline takes the form in all human action (such as the bathing scene above) disposed toward the correct functionality of basic principles. The daily connections we make always include the distinct possibility of their recognition, when we make those connections in a mindful state. From any common sense, or ‘more reasonable’ position, it would be more productive to view reality as possessing at every level the same innate values or principles consistent with our ability to measure, or recognize them. To view reality as having ‘different’, or antagonistic properties, is simply a misguided view of ‘what is’. That form of perspective is counter productive when it attempts to establish mythical dichotomies as realities in their own right. When the realities of principles are made transparent, we can then ‘more reasonably’ make use of them to further their basic existence. Here we use reason to exemplify their necessary function, and once established it becomes (if necessary), ‘more reasonable’ to locate them in all things. ‘More reasonable’ seems to have the particular philosophical motivation, not toward simple, sensible, and reasonable evidence, but more likely toward that ‘immaterialism’ ideology, and continually seeking for an elusive protracted answer is hardly ‘more reasonable'. Since we are apparently confined to a human perspective, we must settle with the latter position: the apparent state of representation of the world. The de-materializing of any object through the practice of ‘perceptual illusion’ is an attempt to deny the reality that exists. Where perceptual illusions are concerned, innate direct communion with that which is, suspends the effect of such illusions. All the properties in a chair are recognized as the reality that exists. That is materialism. A chair does have the principles of form, design, structure, colour, substance, etc. However it is analysed – it is a quantifiable reality. === Truth and Reason. === The reason could easily be defined, and validated, as the correct application of common sense. More expressions of common sense can only endorse the completeness of any concept. A true experience of reality does not require endless explanations as to its ‘wholeness’. It just is. Truth is in reality a network of implicit principles in which it is the predominant energy in each of them. They are identifiable by their interdependent nature (see network below) not the least of which is common sense. Dictionary given definitions of ‘truth’ place it in a very common sense acceptable category. One of which is ‘accuracy of representation’. Note how the two definitions in this paragraph coalesce. The human drive toward recognizing and understanding the place of principles (constants) correlates to the energy we expend on questioning ‘who we are". The constant principles of action, reaction, interaction, are the automated natural impulse toward ‘establishing’ a human reality, and human identity. The process of evolving within that process has an egalitarian dynamic that powers it. In essence, it is a natural gift that we must accept. Each life and its identity contains all its personal experiences which can never be known to anyone else. In a sense, we can never “know” another person. Their life is sacrosanct. We can know a lot about them, and there it ends. Truth is at the top of the gradient scale that measures the veracity of all things that are complete and related and paradoxically all reality is the truth. It gets back squarely to ‘who we are” and where we exist on that scale. To view gradient scales as having no truth to their structures is denying truth itself. For the entire interconnected, interdependent network of principles, each of them has a gradient scale whereby each measure expresses truth in its own manner. All forms of leverage, from the minuscule to the lever that will move the world, are in of themselves, true and exact at that point. It is the only way we can recognize their existence, and use that complete truth at that time, to move up the scale. Time is the relative measuring stick that determines the amount of knowledge we can absorb. Consider the advanced extensions to the Archimedes principle of leverage throughout time. Network scale example. Truth Knowledge Common sense Responsibility Reason Understanding Intelligence All interdependent, and interconnected with all other principles and absolutes. No ‘thought processes’ or ‘mind’ constructs can create reality. All we can ever do to gain knowledge is ‘act’ react’ and ‘interact’ within the confines of our immediate reality. The quality of that action is determined by the nature of available information. When there is freedom from embedded thought processes, there is a natural human ability to relate to the existence of truth as it is expressed in reality, and our brain records it accordingly. Thus, the principles of civilized societies evolve. Where there are predominant belief systems, the implicit energy will naturally direct itself toward human standards that blend all ethics together. That implicit energy will find its true home in the principles it seeks. The connected strength of those principles offers sanctity of experience that demands no sacrifice. Everything that is, must of necessity, have a true comparative value (not a distorted dichotomy value) for honest recognition to be realized. which is to ‘know’. All things are relative but only within their own true scale. It is the process by which we can identify reality, as it is. Principles cannot operate on any scale practicing negative discrimination. Thus a healthy individual can be at the top of the scale and someone with various health issues can be near the bottom of the scale. But that is how Healthy they are. There can be no relativity when ‘mind’ or ‘thought processes’ believe in mythical dichotomies. When human perception is distorted by such beliefs, they create a false reality and deny access to the true state. Where there is a network of connected basic standards that are universal, then it is possible to use them and be nourished accordingly. The scale of natural human progression provides recognizable evidence that we are constantly developing. Reality is the direct and conclusive evidence of possibilities realized. Therefore, the reality is always the source of all possibilities where truth exists. When the truth is used as the universal measure of ‘what is" there can be no discord as to its accuracy. It can only measure the principles that are implicit in everything there is, its natural milieu! Truth can only deal with ‘something’, it cannot measure an imaginative negative counterpart. Truth is the constant implicit property in every universal scale of principles. Thus, reality becomes transparent. Have writ large on the value of distorted comparative perception judgments. The standard of correct knowledge always carries with it, its own appraisal. Where principles are concerned there is an obvious scale of identification (e.g., leverage and the numerous references) that is all-inclusive and provides us with evidence of its existence. We could say with some truth, that the industrial crane has more leverage than a child’s see-saw, but we cannot deny the truth at the lower level or the reason applied. Where principles are concerned, truth is not a possibility, it is a constant reality (e.g., leverage). When mythological dichotomies are recognized and established for what they are, the process of ‘ironing’ them out and experiencing their constant reality will translate into the reality, which they are, and used accordingly. To evoke that new sense of reality, the mechanisms of ‘selective immaterialism’ need to be dismantled. Where human experience is presented with something it does not understand and is unable to appreciate the principles involved, the reaction can invoke a sense of fear. That condition can be a primary breeding ground to establish a language of dichotomies and put a selective name to something it does not deserve. The diffusion of a false singular dichotomy into the natural healthy state of the common good puts responsibility into its proper place. Within the process of diffusion, there is the natural and equal absorption of our true reality. The transition between separation and inclusion will be a seamless process because it is our natural state. Objectivity and Commonsense: Explore the plethora of principles - truths - constants that are the mark and phenomena of Homo sapiens. We cannot have any doubt about our existence in this present reality. The truth of reality is and can be experienced wholly and completely by anyone at any point in time. All human progress is the result of such experiences, manifest in principles throughout time and their subsequent ‘use’ evolves exponentially. The overwhelming evidence is our reality, now. A simple analogy of objectivity and commonsense. Somewhere in our early development, someone put the ingredients of a loaf of bread together, somehow baked it, and hey presto, the first experience of a loaf of bread. It is now a form of sustenance, which feeds billions of people. We no longer need to experience that ‘truth’ that ‘knowledge’. It is unnecessary because it is subjective assimilation and the act of external and internal activity. Because ‘our’ brain functions in a manner that can identify the natural elements it exists in –space, time, energy, matter, we learn to ‘know’ and recognize ‘completely’ a child’s see-saw. Knowing is a natural evolutionary function. The quality of ‘knowing’ has its own natural determinants which of necessity contain the measure of principles required for universal recognition. An Archimedes insight (or anyone else’s) could not become a universal reality unless those determinants were in play. I know the very same way we all know – by experience via action, reaction, and interaction. Truth persists and what Archimedes experienced was true and complete. Any experience of any truth, principle, or constant can be as ‘complete’ within a grain of sand, or knowledge of a pyramid. All experience of that nature is an experience in ‘time’, and when it is the truth, we use it accordingly. There is no mechanical translation, or opinion of ‘necessary factors’ as they are constant universals. That, which is ‘complete’, is transparent universal knowledge e.g., the principle of leverage. The precision of terms must include ‘necessary factors’. Necessary factors translate into a common universal language so there is no loss of meaning. All truth – principles-constants – absolutes, that stands the test of time we use accordingly. Thus, human societies evolve, and we evolve without the necessity of having to re-experience any of the principles we recognize and establish. That simply would not be a ‘natural’ commonsense proposition and an entire waste of unnecessary energy. The truth of any principle at any point in time, and at that point in time, can be experienced completely by anyone. Whatever it may be if the principle is established – from then on, it will evolve. There are some misguided notions that ‘truth’ ‘knowledge’ and ‘experience’ have some secret value that is unattainable by ordinary experience. It is a ‘natural’ evolutionary reality that what we ‘know’ becomes useful. We have a mutual responsibility to recognize, use and honor the principles, which are the common property we share. = Human consciousness. = [[File:Lane past Coombshead Farm - geograph.org.uk - 589598.jpg|center|thumb|369x369px|"The roads half travelled when you know the way"]] [[File:Sombrero Galaxy in infrared light (Hubble Space Telescope and Spitzer Space Telescope).jpg|center|300x300px]] === The Universe === ''<big>We do not own Space</big>'' ''<big>We do not own Time</big>'' ''<big>We do not own Energy</big>'' ''<big>We do not own Matter</big>'' The human capacity to understand the question of sovereignty or ownership of Space - Time - Energy - or Matter can only be accepted when any basis of dispute includes two dispositions - human and spiritual indigenous ties throughtout history. With the evolutionary appearance of indigenous peoples throughout this planet, their way of life should make it paramount that their existence be recognised as a natural law that has providence! Their culture and way of life has its own identity in which Space - Time - Energy - Matter is expressed as they experience it. That proof also lies in the existence of caves thousands of years old, and the existence their art and culture. So the constructs defined below are a new approach to understanding the concept of ‘knowledge’ and its proper place in an evolutionary expanding universe. Knowledge acquisition requires appropriate recognition through action, reaction, interaction, in which proper perception and comsciousnessvalues are applied. That form of construction requires the dismantling of previously embedded information. This requires a new direction to formulate a sound basis from which to extend. Construction of an analytical methodology to establish a form of ‘knowledge’ that is best suited to distinguish in a contemporary reality. A reality that adequately conforms to common notions of that which is true, and can only exist without any false relation to that which does not exist. All science needs the certainty that established absolutes provide. A pragmatic construction of real knowledge would propose that all reality expresses a form of evidence or proof and that the observer and the observed contain innate properties necessary to establish a foundational agreement. That form of agreement would necessarily function on the basis that everything that is – is truth. Any other interpretation would be a disconnect from reality, and the interdependent correspondence that must exist for true recognition of any absolute. Controversy will always emerge when the discourse on bifurcation and the introduction of non-existent dichotomies are used as arguments to be explored. Pragmatism would say that the human species would need basic properties to formulate any form of reliable epistemological analysis to explain and simplify the reality that forms their existence. That reality could well be recognized initially as the absolutes of space-time - energy-matter, through innate perceptual data that corresponds to an outward structure that is constantly evolving. That which is absolute is the determinative factor in establishing the existence of truth. When a chair, is a chair, is a chair, its recognition is established when we ‘commonly’ apply subjective and objective measures to that which is truly external to that which is truly internal. That which does exist becomes ‘common’ knowledge, and accepted as being true. Thus, the distinction between a priori knowledge and a posteriori knowledge no longer exists. Where there is consensus, everything is. The formulation of absolute criteria that offers ‘simplicity’ as a tool to measure all and everything, dispenses with the confusion of ‘difficulties’ historical philosophies engage in. It also offers an observable synthesis that clarifies the confusion. Within the combined properties of those absolutes then everything potentially would be. Everything that is by that definition is original, ipso facto, everything that has no false relationship, and therefore true (no dichotomies). Given contemporary human development, it would be ‘common’ pragmatism to accept the proposition that we exist within the absolutes of space-time – energy-matter. Within those absolutes and their innate properties, it would also be pragmatic to assert that ‘everything then is’ (whatever everything may be). Pragmatism would also dictate that ‘everything that is, is its own form of truth, and must contain available constructs of meaning. Therefore knowledge and understanding would be obtainable to that common experience, and at whatever level that experience is activated it is in interdependent unison with the source. The continuing establishment of basic absolute principles (their generalities and their specifics) forms our reason. 1+1= 2 is a ‘simple’ but true universal constant generality. That form of generality is used because we recognize the specific principles of a balanced equation that adds up and makes sense. Simple generalities with their inclusive specifics form the foundation of human reason and its constant evolution. Simplicity is the bane of a ‘mind’ that must have difficulties. = Availability. = The unifying feature that makes ‘knowledge’ available to us all, are the innate universal principles in all things. Archimedes established the principle of leverage. To paraphrase - ‘give me a lever, and I will move the world’. Once the law is established it can then be put to good use. The principle of leverage is manifest in countless ways, and put to good use! The principle of leverage is a constant available to us all, and always has been. Through his application Archimedes conveyed his insight in practical terms, and made aware of the principle and the laws governing it. We now use those laws. We no longer need to philosophise on its existence as a truth. Similarly whenever 2+2 = 4.  Whatever we use to make the equation – apples, oranges, bricks, the mathematical equation is a constant, and the principle of correctness applies. Here 2+2 =4 is empirical proof that the principle of correctness and agreement exist. Principle, or law: ‘a fundamental truth used as a basis of reasoning’.           If it were otherwise we could not measure anything. That which appears abstract then, requires no implausible argument as to its non-existence. Such is the nature of all universal principles, they exist whether the notion of a ‘mind’ can perceive their existence, or not. Because ‘thought’ does not create the reality of principles – universals- truth, it cannot from its mythical standpoint understand the simplicity of a Reality where ‘everything is’, nor the simple and factual conclusion – how could it be otherwise! Where ‘everything is’ evidently encompasses the whole evolutionary dimension. It is not an ideological enclosed static that stultifies expansion of an unfolding Reality. Any pure knowledge experience that ‘everything is’ ,(quite apart from the common-sense truth of the statement) is to experience the Absolute in any immediate part of anything that exists, which establishes its own truth forever. To examine a road code of law with that knowledge, and view the actions of drivers at traffic lights, it is more than reasonable to conclude with some conviction that there is  to some degree, Agreement – Knowledge – Understanding, and Conformity to that code of law. It becomes a ‘more reasonable’ proposition within Reality to understand that that code of law is multiplied exponentially, and the principles practised,  wherever drivers, motorcars, and traffic lights exist. It is the nature of the type of knowledge we are measuring that determines the measure of reason that can be applied to any given form of Reality. We can conclude that 2+2 = 4 is a reasonable mathematical calculation that contains the principles of Agreement – Understanding – Conformity. Given the accepted knowledge of these innate principles we can with more reason apply such a calculation Universally. Knowing is agreement with ‘what is’. Knowledge is not the attempted denial of any existing reality. That is a contradiction in terms. Mechanistic observation is akin to viewing from the outside, a straw in a glass of water. The straw always looks bent, but when removed from the glass we realise it is straight. To claim an experience of that which is Absolute, is not a claim of an experience from a higher domain, or an  isolated incident – it is common-place, numbered by just how many we are. Attempting to denigrate such experience is denying the everyday actions that contain the innate principles of a constant Reality. All life functions within the constraints of the laws that are the constructs of Nature and Reality. All life is an expression of the Absolute. It is when that expression is realised, not only in an instinctive sense, but in a real sense, that we penetrate reality beyond a comparative framework of mythology. To claim that you ‘know’ intrinsically what the principle of leverage is, or the principle behind the mathematical equation 2+2=4 is to claim experience of the Absolute. The Absolute is not some abstract esoteric truth – it is that which is immediate. Whether in awareness or not, we constantly comply, to some degree, with the laws of a constant reality. Therein lies the difficulty for a comparative framework mythology – the Absolute is everything! = Basic Equation. = However much the simplicity of the equation is, it contains the properties of correctness – balance – equality – mutual identity – meaning, which in its ‘simplicity’ presages all future mathematical equations. In that universal meaning, there is particular knowledge of consistent truth. That form of ‘simple’ consistency, creates its own natural equilibrium, and its ‘usefulness’ evolves exponentially up the reality scale. Here is where we need to give proper credence to ‘that which is’. All generalities have profound and specific principles as their ‘common’ identity, which are absolute. Unless those components are recognized, both objectively and subjectively, they are reduced to a comparative value spectrum (using dichotomies) as a misguided ‘simplistic’ factor. Deductive reasoning is then deprived of all value, and leads to the inevitable spurious question ‘how do we know? Given the above criteria to establish a correct basis for knowledge that is recognizable, and of a kind that can be used universally, ‘simplicity’ can be recognized as a tool that promotes its own established formula. That which we constantly use. Everything is the truth with regard to the methodology. How that truth or generality is expressed denotes the measure of the principle that is at its core, and forms that measure of reasoning we enjoy. Fortunately, although the truth is an innate property, it is not a ‘personal’ property per se, nor is the ‘experience’ of its reality. Its natural evolution is progressive. In that progression we are in common, the beneficent recipients that ‘evidently’ conform to its constant existence. Philosophical dissertations have become a monopolistic form of opinions that always seem to presume the ‘rightness’ of difficulties in establishing the source of our being, and are unable to put in ‘simple’ terms the question of ‘who we are’. There comes with that the denial of evidence that permeates human history, which establishes the principles of our ‘common’ reality. Those opinions carry with them a colossal library of questionable erudition that becomes embedded, using questionable values to support their argument. The most distinguished opponent of such arguments (Ludwig Wittgenstein 1889-1951<ref>{{Cite book|url=http://worldcat.org/oclc/1203018418|title=Tractatus logico-philosophicus|last=author.|first=Wittgenstein, Ludwig, 1889-1951,|isbn=978-1-78527-656-9|oclc=1203018418}}</ref>) proposed that language logic was a necessary tool to dismantle the convoluted ‘mind’ propositions that have permeated the philosophical hierarchy. Those ‘mind’ propositions only served to construct meaningless concepts as to ‘who we are’. Although he gained prominence in philosophical circles, his work was directed more toward academia. In his Tractatus Logico – Philosophies he quotes: “The limits of my language mean the limits of my world - What cannot be shown cannot be said”, and “There can be no representation of the logic of facts”. Having a belief in mystical truths that were inexpressible, his statements above are indicative of the embedded language of dichotomies. His form of ‘knowledge’ ‘philosophy’ gave (without question) the concept of ‘mind’ credence to formulate its own logic to clarify its own form of reasoning. Wittgenstein, by not recognizing within the language the distortion that dichotomies create, was unable to approach the evident constructs of true meaning that lie within the interdependent relationship of absolutes, and their constant existence. Evidential reality is all there ever is. The ‘meaning’ or ‘knowledge’ that becomes evident in reality is ‘commonly’ accepted and used accordingly. To repeat, the evidential reality is all there ever is. In that regard, the exponential drive toward ‘difficulties’ amassed a historical discourse of misinformation, which is used to address the very ‘difficulties’ created by spurious value systems. In effect, dealing with ‘nothingness’. We have managed to turn ‘truth’ into a problem, into a difficulty, when the truth is simplicity itself. An oxymoron of gargantuan proportions. This contemporary malady solidifies a diversion that discounts the reality of continuous progression. Progression in which ‘common sense’ is a motivating principle that promotes human evolution. The consistency of specific principles allows us to achieve correspondence. Real knowledge is not a fabrication of convoluted prescriptions. It is the ‘coalescence’ of ‘what is’ to internal reality. That established, evolutionary progress is assured, and reality factors are recognized for what they are. True meaning can best be attained by the interaction and interdependence of natural principles and so recognized as such. That meaning which contains all the specific components of reality is experienced as ‘true knowledge’, measure by measure by anyone. That form of ‘experience’ is not a ‘mind’ process, but a very natural state of realization consistent with our level of action, reaction, and interaction. A correct and pragmatically form of knowledge-seeking foundational answers to perennial questions would seek a direct passage to our ‘commonality’, the beacon that offers guidance. Consider the quantity and quality of knowledge we all pursue that has meaning and usefulness. To posit the notion that there are no dichotomies is a cataclysmic proposition that seems nonsensical to established embedded constructs of knowledge. Constructs of knowledge that offer only confusion, and continually pose impossible questions, whose absence would provide clarity. That absence of confusion would dispel and dismantle a reality of ‘mind’, which functions on its own selection of problems. To address reality as having only absolute constructs dispels the confusion of duality and its inability to ‘use’ relativism in its proper fashion. All general absolutes contain specific principles representing facts; the essential properties that confirm reality. This reference directs the observer to observe, and go beyond the restrictions of a ‘mind’ governed by dualism, monism, or any other spurious form of philosophy that distorts the very reality it exists in. Where there are no dichotomies, all we can deal with is ‘what is’, and the logic of ‘necessary factors’ thus destroying the possible inclusion of anything described as a “paradox”. = Pure Experience. = To design a chair our brain requires to exercise the qualities and properties necessary for its manifestation e.g., strength, balance , design, functionality etc, etc. A chair, is a chair, is a chair, the product of innate knowledge.   If we did address any tentative agreement that ‘I’  is a ‘fiction’, could not our ‘conscious experience’ of that ‘fiction’ be just as fictitious. It would follow that whatever perception of Reality we experience must also be fiction. Our contention, as always, has been that ’I think - therefore I am’ by Descartes is the greater fiction for reasons already explained. To consider to whatever degree that we can function on the basis of a fictional ’I’ precludes any attempt to honestly address ’who we are’. Saying that ‘conscious perspectives’ are limited and inconsistent with apparent reality are quite correct. It then brings into question the validity of ‘conscious perspectives’ to guide us toward ‘what is’. The entangled fictional relationship between ‘mind’ ‘I’ ‘thought’ ‘consciousness’ impose formidable barriers to that which is evident. Base observations on the construct and interpretation of what ‘knowledge’ is. Human experience is limited by its mechanical interpretation of Reality, especially ‘cogito ergo sum’. If everything that is, is its own measure of Reality (the differences) then everything must be measured, at whatever level, as being that part of the whole with all principles intact, making that measure available to be experienced as the Absolute. With absolutes there is no antagonism. We cannot exist or experience anything without a Universal complementary source of identification. It is notable that within the structure of Cartesian dualism, Descartes' personal address to innate knowledge he attributed to ‘thought’ which he identified as being distinct from his body. How different Western philosophy may have been if his attribution had been toward his brain and the existence and evidence of other physical entities that functioned every bit as efficiently as he did. The premise that Descartes operated from ‘never to accept anything as true’, was simply a wrong ended approach which brought him into conflict with his passing acceptance of innate knowledge, that the idea of God was innate to his being. To view the proposition that ‘everything is true’ allows reason to seek and identify that measure of truth. No quest can be productively based on cynicism or denial, nor adherence to belief systems that separate experience, knowledge, and Reality. We have the obligation to question whatever reality has placed before us , but if we constantly deny its existence and attempt to ‘disappear’ it from our experience, then we are in danger of never experiencing that reality. Not experiencing Reality as it is, is equivalent to not experiencing ‘who we are’, and is indeed the only human source and validity of truth, although Descartes held the erroneous belief that such knowledge was independent of any experience. That belief we suspect was some form of impetus toward his ’cogito ergo sum’.                   Knowledge and experience are co-existing ‘necessary factors’ So long as anyone believes that human experience is based solely on indirect conscious interpretation (mechanical disposition), therefore any ‘knowledge’ derived from experience will be incomplete. Clearly it is the quality of ‘knowledge’ that one experiences (e.g., Archimedes) that leads to a common certainty of evidence realised through direct experience. That quality of knowledge can be available when we observe directly the activity of drivers at traffic lights with the knowledge that it is a very common activity recognized internationally. In every case we can logically pronounce the premises to be true, therefore we have a conclusion that is also true - whether that conclusion is defined as Mutual Agreement, or Common Acceptance, it does not matter - they are mutual principles. It is this form of logical knowledge of innate principles that is the precursor to knowledge of the Absolute logically defined within all reason for ‘what it is’ . Where the basic premise is true that there is ‘Mutual Agreement’ between a multiplicity of drivers at traffic lights then we can with certainty conclude that the same principles exist Universally. We can also draw concrete conclusions, and establish knowledge, that it is not ‘absolutely necessary’ to experience by observation the multiplicity of drivers conforming to their particular road code. We have already established that knowledge. Knowledge and experience are not separate philosophical theories. One cannot be without the other.. Everyone has the potential to experience the Absolute paradoxically, in part or in whole. Everything that is, must contain the properties of the Absolute, otherwise nothing could be. To ask questions about human experience based solely and inevitably on our interpretation of ‘knowledge’, and co-existing with that, its particular meaning in human existence. So long as we can only deal with our conscious interpretation as representing Reality then we derive functionally less meaning than we are entitled to. When we see other humans consistently using levers to open crates then we can recognize a ‘social intelligence’ operating which equates to understanding that is not based on opinion, but is a clear expression of human activity that has correspondence. All of the principles involved in that experience can coalesce to provide that form of Reality that requires no interpretation. It becomes recognizable knowledge. How we understand that knowledge is through the realisation and identification of the principles involved, which become immediately transparent. The Absolute could be categorised as a knowledge experience that encompasses all and everything. Whatever is manifest is that measure (complete in itself) of the Whole with all its principles intact. Where there are at least two actions that are identical we can reach a common-sense conclusion that a definitive principle is operating. When that corresponds with innate knowledge then we have the complete cycle. The definition itself is language opening the door to an experience of Reality. No one can know in isolation. An imaginary ‘I’ restricts any experience of who “we are”,  and is not a necessary part of human experience.   Explaining experience beyond imaginary thought processes requires a definitive language that deals with the principles of Reality itself. Pure experience. The world-wide disposition that has no grounding in Nature and Reality becomes captive to any mythical fear that  offers a target to give some form of direction or stability. There is nothing more simple than to make Reality transparent - its evidence abounds. We can pronounce the principle properties that provide guidelines to its existence whereby the reality is made apparent. = Expansion. = To address concerns on ‘negativity’. Negativity is in essence the inability to establish a measure of Reality. Mechanistic processes of denial are the attempt to understand and make transparent that which is apparently unexplainable, and resolve a condition whose energy is driven toward finding that core of affirmation. The evolutionary principle from all available evidence is that human beings as a species progress. This seems a paradoxical contradiction to the embedded proposition that we can never know the ‘truth’. The consequence of such a traditional premise is that denial and negativity both hold paramount positions. We are conditioned to accept the premise that there is in fact no premise that will enable us to go beyond presently accepted norms of experience. We are conditioned to accept that the ‘truth’ is inexpressible. The evolution of the human species is constantly subject to contemporary ingrained social habits, which give some kind of credence to that particular point of existence. Indirect conscious interpretation classifies itself as a solid perspective to govern and justify human activity, which in many historical ways has proved disastrous.                   Our continued intention is to expose detrimental barriers to the realisation of ‘who we are’, and in that process establish a smoother, more realistic approach to ‘who we are’. The Archimedes legacy. When we establish knowledge of something that exists through a multiplicity of experience and evidence, then from every reasonable standard we can establish that it is true, ergo that which is true is Absolute. The principle of leverage is well grounded in social intelligence, and our natural knowledge of that does not need erudite explanations of its presence, nor any ‘conscious interpretation’ to realise its existence, or its practice. Evolution eventually removes restrictive passages to direct experience, the very purpose of evolution. The principle of leverage is not a matter of opinion, it is the realisation of actuality and our continued ‘more reasonable’ response each time the principle is applied. Children learn to speak their language primarily through experience without any direct, or indirect conscious interpretation, and so, universally we ‘know’ the most powerful means to communicate. Were we to move 50 miles in any compass direction from the town we live in, there is a certain predictability that we will meet others who speak the same English language that we do. If in that experience we find that these premises we have drawn about our travels were true, then the conclusion we would come to in particular, is that when we communicate we make known what we know. …………………………………………………………. '''Please note the date:''' '''Oct 2005.''' I am offering up this older material below to provide insight as to the progression of this work. There may well be some duplication to date. During this period my wife and I worked in collaboration to ensure an equality of experience. = Stepping Stones 1. = There is nothing other than what is – there is no hidden Reality that we need to seek, it embraces us at every turn. The Archimedes experience is the pure experience of Ultimate Reality, which provides indisputable knowledge. Reality is the source of complete knowledge, it is the constant source that has provided us with all human development, from the writings of William Shakespeare, to the computer development of Bill Gates. What they have produced is now an evident part of our reality that we can engage in. We can experience ‘mutual agreement’ through epiphanies, insights, enlightenment , understanding, Eureka moments etc, they are all one and the same. Implicit within the macrocosm is the microcosm – it cannot be otherwise. The more we conform within the microcosm the more we begin to appreciate that Reality contains everything, and that we can realise through experience its manifestation. Each Eureka moment is that personal point of experience that connects us with the Truth. The principle of leverage was always available, it took an Archimedes to explain it to us. Each Eureka moment necessarily engages with the reality of complete knowledge, and utilises its share at that time. When we have complete knowledge of who we are in that personal moment, then we understand that these, egalitarian properties, are rightfully shared by everyone and that we have experienced that which is infinite. It does not mean that the process of evolution is over – it has only just begun. It does mean that we can no longer continue coasting through this existence in a near comatose state. If there is a hypnotic fixation in holding the principles of Reality as being separate, and different, then the potential realisation of their immediate unity, and communion, becomes problematic. Knowledge, and experience are one and the same – they are not different!! Experience = Immediate knowledge of basic reality that is factually correct, and that we can reasonably use. Knowledge = Immediate experience of secure, and accurate information that is constantly stable, and sustains principles. Reality = Complete Knowledge. As the microcosmic part of the total macrocosm we are immersed in reality. The real question should be, ‘how can one not know Reality, or ‘who we are’. Mutual agreement is evident when we know we can go to the bank, and deal with money transactions. Mutual agreement is evident when we know we can go to the supermarket and exchange money for goods. Mutual agreement is evident when we know we can send our children to school to enhance their education. Evident proof is validation of what is – it is not a matter of anyone’s opinion, nor is it an assumption of ours. Neither do we assume, or offer any opinion, on the Universal Reality that there is ‘mutual agreement’ that we need air, food, and water to sustain us. Evident proof is also the basis for the mechanics toward realisation of ‘complete knowledge ‘ of who we are. Reality can be realised through concentration on its basic principles. We use language to express our understanding of who we are.It is relatively easy, it is reasonable, and it is responsible. We convey through language our measure of intelligence, and to the best of our ability conform to the basic rule of communication – ‘we make known’ Implicit within that exercise is ‘mutual agreement’. We may differ in some specifics, but we meet the basic obligation of communication – ‘we make known’, and always we progress to some degree. Simultaneity is one of the constant principles that we all share and they come from Here, Now, the Present, where they have always been. Everything is. Our being is always engaged in the present, and we each have an obligation to understand our relationship to what is. The present is the only point of contact we can ever have with Reality. To some degree or another, each one of us is directly connected to Reality (we do not have any choice in the matter), and we can potentially evaluate ‘what is’ through the utilisation, and examination of factual reality. We are the microcosmic part of that Universal Macrocosm, and because we already have that innate information it is a matching process when we have a Eureka moment, an epiphany, an understanding beyond question. Nothing enters our minds - we already know! Everyone has innate knowledge of the principle of leverage. It requires correct examination of ‘what is’ for realisation to occur. It is then a relief to have ‘mutual agreement’ on the things we would wish to make transparent to others. To use a traffic analogy, it is evident that there is ‘en masse’ mutual agreement when we know to drive off when the traffic light turns green. Mutual agreement is translated into people obeying traffic rules (otherwise chaos). Two cars, two drivers, sitting directly alongside each other at traffic lights, discuss their understanding of their Road Code in this particular position, and what they should do. When the light turns green there are a myriad of principles that apply when they drive off simultaneously. They have both demonstrated their ‘complete knowledge’ of the significance of the green light from this perspective. There is Mutual Agreement. There is Predictable Conformity. There is Common Ground. Each one complements the other. They are both right. One more remove: From an outsider’s point of view – they both know! The green light could be categorised as a Eureka moment, it sets in play all the above principles, whether the drivers are aware of it or not. From the perspective of two outside objective observers who know the traffic rules, if asked, did the two drivers at the lights obey the rules – the answer would be yes, there would be mutual agreement. They have complete knowledge of this particular circumstance concerning drivers, and green lights. Could it be that certain schools of thought are curtailed by a questionable refusal to recognize what is, and have a preference for creating a difficulty where none exists! No one can examine what isn’t! There is no such thing as ‘nothing’. Something is – what is it? A Scottish engineer functions on the same principles as an Italian Pope. Because Archimedes was prominent as a mathematician, his realisation of the principle of leverage, and his understanding of the difference in water displacement between silver and gold was widely reported. This does not mean that realisation of ‘what is’ is an exclusive experience. As said previously Archimedes did not realise something new – it has always existed, and all forms of life would have utilised the leverage principle to some degree or another (watch a bird build a nest). At that time there were probably many thousands of people who had some understanding of the principle, but Archimedes was the one who made statements about it. As in any Eureka moment, we can experience infinity and who we are. It is mutual agreement (an understanding) between the part, and the whole. It is when the principles are in unison Eureka! Reality is there to be examined, and experienced, it is not separate from us, nor should we try to make it so. = Stepping stones 2. = Knowledge is not conditional by the activities of what may be called ‘thought’ or ‘consciousness processes’. Real knowledge is that which is available to all, and to be shared by all. It cannot be contained by the ‘experiencer’ and then not ‘known’ by the accident of experience. It is the actual innate experience itself which conclusively establishes the truth. It can only deal in the truth which is its modus operandi of dissemination. For me to say that ‘everything is’, is a statement of fact which cannot be denied, and an intellectual dishonesty to attempt to deny the evidence by philosophical machinations. Hostility toward the truth leads inevitably toward attempted negation - looking for nothingness! ‘Being here’ demands its own recognition - attempting to deny it is simply perverse. Knowledge is the realisation of ‘what is’.. Rene Descartes ‘I think - therefore I am ‘did no service to human evolution, or education. It established in Western societies especially, the culture of individualism, with the precursor that so-called ‘thought’ was the inward evidence for existence, and for the following unfortunate claim that we have a ‘mind’, or to use the euphemism, a soul! Experience is true knowledge. When that experience marries up with its innate counterpart then recognition is realised (cognition). In simple terms, a light goes on in the brain. There can be no real knowledge without truth. All thought qualifies experience and attempts to reduce truth to near nothingness which is a widespread conditional activity. We cannot manufacture knowledge, or the principles which are its properties. No matter the amount of correct information anyone can ingest, it does not become knowledge until there is tripartite coalescence between inherent knowledge - ingested correct information - and ‘what is’. Then we truly recognize that which is Absolute. Within Nature we have the distinct privilege of evolving in a Universe that can only recognize the attributes of social cohesion. Knowledge is not anyone’s personal possession. Whatever measure of experience we may have of it, it is only available as a Universal sharing experience to be beneficially used. Human activity whereby we witness people using tools for leverage, or drivers at traffic lights obeying the rules of the road, are observable markers that contain the properties for understanding our own reality. Unless seen for what they are, they are only mechanical platitudes with an equally mechanical response. We could rightly claim that that at least is some response, but of no real value. The natural process that operates when we see that which is innate, overrides any erudite explanation from an academic base however intellectual its original source. ‘Thinking’ for oneself cannot make judgments about a ‘natural’ experience. When we see human duplicate functions in operation then we are in communion, and at another level we recognize who we are. When we actively see the activities of the human brain in action we are not dealing with any internal ‘will - o’ - the wisp’ that no one can ever experience. We exercise that prerogative (human activity) at every moment in time, but quite apparently without that focus of attention that denotes realistic recognition. To seek identity in sectarian, or secular belief systems to overcome the contemporary feeling of loss of identity leads to the acceptance of anything that offers some form of stability. That is then used to strengthen that which is euphemistically addressed as the ’self’. To retain that security the acceptance of information transmitted throughout generations, is absorbed into the culture, and defended to the death against those who would question that belief system. The greatest knowledge we can ever have is our own and it has the potential to transcend all else and provide insight into infinity. The most tragic human condition is the lack of experience of identity in a multiplicity of identities in which we all share. The real problem is not one of ‘identity’, but a lack of ’communion’. Whether we like it or not, whether we are aware of it or not, the principle of ’communion’ must always exist to some degree for evolution to proceed. It is within the experience of that principle that we understand the fallaciousness of that much heralded ‘self’ which draws down so much energy in an attempt to establish itself as a reality. Within positive language structure possibilities (no dichotomies), there should be the disposition toward the realisation that our relationships to cognize into ‘communion’ must be addressed as specifically dependent. Social attempts to be ‘independent’ are the very remove from reality and signify reduction attempts toward nothingness. Adherence to, and the cultivation of faith and belief systems give little elbow room for any factual occurrence to be anything other than a comparison to the myths that are held. The cultural and educational socialisation of generations of children must carry with it, its historical belief systems that overwhelm the natural instincts. Observe an animal out of its natural habitat and locked in a cage for its entire life. It would be a salutary exercise if we could dispense with the term ‘mind’ from our vocabulary and magnify the use of the word brain to promote a realistic discussion on ‘who we are’. My action of levering open a wooden crate and knowledge of it is one and the same. Our remarkable brain functions like that, the purpose of a brain, the natural repository of innate knowledge. The assertion of principles is critical to avoid all activity being submerged by questioning their very existence, and being unable to see directly. It would be a rarity today, if anyone using a lever to pry open a wooden crate would have the same enormity of experience that Archimedes had, nor the need to make pronouncements about it. It has all been done prior to our awareness of its value with the accompanying data attached. Our brain knows the value of a lever and activates our body accordingly when needed. It could be categorised as evolutionary transmission. The observance of someone prying open a crate with a lever, or drivers conforming to the road code at traffic lights, is a function of the brain in action, not a mythical entity in a singular locality that denies its own senses. When the brain is not burdened by distorted belief systems it then has the potential to experience ’that which is’, which is always constant. When we understand the function of a lever, or the presence of traffic lights, then we can activate the principles involved because we already know how! The negative impact in the use of dichotomies in language lies in their distraction from the truth, as our brain processes the words we use in relation to Reality. The tendency to attempt to separate inherent truths through the words we use disrupts that natural correspondence necessary for identification. A chair, is a chair, is a chair. = Stepping stones 3. = Where principles are concerned the constituent linkages in language are identity markers to that which is real - reference points. Without dichotomies there is no separation, or ambiguity between what we experience, and ‘what is’. Philosophy in its attempt to address something through denial is an elementary confusion. To say that that is a chair, and then attempt to deny it invoking philosophical theorems concerning the human ability to experience it, is a severe contradiction on the existence of the object , and the observer. When this form of contradiction is then taken as a constant, it then precludes any common-sense and definitive answer to the existence of a chair. '''For philosophers, George Orwell’s ’to see what is in front of one’s nose needs a constant struggle’ would be apt.''' Real concepts cannot exist in any mythology, therefore all that we experience is inevitably the truth that is there to beproperly categorised for what it is. The proper use of language in this context will identify whatever it is to correspond with present reality. Misuse of language (dichotomies and mythologies) leads only to the acceptance of a fractured state where nothing is whole and represents confusion. The dissipation of the supposed problem is never realised. Fiction has been elevated to the status of an accepted reality. Very early evolutionary physical dangers allowed the development of fictions that offered some form of imaginary protection beyond limited physical ability. That contemporary humanity endorses the mythology of ‘I’ is testament to the psychological fear that still exists and requires its proper recognition. Emphasis must be placed in the relationship between language and reality for understanding to proceed. The persistence of dichotomies has their own persistent confusion which then promotes a false reality through misleading information. Microcosm and macrocosm are one and the same in a Universe where ‘everything is’. Isolated viewpoints are exactly that, and are unable to view the expanse in which we are encompassed. We must learn to view reality through both ends of the same telescope. When we understand the extensive scope of ‘truth’, then we know that its values and properties do not change - which relates to ‘completeness’. Philosophical, ideological, and intellectual endeavour , try to shape the structure of ‘what is’ based on pre-dispositional knowledge, which can only ask the same questions, and look for the same answers. Not to experience that which is absolute or whole is the normal result of the confusion of language which has no correspondence to that which is real. To discuss with a philosopher the possibility that ’mind’ per se does not exist, and to dissolve it as a concept would place them in a realistic position, would indeed be a difficult proposition. The strength of that difficulty lies in another imaginary concept, that that ‘mind’ represents ’I’, and it is anathema to that fiction to consider its own demise! = Stepping stones 4. = There are no dichotomies. Everything is, and everything that is, is complete, everything is an Absolute complete Reality. You are experiencing your measure of that reality. It cannot be otherwise that you are experiencing that measure of completeness. When we come to terms with it we have the innate capacity to see the Absolute in a grain of sand. That is knowledge. Belief in dichotomies is the mythical barrier to that particular experience - which is only denial, supported by erudite protestations that human construct dichotomies exist. At a mechanical level Intelligence and Stupidity appear to be separate identifiable conditions, and they appear to be antagonistic. Stupidity is in Reality a measure of the Intelligence which is always constant. If someone was in a state of mythical utter and complete stupidity we would not attempt any form of emancipation from that condition. We know that that is misguided and proceed with techniques to advance intelligence. Consider the proposition that there are no dichotomies, and within that possibility all questions become irrelevant. Presuming that there are no dichotomies allows the process of establishing ‘necessary factors’ to proceed, and allows each measure of wholesomeness to be realised. Experience is the criteria for knowledge. Some Reality experiences were simply transposed into particular belief systems and elevated into a pseudo spiritual dimension, or a philosophical conundrum. Where there is a belief in a divisive fiction (dichotomies) there is automatic mechanistic restriction to that which is Real. There is a capacity beyond ego and intellect which can commune with ‘what is’, and recognize its properties. Reality is constant. Within the accepted comparative framework there is the view of principles as having different divisive categories e.g., as above, Intelligence and Stupidity, and classify them within ‘thought’ structure as dichotomies and give credence to them as being an antagonistic reality. The consequence of that, is, that one is always a remove from recognizing the structural properties of immediate existence. Any construct of knowledge necessary to evaluate ’what is’ will address the properties (principles) that are the constituent constant markers available in that which is the microcosm and the macrocosm. That identity (the Absolute) is found in any sphere of Reality.Everything is - and everything that is, must be experienced for what it is, and not for what anyone denies it to be. There is no mythical human construction that can deny ’what is”. Everything is - without dichotomies. To repeat, we do not have the ability to create ‘nothingness’ - ‘that which is’ has no imaginary comparative human construct. To attempt to deal with such constructs, and give credence to them is always the denial of ‘what is’, and adherence to ‘thought’ processes whose only purpose is to cement that activity. Indeed realising that the concepts of dichotomies are human mythical constructs, denying true perspective, is the beginning of insight. The dissipation of such processes through addressing the principles of Reality allows us the potential to experience directly ‘what is’, in simple terms -the truth! Intelligence is a ‘necessary factor’. Addressing stupidity is a denial of reality at whatever level we find it. Intelligence and Stupidity are not antagonistic, they are one and the same principle with measurable degrees of existence. Only from a comparative framework standpoint is credence given to any mythical form. The above observation is not negating the process, it is questioning the markers which evolve into imaginary separation (trapped in a comparative framework mythology). That particular process can and does create a false mythical reality that appears divisive. We cannot exist within a divisive reality! Reality must be complete for us to recognize its existence. Where there are no dichotomies within the premise that ‘everything is’, there exists no antagonistic position. The distinction between human constructs of positive and negative are matters of mythical perspective wherein no experience of the Absolute is available. It is because the human ’mind’ per se places its own construction on its immediate experience, and must have its particular interpretation based on what it considers ’knowledge’. There is a difference between ’mechanical knowledge’, and ’pure knowledge’. From the mechanical knowledge standpoint which can only deal ‘in indirect conscious interpretation’, it is quite correct to say that that form of knowledge is incomplete, and it always will be. Pure knowledge experienced via our brain knows no separation, nor antagonism, and is responsible for our ability to recognize the actions of others who may pry open wooden crates with a lever, or drive off uniformly at traffic lights. Within that cohesive activity it precludes ’a matter of opinion’ and by themselves can become subjects of a pure knowledge experience. To repeat, it is a form of ’communion’ with ’what is’, and available to all. Where drivers at traffic lights universally conform to their particular road code, and where universally there is a language which identifies their activity as Mutual Agreement, or any other logical definition, we can concur with the common-sense conclusion that we have universally established that within language and common activity, there is indeed a truth formed. The coalescence between universal language and universal activity are the logical constructs that create civilizations. There is a vast social network of common activity that solidifies the logic into an honest and persuasive conclusion that confirms innate common principles –knowledge. = Stepping stones 5. = The Art of making sense of everything. How to understand principles. # Principle. A fundamental truth or proposition that serves as the foundation for a system of belief or behaviour or for a chain of reasoning. # All principles are interdependent, interconnected, and infinite. # Each one is dependent on the other two. Examples of a principles template and how to define them without dichotomies. Communication. Truth. Standard. Proof. Express. Contribute. Mutual. Direction. Advance. Comfort. Organize. Certain. Immediate. Interest. Improve. Present. Constructive. Gain. Trust. Progress. Source. Knowledge. Basic. Original Reality. Awareness.Freedom. Purpose. Connect. Understand. Support. Peace. Cause. Unity. Ability. Rights. Honest. Discover. Positive. Energy. Balance. Good. Courage. Willing.  Control. Use. Association.  Observe. Reason. Easy. Wealth. Simple. Law. Increase. Order. Flow.Co-operation. Exact. Quality. Accuracy. Strength. Responsible. Operating. Creative. Measure. Recognition. Accept. Constant. Obligation. Include. Dependence. Relationship. Value. Success. Principle. Equality. Stable. Share. Love. Sustenance. Action. Identity. Intelligence. Education. Secure. Facts. Agreement. Information. For. Rules.Clear. Yield. Example: Success = Securing facts                = Responsible co-operation               = Constructive knowledge So success by definition is : Securing facts through constructive knowledge and cooperating  responsibly. All definitions of success from your template are infinite. You will find your own suitable definition. ==== There are no dichotomies! ==== Any principle is correctly defined by any two other principles. You create a new language of Absolutes. Using conjunctions you can write your own book. The man whose book is filled with quotations has been said to creep along the shore of authors as if he were afraid to trust himself to the free compass of reasoning. I would rather defend such authors by a different allusion and ask whether honey is the worse for being gathered from many flowers. Anonymous, quoted in Tryon Edwards (1853) The World’s Laconics: Or, The Best Thoughts of the Best Authors. p. 232 Amen to that! “One is not born, but rather becomes a woman” Simone de Beauvoir. “Time does not change us. It just unfolds us” Max Frisch. We experience ourselves our thoughts and feelings as something separate from the rest. A kind of optical delusion of consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Albert Einstein, in One Home, One Family, One Future. = Who we are. =   It is notable that within the structure of Cartesian dualism, Descartes' personal address to innate knowledge he attributed to ‘thought’ which he identified as being distinct from his body. How different Western philosophy may have been if his attribution had been toward his brain and the existence and evidence of other physical entities that functioned every bit as efficiently as he did. The premise that Descartes operated from ‘never to accept anything as true’, was simply a wrong ended approach which brought him into conflict with his passing acceptance of innate knowledge, that the idea of God was innate to his being. To view the proposition that ‘everything is true’ allows reason to seek and identify that measure of truth. No quest can be productively based on cynicism or denial, nor adherence to belief systems that separate experience, knowledge, and Reality. We have the obligation to question whatever reality has placed before us , but if we constantly deny its existence and attempt to ‘disappear’ it from our experience, then we are in danger of never experiencing that reality for what it is..   For anyone to say that ‘everything is’ is a simple linguistic absolute that no amount of ‘more reasonable’ requests (above) can deny. Those requests only appear to be governed by the difficulties of ‘mental complexities’, and embedded ideologies. To accept that ‘everything is’ as an absolute, is a realistic basis to establish any reality, and comprises the basis for reason to be activated. If there is ‘nothing’, nothing can be achieved. Within that which is Absolute there are no dichotomies. Therefore there are no antagonistic positions available. Everything that is, is a measure of the Absolute.  We are always in the present, everyone and everything. Instant elementary ‘knowledge’ which we all share, and must admit to. In being alive, we do not have the ability to not be here, and we do not have the ability to not know! To have a problem in addressing what Truth and Knowledge are, to the point of denying their existence, then that problem exists well below the scale of Reality. To repeat knowledge is not the proprietary right of any individual, it is enshrined in the principle of agreement that we mutually exercise to establish its own reality. The reality of experience is not, nor ever will be, a personal possession that we can have and hold. Its reality becomes more alive when we see the same activity being practised by others. Then we know we are sharing that reality, and that experience. We cannot "have" the principles that exist, but when we undrestand them then we are obliged to use them with integrity. That form of integrity in any language, is an added foundation stone to any belief system Mechanistic ‘I’ has no concept or understanding of ‘pure knowledge’. Only when we break free of the myth that some clarity becomes apparent, and we have the opportunity to engage with what is real. Philosophy it appears to me is constrained by individual ‘thought’ processes, which (without experience) cannot escape from that individuality. Those ‘thought’ processes conjure up a human history of inflexible, and impossible propositions which only serve to protect that individuality. Descartes ‘cogito ergo sum’ has compounded the difficulties by strengthening the incorrect premise of a false individuality. ‘I’ is a phantom consciousness much like a phantom pain experienced after a limb is amputated. The brain registers the pain signifying that something should be there. Likewise our brain has that same relationship with Nature and Reality. It is analogous to our brain dealing with a ‘phantom reality’ knowing that something is missing but is continuing to evolve to establish the whole. There exists a ‘phantom chasm’ between our brain and Reality and an understanding of its properties. We are robbed of real meaning. Evidence, recognition, and the truth are the principles it uses to reform. Within their structure is the meaning of reforms. Nature does not impose any morality on us, the principles implied in morality are there for us to understand and use. Our brain has the capacity, once reality is correctly examined, to recognize ‘that which is’. Once realised it becomes embedded. That ‘phantom consciousness’ is an experience removed from its proper environment. It takes its proper place when we experience reality for ‘what it is’, which provides the totality of meaning. True experience allows us entry to the quality of knowledge that is a continuous reality. So long as anyone believes that human experience is based solely on indirect conscious interpretation (mechanical disposition), any ‘knowledge’ derived from that experience will be incomplete. That form of philosophical negative conclusion can come down to not believing that Reality exists (a chair is not a chair, is not a chair etc,), or that our experience of ourselves and others is real, and discount any other form of knowledge that threatens that belief. There is an intellectual dishonesty in denying the existence of principles. Hostility towards the truth leads inevitably to negativity. Being here demands, not denial, but the right to be recognized. We are the recipients of a ubiquitous communication system – making known. We can only understand that which we know. We make known all the time. Knowledge of Reality – Truth – the Absolute is a collective inclusive experience of the principles we share, and never the property of any individual. To ‘know’ ‘who we are’ is an inclusive experience of the principles involved. Never ‘cogito ergo sum’.   Philosophers in investigating the nature of knowledge and the Universe, firmly established for themselves that the source of reason and logic was located in a mythical concept ‘the mind’. From the wrong basis evolved elaborate and metaphysical constructions which removed the investigations further, and further, from the truth. To comprehend the material world, and give it credibility, the recognition of implicit principles is paramount. We need to construct a language that provides that form of recognition. Any philosophical theory of ‘mind’ that will deny the evident structure of solid objects is misguided by the injection of a mythical entity (mind) that determines that seeing solid objects is a ‘perceptual illusion’. That form of determination is singularly narcissistic, empowered by the self-induced threat that venturing into a ‘materialistic’ world is a loss of that illusory self, and all the belief systems it has constructed to protect it. That erroneous established view that not addressing ‘materialism’ as a profound Reality, and as only a ‘perceptual illusion, is compounded by the belief that that form of illusion is implicit in every human view available. We cannot manufacture knowledge that leads to a mechanistic understanding of ‘what is’ , nor the principles which are its properties – however much dogma is practised. We can only aspire to relate to ‘necessary factors’ that are the implicit fundamentals of existence. For me to use a lever to open a crate is a form of communion with Archimedes through the principle he enunciated. It is now not ‘necessary’ for me to go through the same experience as Archimedes to establish that ‘necessary factor’ or ‘principle’. It is now common-place, and common-sense to utilise the principle. The extract below provides some explanation of the brain processes in action Universally, and coincides with any reasoning on the observance of the leverage principle, and the actions of motorists conforming to the Road Code wherever traffic lights exist. ''The right-to-left shift of mental control looked increasingly like a universal phenomenon, capturing the essence of every learning process on every time scale, from hours to years. An individual faced with a truly novel situation or problem tackles it mostly with the right hemisphere. But once the situation becomes familiar and is mastered, the dominant role of the left hemisphere becomes evident. It looked like the empowering patterns capturing the essence of the situations (or rather the whole class of similar situations) were, once formed, stored in the left hemisphere. (The Wisdom Paradox. Professor Elkhonon Goldberg. P202)<ref>{{Cite journal|last=Vandermeulen|first=Jo|date=2008-08|title=Verstand komt met de jaren|url=http://dx.doi.org/10.1007/bf03077135|journal=Neuropraxis|volume=12|issue=4|pages=137–139|doi=10.1007/bf03077135|issn=1387-5817}}</ref>'' = Limitations. = However limited our view of connectedness is, or however tenuous the reality our experience is, ‘everything is’, and everything is connected. Innate knowledge and the fundamental nature of Man is the prior source of knowledge that seeks and identifies that connectedness. Attempting to address what we don’t know is that mythical infinite regress toward that parallel mythical ‘nothingness’. To always address what we know establishes Reality. To establish knowledge of principles, start from ‘we are here’. The natural principles within the diversity of human culture and activity when recognized as mutual ‘necessary factors’ will have the effect of enhancing and directing vital energy toward the very process of  the communion we seek, and the gradual penetration of a reality that always exists. Knowledge of Reality is not ‘different’ in other locales. The fundamental principles are the same. Addressing ‘what is’ instead of denying ‘what is’ is the basic construction of real knowledge. Within the structure of the Absolute we are all the same with a magnificent differential in our expression of the principles of necessity. That expression is our ongoing effort toward its own experience which gives it life and meaning. That experience in turn exposes us to an immediate Reality that is in communion with the fundamental structure of our being. All that we can contribute toward that is 50%, the other half is in our momentary relationship with Reality – then we know! That form of knowledge is always available through that form of experience, and it always comes in the form of confirmation which reforms. Until that experience our prior condition appears mechanistic, without direction, or understanding. Reality, life, is not mechanistic. We are the recipients of innate principles with the constant potential to experience those principles in action (Archimedes et al). Dogmas, ideologies, are the restrictive practices used to blur the recognition of principles operating to a level that understanding of that common and constant activity is virtually denied. Our natural capital (principles) is degraded to the point that their factual evidence is reduced even to the point that they are categorised as a ‘perceptual illusion”. We can trust facts 2+2=4. Simplicity has its own majesty. Anything circumscribed by reason requires control of our emotions. To comprehend the material world, bring it alive, and give it credibility, the recognition of its implicit principles is paramount. We need to construct the language that provides evidence of that Reality. Any philosophical theory of ‘mind’ that will deny the evident structure of solid objects, is misinformed by the injection of that mythical entity (mind) that determines that seeing solid objects is a ‘perceptual illusion’. That form of determination is singularly narcissistic, empowered by the self-induced threat that venturing into a ‘materialistic’ world is a loss of that illusory self, and all the belief systems it has constructed to protect it. The erroneous established view that not addressing ‘materialism’ as a profound reality, and as only a ‘perceptual illusion’, is paradoxically compounded by the belief that that form of illusion is implicit in every other human view available, thus it then makes its own sense, form and justification to the illusion! The evident question we must ask, ‘how does a ‘mind’ conclude that ‘immaterialism’ exists universally? Surely it is a simple but massive contradiction in terms. If there is nothing there but ‘perceptual illusion’, how can you attribute it to other ‘minds’. The oxymoronic effect of narcissism is that it is the very denial of ‘who we are’. Man is not composed of an overwhelming self-love. That mythical embedded belief cannot consider the possibility of underlying principles that are the real life force of Man in his relationship with Reality. The truth of that, is that humankind (in spite of itself), evolves towards its own Reality. The only human values that exist, lie in Man’s recognition of the principles involved that provide human direction. Our ‘material brain’ is a product of Nature's evolutionary process, and has innate within it the same principles that exist in all matter. That ‘which is’, is the truth, and our brain evolves to process that at every level, and we constantly manifest that in every action we take – whether we like it or not. The fundamental similarities between human beings is that we are not only evidently human, but that we also function and construct societies that we recognize as beneficial to our immediate well-being. All social function is determined by our brain capacity and its ability to postulate the relationship it has with Universal principles. = Illusion. = Considering that we can contradict things is an illusion. We can never contradict the truth. We do not have the ability to create proprietary constructs of reality. That ‘which is’, can only make its basic properties transparent to us through direct experience. Imaginary concepts must in the end conform to a measurable construct that we can identify. Within the structure of any philosophical theory of ‘knowledge’ it must contain the basic elements of truth at all times, or there is nothing!! To say that ‘everything is’ is motivated by pure reason experience as an objective, and subjective reality and as an axiomatic grammatical premise that no amount of mental acrobatics can deny. We can only deal with ‘something’, whatever it may be. There is no metaphysical construct that can provide evidence that ‘nothing’ exists, outside a mythical mind. Explaining experience beyond ‘thought’ processes requires a definitive language that deals with the reality itself. We all Know. It is innate. The ‘difference’ between us is only the measure of the knowledge that is made manifest, and that knowledge continually proliferates. The ‘individual perspective’, and the illusory ‘I’ which dominates, is the barrier to any relation to ‘what is’, and the malady of never experiencing the truth directly!   Truth, knowledge, agreement are the abundant and embedded Absolutes that form the structure of human evolution. That we constantly utilise and improve on their use is evidence of their reality, and the material transparency within every social structure. The survival and proliferation of such realities should be the evidence to establish that ‘that which is’ is Absolute. When we focus our ‘perspective’, opinion, or a hypothetical consideration of a space, time, or identity to question a Universally accepted fact, it is hardly a categorical argument to dismiss that which is true as nonsensical. Any denial that 2+2=4 is a fundamental truth hardly takes into account that the reality of such basics are vital to the success of higher mathematics. Unless the basics are continually correct, and evidently so, then no correct solutions could evolve. We know that within any basic structural ‘use’ that the calculation is correct. We commonly accept its correctness as an embedded reality. All forms of lower or higher mathematics would have the axiomatic principle of ‘correctness’ as their basis to extend from. Also, they would have as an axiom that the reverse is true. The 2+2=4 is, in its reality, the epitome of balance and construction. The 2+2 reality forms its correct conclusion when the principles of mathematics are propounded and they conform to transparent truth and arrive at 4. Only when it ‘adds up’, does it become a truth that we all recognize. Our greatest ignorance is taking for granted the proliferation of such truths through an ideological blinkered perspective. Because truth takes a commonplace form it is no less fundamental. Unless there is correct knowledge as to the existence of fundamental truth, that ‘which is’, goes unrecognised. That form of truth must be applicable to all. Truth exists in everything – it is an evidential reality. Searching for an esoteric truth is chasing shadows. Every truth is a ‘necessary factor’, and fundamental to our existence. Because of the imposed limited perspectives (via education, ideology, beliefs) that which is evidently true, and transparent, is delegated to a position of simple practicality with conditions placed on it which further deletes its substance, and we have the awful predilection of conforming to the attempted destruction of that which is true. Do we have a problem with seeing something, which is correct, as also being true? All truths are fundamental. They are not subject to attempted denial because of any diminished realisation at any point in time. Where there is reasonable evidence of balance, equity, and agreement we can conclude that a truth exists. Once innate information of that truth becomes transparent, it becomes an embedded useful human utility that must have some measure of fundamental truth as their starting point. From any common-sense, or ‘more reasonable’ position, it would be more productive to view reality as possessing at every level the same innate values or principles consistent with our ability to measure, or recognize them. To view reality as having ‘different’, or antagonistic properties, is simply a misguided view of ‘what is’.  That form of perspective is counter productive when it attempts to establish mythical dichotomies as realities in their own right. When the reality of principles are made transparent, we can then ‘more reasonably’ make use of them to further their basic existence. Here we use reason to exemplify their necessary function, and once established it becomes (if necessary), ‘more reasonable’ to locate them in all things. The dematerialization of any object through the practice of ‘perceptual illusion’ is an attempt to deny the reality that exists. Where perceptual illusions are concerned, innate direct communion with that which is, suspends the effect of such illusions. All the properties in a chair are recognized as the reality that exists. That is materialism. A chair does have the principles of form, design, structure, colour, substance etc.  However it is analysed – it is a quantifiable reality.   = Human representation. When we understand the validity and existence of principles in all things, it is easy to understand that ideological dogmas are never the foundation for real knowledge, or that direct experience of ‘what is’. Our real perspective is not some individualistic experience that confines us, it is that expanse in which we exist that offers us the view of that expanse. Everyone has the potential to go beyond their ‘apparent’ human perspective limitations. Shifting our sense of perception toward that which is basic, paradoxically extends the experience of that which is true. Let general knowledge be directed toward the performance that identifies the measure of principles that are enacted. Therein lies the production of knowledge that offers a sustainable growth of that vital universal aspect of knowledge, where, reason and truth, can prevail. Any correct definition is language itself, opening the door to that reality experience which is critical. Only when we know and experience that the same reality (with all its principles intact) exists for all of us can we then recognize the mythical distinctions that are taken as being real. The majestic experience of that reality goes well beyond historical beliefs. Exploring simple ‘necessities’ is not based on any sacred text, but the privilege of recognizing a sensible evolutionary path through life. Whatever may be in the future, is implicit in the material world now, and it has always been so. 2fhkwn84gwmegj5jdvg9yrfgntqirzm 2416465 2416464 2022-08-19T09:28:47Z Hamish84 1362807 /* Leverage. */ wikitext text/x-wiki = <ref>'''William Shakespeare''' (bapt. 26 April 1564 – 23 April 1616</ref>A message. = What humanity needs is not any individual approach but a governance powerful body of excellence that has modern technology, knowledge, and freedom they can use to disseminate clear information. Clear information about a new language structure of absolutes beneficial to an International forum, and eventually to reach a Universal status.Its benefits reach toward conceptual language on a planet that speaks more than 7,000 languages. No matter the language spoken the concepts of: Air - Food - Water are recognised. The overall development of conceptual language can only be beneficial and will be as appropriate to Absolutes definitions. Reaching for the Stars might show us the way. Universal Language of Absolutes. "A very grand title but it took many years to explain its value" = Our history. = Born in Scotland in 1927 left school at 14 years of age. Married at 21 years of age and we had two children. We emigrated to New Zealand in 1953 and lived there for approximately thirty years. During our stay there I did a Liberal Studies Course at Canterbury University Christchurch and graduated. We have since had ten books published through Google books on the subject matter at hand and my wife Jean Caldwell McMillan is the co-author to most of the work presented here. My wife Jean was an avid reader of many works on philosophy and psychology. She was influenced by the works of Erich Fromm. Jean died 9th of January 2011. To refresh the original purpose of  our  earlier writings my wife and I  went on an odyssey  looking for any data, ancient or otherwise, on human consciousness, specifically related to Alzheimer’s disease. Now at 95 years of age (well past my used by date) it may well be that I am a candidate with a focus on my own pending dementia. If so, then the theory and the method I now write about is holding it at bay. To address the health of my mind in this way could be the catalyst that retains its own functional activity. The creation of a  semantic template is well documented below. No semantic definition of absolutes or principles can be ill-defined. They are always interconnected, interdependent and infinite. Each configuration constructed by anyone has meaning particular to them, although its value is universal. That  is why it is never personal property! [[File:JimNJean.png|center|thumb|228x228px|Jim and Jean]] = The Beginning. = "The road's half traveled when you know the way" [[File:Tree lined path - geograph.org.uk - 2269906.jpg|center|thumb]] Oxford dictionary definitions: Principle: "A fundamental truth used as a basis of reasoning". Absolute: " Complete - Entire - Perfect - Pure. These particular dictionary given definitions offers us guidelines to ‘existing conditions’ necessary for complementary understanding, and experience. We can only examine that which is real, basic evidence, that is fundamentally true, and we must ‘use’ it, to establish that which is reasonable. The general consensus is that there are no Absolutes. The following material is set out to show the very reverse is true and that everything that is is Absolute. Establish that there are no dichotomies that will leave the primary terms alone to create a semantic template. There are no dichotomies. Mythical dichotomies distort Reality. Everything is: The computer you use today has always existed, it is the arrangement of particles that have materialized it. The subject matter "Universal Language of Absolutes' is promoted to provide a new understanding of spoken language. This understanding was initially constructed by the cognitive experiences of both my wife and myself many years ago. Just like the principle of a jigsaw puzzle, meaning lies significantly in the fact that all pieces of the puzzle are interdependent and interconnected. When completed they provide a picture of the whole. We have endeavoured to produce a picture of the evolutionary process of language in human history because the evolution of language prefixes all modes of thought in human culture. The material directs the reader towards a new view that all that evolves is in a vertical direction, not the linear direction commonly understood. Human consciousness is of itself the phenomenon of evolution and to recognize its existence is part of the process. Shakespeare expressed this succinctly through the voice of Juliet who proclaimed, “a rose by any other name- would smell as sweet.” [[File:Comestible rose in the Laquenexy orchard garden, Moselle, France (01).jpg|thumb|center|237x237px]] === Conceptual language. === My wife and I recognized how profound the extension of this observation would mean conceptually. Of all the languages spoken on this planet, it would be fair to say that all of them would contain the properties of, air, food, and water conceptually, etc. This is a form of consciousness equality that is available to us all. It points to the reality of our constant relation to each other and our existence. We can never exist in a world of individuality, but only in relation to the consciousness of one another. Consequently, that exceptional experience can only be shared superficially. We cannot ‘know’ any other life experiences other than our own introspection. <gallery> File:Einstein 1921 by F Schmutzer - restoration.jpg </gallery>" ''Albert Einstein 1921. We experience our thoughts and feelings as something separate from the rest. A kind of optical delusion of consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us."'' ''Albert Einstein, in One Home, One Family, One Future,p.99'' Einstein came very close. In reality, every human being has a backpack from the day they were conceived. In the backpack every experience in their mother’s womb is experienced. At birth and throughout their lives, everything that happens to them in life is registered and creates their personality. That life with all its experiences can never be known to anyone else, consequently, we can never “know” another person. It creates equality of consciousness that we must understand. We can know details about a person, but that is all. That life is sacrosanct. Who we are really goes beyond normal human experience and into the realm of the Absolute. Werner Karl Heisenberg (physicist). The Heisenberg Uncertainty Principle. As a young layman with no knowledge of Heisenberg but interested in principles it seemed to me that the Uncertainty Principle was just a contradiction in terms. In later years I found that Heraclitus describes life as being in a state of flux a replica of the Uncertainty Principle which in fact can be defined as an absolute state. Within the context of knowing who we are and the backpack we carry our life in, we can never know each individual life as that life experience is singularly their own and sacrosanct. It now seems that the Heisenberg Uncertainty Principle can fall into the category of being an Absolute. Evolution proceeds in advance of our need to evolve. In our pure active state, we are.There is no static end (an abomination) - only beginning. As we cannot know what tomorrow will bring, living with expectations is rather futile. Nature has its own agenda. Zen Koan recorded 1228: 'An instant realization sees endless time. Endless time is as one moment. When one comprehends the endless moment He, or she, realizes the person who is seeing it.' We do not own Space. We do not own Time We do not own Energy We do not own Matter [[File:Universum.jpg|center|thumb]] "What we call the beginning is often the end And to make an end is to make a beginning. The end is where we start from We shall not cease from exploration And the end of all our exploring Will be to arrive where we started And know the place for the first time" T.S Eliot = Everything is in scale. = The present moment is the point in which Eternity has placed us – we all live in that moment, and whether we like it or not, we exist in it, experience it, have knowledge of it, and we all share it, measure by measure. There are no dichotomies. Illusion is a measure of Reality, as Stupidity is a measure of Intelligence If one keeps measuring illusion it is an attempted downward spiral to nothingness.There is no opposite to Reality – that illusion is a measure of Reality. There is no such thing as ‘nothing’ in the elemental construction of Homo sapiens. All the innate ‘something’s’ are the fundamentals of our being human and all our experiences. The correct use, and understanding of who we are, is an extension therefrom. Does it require any interpretation on anyones part to say ‘we are? Any attempted denial of that statement would be perverse use of the language, and delusional. === Try saying ‘we are not” === ‘We are’ is the foundation of all affirmation, and within that spectrum, we can know, and be. ‘We are’ is self-evident Truth. We can neither know, nor experience what isn’t. Eternity is the here and now, that is why it is possible to explain the experience of Eternity. Nothing is ever lost in Time. We are located in a vast Universe. ‘twas a moment’s pause,- All that took place within me came and went As in a moment; yet with Time it dwells, And grateful memory, as a thing divine. Wordsworth Prelude, Book V111 We already know – the ''basic'' condition that must exist for us to re-cognise. It is at that moment of pure affirmation, when all that is, is manifest. Unless there was mutual identity we could not know anything. It is why we are urged to evacuate the Platonic cave. Sadly most prefer the shadows rather than confront who they are. We already know – the basic condition that must exist for us to recognise. It is at that moment of pure affirmation, when all that is, is manifest. == Lost Shadows == [[File:People Shadow.JPG|left|thumb]] The shadows move Lost in confusion Lost in despair Imagination shrouds the real Looking back Looking forward Is this the Centre? James Brines. = Basic Principles. = We are all in the business of living and attempting to understand the principles involved in that human process up to the end of life. The implicit principles necessary for life eventually disappear and all measurable criteria pronounce a body to be devoid of life. Throughout historical agreement we know what that means, and we act accordingly on *common knowledge*. We know that dying is a necessary factor of life. It is a Natural law that if we live - we also die. Natural law is Universal, for us to *know* that a body lives; we also *know* that a body’s life will end. Albeit that reports tell us that today millions of people die of disease, starvation, wars, we of necessity accept that as the ongoing reality because again we are universally connected and know the results of such carnage. Because it is in our realm of common knowledge we have graveyards, crematoriums, undertakers, doctors who pronounce bodies to have died. We understand the consequences of leaving such bodies unburied, the diseases that would prevail. Again, please explore the definition of principle (Universal principle) and try to go to the limitations of the definition without using mythical dichotomies. [[File:( A great picture of outer space ).jpg|center|thumb]] Principle: A fundamental truth used as a basis of reasoning. All of these questions are based on singularity (the Cartesian dogma) If Descartes had only introduced inclusion into his musings (they were taken as conservative singularity) he may have realized the difficulty of addressing thought as reality. He then may have quite easily concluded that universally - *We are!!* That pronouncement is inclusive, and conclusive in every way, *we are - and we know!!* Because my knowledge is not a private, personal piece of property concerning principles, but Universal (Archimedes et al), then that innate knowledge has completeness we can share. Whatever identical resonance we may be able to share (concerning completeness), that can only be accomplished by understanding the principles involved and their constant relationship to each of us. Clearly the plethora of present and past discoveries establishes the existence of that which is fundamentally true, and the foundation for law. The principles are established, and always have been, we are in the business of making them transparent and complete. However tenuous the link we are all connected through communication, the air we breathe, the ground we walk on, the universe we live in, the common principles we live by. We all must have sustenance to survive, or we will not live. (See above) <nowiki>*</nowiki>There is nothing else to experience.* When we actively explore the reality of anything, all principles involved in that exploration are complementary, and honest, and we understand the wasteful divisive mythology that people attempt to attach to them. We cannot */partially know/* the truth, it must be complete. Dichotomies attempt to deny the existence of truth, and are misleading. How do we more reasonably completely know anything? The complexity of language systems with contemporary usage requires new and creative structures to provide clear information. Internal and external reasoning capacities can only develop in concert with Man’s recognition of the principles that essentially form our lives. The inevitability of human consciousness rising beyond its historical beginnings posits a future outside our normal perceptions, and  a factual reality that points to the existence of new perceptions that are infinite.   It is natural to know when we are no longer trapped in any mythical ideologies that gives credence to dichotomies that stifle the recognition of simple principles. There was no cause for knowledge to be established - it has always existed.* Ask yourself - ‘how do you know to ask any questions at all?’ There is no hidden dimension or mystical world. The only philosophical reality is ‘that which is’. To access that we need a new structure to explore ‘knowledge’, a new transformational language. Real knowledge is not amassed information, nor is it the establishment of dogmas, isms, or mythical belief systems. [[File:FAIR data principles.jpg|center|thumb]] Basic principles are the source and foundation of all /*knowledge*/. Until that is recognized, extension from mythical sources only leads to a denial of one’s own senses. Trying to conceive of a contemporary world without principles is to posit a world without reason, or intelligence. The principle of pure knowledge (Truth) could be said to exist in another dimension given the present state of human understanding. To progress that mistaken belief there would be strong support in the need for a comparative reality. Curiously it seems that philosophy (the seeking of knowledge) constantly discounts any knowledge that does not come within the sphere of established philosophy, and the comparative reality dictum. That consequence profoundly distils the purity of any experience and alienates the observer in their confrontation of that which is real. The measure of that ‘comparative reality’ knowledge bounded by dichotomies is so restrictive that it lies in a mythical dimension where denial of its very purpose is the order of the day. Evidently it will not allow doubt to undermine its own denials. A basic principle of Nature is /*knowledge*/ and it constantly communicates innately in every living structure. How to grow, develop, and disseminate. Knowledge is reciprocal truth that depends on our relationship and the recognition of principles operating. Knowledge (unlike information) is not stored in an individual box; it is ubiquitously manifest in everything we do. Knowledge is the experience of a positive reality, and its true construction is a dependable source of secure information (not to be misused). We daily have the opportunity of witnessing ‘knowledge’ in action as expressed by the ‘the principles of knowledge’, namely the principles themselves. The questions lie in a continuing mythical belief in a mystical unknown (the Cartesian stance) which because of its non-existence can never be known. It is a belief that is detrimental (because it attempts to deny all existing factual knowledge) to dealing with Nature and Reality and the fundamental necessity of our relationship with them. Making that relationship transparent is our basic obligation and the ongoing evolutionary activity. Although there is an obvious avoidance to address the definition of principle itself, it is a factor that must be paid attention to, to realize that ‘common knowledge’ is the only reality that exists. Amazingly although there is avoidance of principles - truth - reason etc, there is acceptance of the Cartesian dogma,. Paradoxically this determined acceptance of Descartes supports the reality of innate knowledge existing (I think - therefore I am) which establishes for him innate knowledge; however mistaken he is concerning the interpretation of his experience. = Leverage. = Long before I read of Archimedes and his various principle discoveries I was using the principle of leverage in a variety of ways, prying lids off boxes, moving articles with a lever well beyond my physical strength to do so without said lever, and I knew how to do it. Transferring that knowledge to a student or apprentice is relatively easy because innately they also /*know*/ how to do it. Every aspect of human industry uses the principle in a myriad of ways because it is our obligation to constantly progress the principle and confirm the constant utilization of knowledge. We wholeheartedly adopted Archimedes principles (et al) because we recognized their fundamental utility. It is preposterous to question the widespread /*factual knowledge*/ of all principles, more especially so when we cannot escape their ubiquitous daily existence in all our lives - Nature and Reality do not lie. [[File:Archimedes'-Lever.png|center|thumb]] One may abstain from admitting their existence. To do so is simply attempting the impossible, and is devoid of all reason. Real knowledge has been put into the realm of the mystical unknowable even to the point that knowledge practitioners go to the outer extremes and deny the gift of their natural senses. With their adherence to what they consider is knowledge they become captive automatons to any prescription for life that is expressed in that ’knowledge medium’, which then becomes the authority. When ’knowledge’ is addressed as having a collective source in Universal principles then we have the potential to experience its complete reality (microcosm - macrocosm) without any imaginary, or divisive comparative content. There is then a re-orientation process toward our true being and recognition of our own reality in relation to the natural processes we share. Real knowledge is elementary and Natural. We know, because that which is knowable is constantly expressed by the principles involved. We all share those principles and can correctly infer the most simple and obvious truths. All social life functions by our adherence to the implicit laws operating within them. Seeking experimental contradiction to a fact of life offers us nothing but proof which is the establishing of ‘complete knowledge’ however ‘more reasonably’ one wishes to extend the exercise. The construction of new philosophies must seek a mandate to fully explore the relationship between experience and innate knowledge as the foundation for pure knowledge to emerge. To repeat, knowledge or truth do not reside in any individual domain, nor are they the private possession of any human being. We know, because ’knowledge’ is an innate natural possession that we constantly share - otherwise civilization could not exist. == Knowledge: Evident facts about mutual standards that provide us with security. == We do not become human beings because we can ‘think’; we become more human because we learn to conform to the implicit principles in Nature and Reality. Denying them - denies our being. It is natural to know. There was no cause for knowledge to be established - it has always existed. Knowledge is an evolutionary process. Human beings developed from primitive innate instinctual knowledge to contemporary cultures. Some know more than others through experience, and make that knowledge transparent. Insistence on how we can ’completely’ know is an ephemeral philosophical question that attempts to deny that we can have ’knowledge’ at all, as you understand it. Knowing that we ’know’ the inherent completeness of everything through the existence of principles, is the natural catalyst to make ’that which is’ transparent. Heraclitus: "No man steps in the same river twice " He believed in the "Unity of Opposites (Absolutes). He cried for the needless unconsciousness of mankind.. “Exploration of a mythical dichotomy below for the purpose of establishing principles. Principles that are not a 'mind' construct, but the very essence of our being. Independence, is the curious and dangerous malady where humanity has lionized negative mythology in denial of its own reality. The human fundamental reality has at its base the simple natural law that we are dependent beings. From conception, the human embryo is entirely dependent on the health and well-being of its mother to provide it adequate healthy sustenance to enable its entry into the world. That form of innate dependability the human species carries with it throughout its spectrum of life. Every aspect of human activity is premised on the availability of air, food, and water without which the organism cannot survive (this would be an incontrovertible 'more reasonable' truism or an Absolute). In a perverse way, that which is our natural state has become the target for what appears to be open defiance of the laws that govern our behavior. When a basic premise is either used mistakenly, or deliberately, its consequences can be socially far reaching, for any deviation however far it is extended is a distortion of the truth, and a denial of who we are. The erroneous conclusion through exercising responsibility that we can confer independence to our actions has gained a distorting and ubiquitous influence which paradoxically undermines the very responsibility practiced. Within the context of being dependent we can correctly be responsible for our own actions but with the surety of knowledge that that responsibility is contingent on the measure of life giving forces available that we are dependent on. The mythical dichotomy 'independence' connotes with the myth of separateness, division, alienation, and the force of these particular myths is expressed in wars, genocide, criminality. Alarmingly the mythological term has become a residual in our lexicon and is used more widely with acclaim than its true counterpart. To uphold delusional 'independence' as a value to strive for erodes our human heritage by diverting useless energy toward a dubious goal, and consequences that leave us questioning our means of arrival. Sadly it is a loss of being with the paradoxical view that the energy expended will deliver up a personal reality. The cult of independent individuality with its mythical ideology based on personal intimacy is now taken for granted, which then passes into the acceptance of the spurious dichotomy as a tangible reality. This in turn disposes the adherents to discredit the very essence of their being, and in the process dehumanizes many cultures. The presumption of independent individuality leading hopefully to a superior future is in fact counter-productive to the purpose, and leads eventually to corrupt power, and subsequently the invention of immoral policies that continue the negative spiral, which in the end has no ethical base to extend from. The alienation from our substantial being creates inevitable tension anxiety, and the need to somehow relieve that anxiety with any artificial means available. = Responsibility. = Being responsible for any social activity would best be enacted with regard to the effect it will have (directly or indirectly) on the lives of those who are dependent on a beneficial outcome. To recognize with some significance the basic structure of our being in turn significantly increases the measure of our understanding of human relationships. Being dependent is not a mental construct choice - it is a state of being, and there can be no being-ness without at least one other being, there is then the possibility that the principle of true egalitarianism becomes the manifest reality. Being-ness can only be identified and expressed in relationship. [[File:Togetherness - Wiki Conference India - CGC - Mohali 2016-08-06 7666.JPG|center|thumb]] This is why the cult of 'independence' is eventually so socially destructive, as it creates that alienation which attempts to deny each real human need, and leads to a depravation of honesty. Human relationships between children and adults where independence is the accepted norm is severely undermined when the educational process predominantly teaches an unnatural form of living (either intentionally or unintentionally). The educational process is then reduced to the adoption of a fiction, which in turn puts at risk any educational program. The effect of interpreting mythical dichotomies as described here is symptomatically ascribed to the existence of all other principles that govern life. The construction of any ethological debate should not be premised on comparative perspectives, based on human thought, but rather on the issues that we can recognize as being universally compatible, therein lies the common denominator point of extension. The focus of attention on comparative perspectives denies justice to 'what is'. To contemplate the 'right or wrong' of any circumstance is a deviation from the truth. The correctness of any debate (however minute the finding), is the justifiable extension, and the only true trajectory we are morally obliged to travel. Truth is not defined, nor experienced by comparison, but by 'what we are'. An orange is to an orange, what an orange is to an orange. To define correctly there should be careful and disciplined action toward establishing 'true factors' that we constantly use to promote reasonable standards. = Time. = In that moment of time, we have the potential to merge on an equal basis with the reality that exists, and to know what true interaction is. That is the point of 'direct experience'. It is then that we know the truth about ourselves and the beauty of this Universe which also reveals to us the folly of our present conditioning. In that experience, it becomes very clear that all so called cerebral activity has nothing to do with reality. The fundamental repository of our knowledge and relationship with life is our being-ness, which is not located inside a mental box to be analyzed, accepted, or discarded at will, but the very privileged natural gift of being. What some scholars deal with is the appearance of life prescribed by the illusion of comparative perspective which functions on the basis of dichotomous ideology. It is in effect a denial of our humanity to conclude that all things that exist [from our perspective] exist only in the mind, that is, they are purely notional. It compounds the denial of 'what is', and an extraordinary refusal to observe transparent life. It should be noted that there is ample contemporary exposure to the Cartesian doctrine, and in this regard, I would refer you to the works of Professor Gilbert Ryle, notably his publication The Concept of Mind.<ref>{{Cite journal|date=1977|title=Professor Gilbert Ryle|url=http://dx.doi.org/10.1093/mind/lxxxvi.341.1-s|journal=Mind|volume=LXXXVI|issue=341|pages=1–s-1|doi=10.1093/mind/lxxxvi.341.1-s|issn=0026-4423}}</ref> The heuristic principle applies throughout when establishing our connectedness with reality. It is only through our contact with reality that we can discover, and equate with the mutual structure of the principles that govern all existence. Have already noted that it is also a peculiar form of arrogance that presumes that life is only a notional existence beyond the boundaries of the 'mind in a box' assertion. It would be foolish of anyone to assert that ice cream has a cold smooth satisfying texture and taste on a warm summers day unless they had actually experienced it, preferably on more than one occasion. For anyone who has never enjoyed that experience, it would be foolish of them to discount the very numerous accounts of such an experience that is available just because they had not been party to that event. From a logical point of view, given the avowed experiences of ice cream eaters, we could reasonably ascertain the validity of each experience by documenting their separate opinions. Each participant would have 'direct experience' in the consumption of ice cream, which at that point in time has the potential for that participant to experience the full measure of that factual reality. We have the natural capacity to experience coldness, smoothness, which equates with the reality that exists, and the potential for those realities to unify. It is not a question of how to get outside of our minds (mind in a box position); we are constantly outside our so-called minds performing acts of transparency throughout our entire existence. The belief that our constant engagement with reality can never be based on a rational acceptance of 'what is', is at the least, very sad. The Platonic Cave shadows are a metaphor for the (mind in a box) syndrome. [[File:An Illustration of The Allegory of the Cave, from Plato’s Republic.jpg|center|thumb]] The need to reach simple, and obvious conclusions and accept them for the reality they are provides the opportunity to engage the complete reality of the moment. It is indeed going too far beyond the reality of the moment searching for philosophical profoundness which does not exist, that fails to establish the constant principles that always operate. Pure principles are not amorphous shadowy ideals; they are represented in everything that exists. The only way we can equate our inner knowledge of reality is through direct experience of its truth. [[File:JUL Soul Iris.png|center|thumb]] Within that context then, life cannot take on a notional existence but is an existence that is very real, and that we continually share through our innate knowledge. That our so-called minds are defined by comparison - incompleteness - dualism would have extreme difficulty in pursuing the proposition that we are defined by our direct relationship with reality which is expressed in our innate ability to directly interact with 'what is'. The reality of interconnection, and interaction, are not idealistic concepts of a notional nature, but actual and consistent transparent realities. We do not live in a shadowy world that is hidden from our direct experience, but we are constantly engaged in the process of life, and we do not have the right, nor the choice, to deny it. The man whose book is filled with quotations has been said to creep along the shore of authors as if he were afraid to trust himself to the free compass of reasoning. I would rather defend such authors by a different allusion and ask whether honey is the worse for being gathered from many flowers. Anonymous, quoted in Tryon Edwards (1853) The World’s Laconics: Or, The Best Thoughts of the Best Authors. p. 232 == Create your own semantic template. == That will consist of an alphabetical list of Absolutes that are all interdependent and interconnected. Their unifying construction creates a ‘new’ consciousness meaning. That ‘meaning’ is yours specifically. The greatest knowledge you can ever have is your own! [[File:Knowledge belongs to all of us. 2030 Wikimedia.jpg|center|thumb]] That meaning also creates its own moral construction that cannot be misused . The semantic template is available to everyone, and its dissemination is our responsibility. “Consider that the language structure, concepts, and definitions now in use no longer always deliver, accurate, reasonable, and responsible information. Indeed at times, they can be quite ambiguous. The statement ‘mutual agreement’, and its physical manifestation in whatever form, is its own dialectic, and will carry within it all other principles necessary for the activity to proceed. Given the Socratic assertion that if something is true then it cannot lead to false consequences no matter how circular any argument may be. Then extrapolating the statement into extended definitions must only lead to a better understanding of the inherent truths available. This can promote recognition of the underlying essence of all things, which can become more real than our conventional understanding of Reality. There is a contemporary need to find new definitions, new paradigms to explore the concepts that govern our existence. Examples of principles and how to define them without dichotomies, just add conjunctions to create any sentence. Communication. Truth. Standard. Proof. Express. Contribute. Mutual. Direction. Advance. Comfort. Organize. Certain. Immediate. Interest. Improve. Present. Constructive. Gain. Trust. Progress. Source. Knowledge. Basic. Original Reality. Awareness.Freedom. Purpose. Connect. Understand. Support. Peace. Cause. Unity. Ability. Rights. Honest. Discover. Positive. Energy. Balance. Good. Courage. Willing.  Control. Use. Association.  Observe. Reason. Easy. Wealth. Simple. Law. Increase. Order. Flow.Co-operation. Exact. Quality. Accuracy. Strength. Responsible. Operating. Creative. Measure. Recognition. Accept. Constant. Obligation. Include. Dependence. Relationship. Value. Success. Principle. Equality. Stable. Share. Love. Sustenance. Action. Identity. Intelligence. Education. Secure. Facts. Agreement. Information. For. Rules.Clear. Yield. Example: Success = Securing facts through responsible co-operation and using constructive knowledge to develop your success. There are no dichotomies! Any principle is correctly defined by any two other principles. You create a new language. Using conjunctions you can write your own book. Where a circular argument is based on an untruth, then it cannot lead to a truth. The reverse of that is that when the truth is established, it cannot be denied. Establishing ‘mutual agreement’ as a center from which we can reach out for extended knowledge in its ever-evolving radius, is not a limitation, or a stop, it is only a beginning! When any concept is truly established the superficial exemplification ceases to dominate, and we can truly experience the apparent essence of ‘what is’. Paradoxically to resource innate knowledge, we must recognize our profound ignorance of Nature, and Reality. Completeness does not lie in individuality. This is an extreme form of monastic expectation. There can be no individuality (or completeness) unless there is at least one other individual. This is the true foundation of completeness. Whenever we are privileged to experience that instantaneous essence of one other, then we know in that moment that we experience ourselves. It is complete complementation with the knowledge paradoxically that it is an endless process. There are many paradoxes we live within that strain our conventional views of what is ‘more reasonable’. Any true relationship experience is not based on a causality premise, but on an experience that is necessarily complementation. Individuality in terms of completeness is a fundamental circular argument back to one, which in its form of denial excludes any form of reasonable argument to the contrary. It is a non sequitur, which denies the pressure of facts that are in abundance, despite the evidence of their reality. To observe ‘mutual agreement’ is looking at things as they are. True observation of ‘mutual agreement’ in action is observing essence transparency – it is knowing ‘who we are’. That form of recognition is essence duplication. The proposition that we can observe the Truth may well be the highest attainment of Realities properties, for Truth is knowledge. Consider the hypothesis of a human entity (an individual) being born in a black space with no other form of life in that environment. How could there be Agreement? How could there be Intelligence? How could there be Understanding? How could there be Recognition? How could there be Love? How could there be Law? How could there be Reason? [[File:Knowledge-Reid-Highsmith.jpeg|center|thumb]] All of the above principles are the transparent manifestation of Nature and Realities properties that are constantly evolving. They are ethical imperatives, and we have developed the positive properties of language to establish them for our use. We can only be defined through relationship principles for they offer us the best hope to recognize the factors that lead to complementation. There is a fundamental need to grasp simple common-sense essentials. The Here and Now is not a temporary transitional time phase that we move in and out of. It is a constant certainty that is essential to recognize, so that our focus of attention has a foundation. Centrism can imply a fixation, which also implies vulnerability, which can be perfectly true if it does not lead to extension. To understand who we are, it is essential that we recognize and become aware of the very principles that we operate from. They encapsulate all the measure of any human societies ethics, morals, and laws, which is a continuous evolutionary educational process within which the realization of its total essence is always available. To use the doctrine that reason is a reliable tool to discover Truth – therefore ‘mutual agreement’ in the context ‘correct information’ translates to the Truth to reason! Evolution is a constant dynamic process. The human phenomena of ‘who we are’ is only understood in our union with each other, and ‘what is’. The paradox again is that there never is any separation. Separation is a mythical non-existent. The principles that are our necessities have continuous expansion properties that as humans we are privileged to assist their propagation. [[File:Making Scholarly Articles Findable- Towards Ensuring F of FAIR Data Principles, John Samuel, WikidatCon 2021.pdf|center|thumb]] The human constellation in its evolutionary march must use these fundamental principles to ensure continuity. To maintain coherence and consistency our source is centered in the principles and factors that we have interpreted from our association with Nature, and Reality. Whatever we write that is of any consequence, or at any other time, is written with the hope that stronger interpreters than us overtake what we present. To ‘see’ Reality as we have seen, and be intoxicated by it, as we have been, will ensure its progression.” == Discovery == The consciousness whole is the sum of all its parts and experiences. As we are all on an evolutionary path, our life and knowledge hopefully develop in the right direction. The exploration took us through a plethora of data and opinions about reality from authorities on science, religion, philosophy and metaphysics. Nowhere could we find a definitive conclusive argument, or agreement, that met our needs. For us, the question came down to “Is there anything at all that provides some form of construction, and certainty?” Something that has its own inherent ethical standards. The alternative proposition to that is a nihilistic “nothingness”. A pathological proposition that makes no sense. Heraclitus’s “unity of opposites” seemed the most promising. Our understanding of that now made dichotomies a semantic illusion. If achieved in a mindful way it is the act of uniting them, providing a conscious correct experience of ‘what is’. We live our lives with secondary knowledge that everything that is – is always interconnected and interdependent. Yet our illusory experience belies that form of knowledge. It is here that we understood Heraclitus and his “unity of opposites”. Mentally uniting opposites replaces the existing illusion of their existence – there are no dichotomies! [[File:Heraclitus, Johannes Moreelse.jpg|center|thumb]] Once the illusion is gone a new solution manifests that is peculiar to the mindfulness operator, and belongs to a higher form of consciousness. Heraclitus was known as “The Weeping Philosopher”. He wept for the needless unconsciousness of mankind when the ‘unity of opposites” was always available. He was also known as Heraclitus The Obscure. A title we suspect that came about because the successful conclusion to uniting opposites and replacing the illusion, opened a door to a new dimension. Semantic description at this time may not have been available. This brings us to the ancient Yin and Yang symbol of the ‘unity of opposites’. As separate entities in Chinese philosophy, they are complementary, and in fusion they represent the whole. So as dichotomies they don’t exist. [[File:Ying and Yang Street Art - panoramio.jpg|center|thumb]] The whole is the elemental answer to any fusion of opposites, whatever that may be. Symbolize a line as being infinite in the sense that any line can be categorised as being infinite. Apply an infinite number of points in any line. Intersect any line through any point by another line then we have a specific identifiable point at the intersection, which at that point in time has an infinite quality, yet constant and complete. Any such point has Matter, Energy, Space, and Time, the epitome of the microcosm. We may locate a Reality point that establishes the Truth. Conventional mechanical ‘thought processes’ deal with dichotomies that are based on a comparative perspective ideology, and consequently, skew any real experience of that which is real. We must use correct ‘measuring sticks’ to secure proper standards, but from the point of view that there is a belief in dichotomies, it will always be a compelling argument that aspects of reality can be contradictory. The element of denial within human historical memory accumulates to establishing dichotomies as being real. We are defined not by how ‘different’ we are, but by our commonality of existence. When we locate that Reality point we will then know that the definition in itself has a whole, and complete explanation of ‘reason” in all possible senses. All the reality we can deal with is here, and now. There is no possibility that ‘infinite regress’ (an imagined reality) is any part of our immediate experience. Infinite regress through thought processes, deals with questionable imponderables. It is a descending spiral, which further removes one from reality, which only produces illusion, and correct meanings are always deferred. It is making a holy virtue out of complexity. The epitome of completeness is the active realization of the operational principle. [[File:Brick house 09.jpg|center|thumb]] Conclusion: A brick – a house. Each complete in themselves. A house is not composed of one single brick, but each brick in its composition is complete, and whole in itself in that it has matter, energy, space, and time. In that context, it is a microscopic whole which has implicit within it the macrocosmic whole, a house. We cannot define that which isn’t. We constantly use negative dichotomous terms in language, which are in essence factually indefinable, and therefore non-existent, but they are used as though we can support a view as to their existence. At this time we constantly use mythical concepts as though they had real substance. That erroneous belief in turn diminishes that which is real and compounds the problem of recognition of Reality. The flat earth society no doubt had to be persuaded of the mythical nature of their beliefs. This dictates that we must research ‘that which is” to achieve an understanding that supports that reality. Separation is the mythical measure we use in an attempt to justify the real identity of either ‘relationship’ or ‘completion’, but it has no substance in fact.That we are connected, that we are related, that we are communicating, that we agree that ‘mutual agreement’ exists, all of these factors fall into the category of ‘that which is. There is nothing that is real that is not immediately available to us, there is ‘mutual agreement’. Attempting to view true relationships as having a necessary separation link, or dichotomy is a clear misunderstanding of the nature and completeness of all that we are related to. == Connectedness. == A relationship is defined as we are by the measure of contact (especially homo sapiens) that is apparent. It would be true to say that I have a measure of relationship with everyone who reads this material. No matter how tenuous the link we have a measure of relationship with all life – we are related! Depending on the strength of that relationship defines ‘who we are’. ‘Who we are’ is not defined by any spurious separation from life, quite simply because we cannot be separated from it, we are engaged in it at every moment in time. Any attempt to establish ‘separation’ as a reality is an attempt to deny ‘who we are’, and another exercise in futility! Again artificial interioralisation of concepts or principles leads only to a denial of their external reality. We are all connected by the very simple fact that we all exist on this planet. It is a very simple axiom that all life on this planet is supported by the conditional properties this planet provides. It is also a very simple, and more reasonable axiom to conclude that no matter how tenuous the link that all life in this regard has very concrete and definitive forms of relationship. We all must breathe, we all must eat, and we all must drink, and if you need any further certainty of ‘completion’ relationships, we certainly, all must die! [[File:Universal Theories.jpg|center|thumb]] To set in qualifications from the premise that there is a ‘separate mind’ (a kind of Platonic cave) to get outside of. This premise precludes either in part or in whole the evidence and experience of Nature, and Reality, within which our beingness is located. It would all be beyond our grasp if indeed our conventional concepts of consciousness was adhered to, which in effect attempts to deny us that direct ‘relationship’ to ‘what is’, and the completeness of that experience. Knowing or being and solipsistic theories advise that we cannot truly know each others experience, nor can we penetrate others experience, but in the democratic recognition that we know and have our being in relationship, and the mutual, and natural convergence of everything there is. Homo sapiens (race, color, or creed aside) necessarily conduct themselves in ways that extend recognition, and understanding at every level, without the constant need of ‘completion’ recognition that is inherent in all our interrelated actions. The flat earth society eventually moved on to a realization that their visionary scope was shrunken, and severely limited. They were deprived of a planetary (never mind a universal) relationship that one can only imagine severely curtailed the very expansion of consciousness necessary for humanity to progress. We have evolved some little way because of our understanding of the natural relationship. The centered in the mind condition - which connotes with the separation ideology - screams to be released from that mythology, and engage itself directly in real relationship with everything that is. = Relationship. = Separation is the mythical tool we attempt to use to maintain a false continuity of an imaginary individuality that does not exist. The taking for granted conditional mythologies (the flat earth society) engage the victims in what can best be described as serious problems in recognizing the very limitations that restrict their development. We must learn to view wholes, which equates with viewing ‘completeness’. The whole is greater than the sum of its parts, but the parts are not necessarily separate conceptual parts. We can ‘see’ the whole when we are able to identify the factors that constitute their existence as a complete reality. That which is complete in Law = Agreements that produce secure and dependent outcomes. We know in essence the concept of ‘completeness’, and we demonstrate the evidence in myriad ways. Each act is a microcosm of the whole – view from the other end of the telescope! In the traffic analogy the driver, and all other drivers, conform to the law by driving off when the light turns green. There is an implicit agreement about the value of traffic laws, and traffic lights that control the flow of traffic. At that moment there is a complete relationship understanding of those values. The ‘complete’ or ‘wholesome’ activity of motorists waiting at traffic lights for the green signal to go, and they then move off, validates all the factors implicit within the properties of ‘mutual agreement’. [[File:ParkStreetTraffic2.jpg|center|thumb]] Whatever we communicate for the benefit of future generations should not be based on mythical assumptions, but should be based on necessary factors. It is ordinary life that portrays all the dignity, honor, and the complementary wonder of the human species. We are here – we are present – we are communicating. We have an obligation (which we necessarily fulfill) to make transparent the basic principles that govern our existence. That, which is factual, provides us with a correct motive for behavior, and we do a disservice to Reality when we attempt to deny it. We exist and live in a world where acts of ‘completeness’ expressed in one form as ‘mutual agreement’, are continually enacted. It is the form of expression, and continuance of processes that we constantly use to arrive at reasonable solutions, and we employ factors that are necessary to provide us with a clear, and unambiguous understanding. They motivate reasonable behavior toward activity that we can accept as being a logical process. == Natural Experience: == No form of life can exist in and of itself, it is brought into existence through a relationship with its own environment, or its species. The obvious egotistical monistic nature of oneness (if there was such an entity) could not leave any room for the realization of anything that might disturb its comfort zone. There is no real knowledge where any concentration is on the “I am” syndrome. "No man is an island, entire of itself; Every man is a piece of the continent, a part of the main." - John Donne, 'No Man Is An Island, Meditation XVII - Devotion Upon Emergent Occasion. “We are’ is an inclusive affirmation that deals with “what is’. There is no constructive dialogue, no real understanding, without a relationship. Based on personal experience, we are a distinct, and unique species born of Nature and Reality that has combined to provide us with the innate ability to recognize the very properties that created us, and utilize them through an evolutionary process toward ever-increasing transparency. That transparency can only become available through a matching process between innate knowledge, and the reality we share, a reality that is our heritage. We like all other forms of life are the product of Nature, and subject to its laws, and principles. Necessity created a language that evaluated Reality, and provided us with guidelines to emulate its constant properties. The measure, and quality of knowledge is dependent on the realization of ‘what is’. The crux of correct knowledge is to know the base that we function from. The principles that are implicit within Nature, and Reality we have now translated through the evolved language systems with symbols and definitions that we now use to share the experience. When principles are fundamentally true and recognized for what they are, misguided belief systems will evaporate. To ask what is the source of the principles we present is ipso facto to ask what is the source of Nature, and Reality, and we repeat, that is an exercise in futility, but that does not mean that we cannot recognize that which is natural to us, and express it, as best we can through language. The experience of ‘who we are’ is the Ultimate transparency that transcends all doubt, or denial. We can know with an understanding that is pure and indisputable, that is the motivational drive for evolutionary continuance. To understand who we are we must address Nature, and Reality, and ask ‘what is’ Here, and Now, with an understanding of the dichotomies that exist in language. The ‘Eureka’ moments, epiphanies, enlightenments, etc, are all evolutionary evidence of who we are, and when we can translate them into principles, and concepts, then the assertions of an Archimedes (and many others) are recognized, and properly used. Through Nature, and Reality we can establish what Truth is! Is it true to say that most people conform to the rules of the road? It would be more reasonable to assert that the answer is yes! Consequently, we can say that we have ‘Mutual agreement’, and ‘Co-operative Understanding’ as observable realities. There is no conceptual source through Time, or history where there is an end. There is only ‘beginning’. Here and Now is always the ‘beginning’, and a more reasonable platform to explore than any exploration into the past concerning our true identity. Contemporary terms like Absolute, Complete, End, we use to match our conditional understanding of ‘what is’. When we use contemporary conditional language to address concepts like Truth, Knowledge, Understanding, they are limited by the measure of our progression, but we use them all as stepping-stones. Language is a constantly evolving process. When we agree that there is ‘some certainty’ and ‘limited knowledge’ you have agreed to the concepts of certainty and knowledge as factors that are part of our natural reality. All of us function within the framework of certainty, and knowledge, to some degree. Given that we agree to their existence, these are the factors that can lead us to the experience of ‘who we are’. They are a part of us that can lead us to recognition of ‘what is’, and make a transparent reality of the very things we do on a daily basis. We do not need absolute and certain knowledge to perform everyday tasks, but those performances are structured contemporarily because of our advanced understanding of the things we do, based on our own innate reality. To honestly perceive the consistency of ‘what is’ (to be interdependent, and mutually connected) in interaction, can and does promote the visible reality of ‘who we are’. That visible evidence translates into the knowledge of our complete presence. We know with certainty that our beingness is of pure essence, and from that experience, we are obliged to formulate, as best we can, the structures that are responsible for making that transparent (witness the explosion of human progression, without the necessity in Time of experiencing fully ‘who we are’). To take a more reasonable stand please observe the multiplicity of human action where interdependence, and connection naturally proliferate. To realize that we are ‘interdependent, and mutually connected’ is the realization of a consistent fundamental truth – ‘what is’. Through identifying ‘what is’ as an internal reality we can make transparent the factors that are our natural construct. It is only through sharing this reality that we recognize it!! These factors are not based on any ideology, belief systems, or opinions; they are composed of the Reality that is available to us all. We are unique in that we have the opportunity to be able to use their value in the manner that is implicit within their structure. That use is evidence of our understanding of Reality. What could have happened without the assertion that traffic lights are a safe way to control crossroads, or the assertion that the rules of the road are beneficial for our safety? Any true experience, epiphany, enlightenment, etc, of ‘who we are’ provides fundamental, and indisputable knowledge of that Reality. Applying the recognized principles provided by Nature, and Reality consistently advances the evolutionary process, hence we have Science, Philosophy, Religion, Education, Art, and Law. When there is a Pure Realization of who ‘we are’ through relationship recognition, it is unquestionably the recognition of the encapsulated, and innate principles we all share, and there is no place for the dissolving of another Real identity through that recognition. Indeed it is a privileged insight into the epitome of purity. Nature and Reality can give us direction and guidance to our human existence, and we repeat, it is an exercise in futility to seek any cause to their beginnings. We have proffered the concept of two as a basis from which human reality can be ‘experienced’. To recognize through, and equate with the true substance, and essence of one other is to automatically experience the totality of ‘who we are’ in full measure. This does not mean that the terms ‘totality’ and ‘full measure’ convey an ‘end’ to ‘what is’. We have consistently offered recognizable facts (not assertions) that are part of our natural human activity, and give correct direction and meaning to our basic essence. We do function within the structure of ‘mutual agreement’, and we do communicate and ‘make known’ – basic obligations. These are evident simple examples of innate knowledge, and our understanding of ‘what is’ made transparent. To repeat we could not recognize anything without innate knowledge. All knowledge is a continual matching process 1 + 1 = 2. toward the realization of ‘who we are’ the development of language structures that correctly establish basic reality as it is, provides continuous knowledge that makes transparent the very nature of that reality. It is vitally important to recognize that we have active communal agreements concerning the existence of basic principles and concepts that form the very foundation of our lives. Constant change and movement in Space-Time - Energy - Matter are applicable absolutes to be recognised, which equates to evolution. There never will be a static property involved in the evolution advance. Evolution encompasses its own absolute properties to provide cognitive connection confirmation. The evolution of conceptual language exists to provide natural equality and to promote cognition between language states. No matter the languages spoken the concepts of air - food- water is the same and can provide a gateway to explore the future dynamics of human relationships. === Knowing. === Knowing what all the truth is is not some miraculous state of perception. Nor is it a high academic achievement of amassed information. It is simply an objective common-sense view of ‘what is” and in reality what must be. It is what must be for life to function within the principles that exist that are its natural foundation. We are always of necessity the living expression of a reality that must be experienced in the whole. Our recognition of the same principles operating universally is also our recognition of who we are. That proliferate ‘sameness’ is an evident easily recognizable identity. When a child is afraid of an imaginary monster in the dark, we generally do not accuse them of being absurd, or that they are lying. Appeasement comes with an explanation of reality at that level which is truthful. An explanation, which the child can grow up with, and find comfort. It is simply introducing a child to a level of truth that is more real to them. In every instance, the only reality that ever exists is truth. However, distorted it may be expressed. One of the major distortions as the result of thought processes is to consider that we can manufacture something other than the reality that exists. The ‘fact’ that you recognize contradictory or absurd statements is that behind them there is a measure of truth. The habitual liar lives in a world in which he or she imagines that truth is something they can manufacture. Where human ‘thought’ constructs its reality in terms of dichotomies it can never deal with the truth because it continually makes those comparative perception judgments. Those judgments are always in question because again they cannot deal with reality as it is. No matter how absurd or contradictory any statement is, that is the measure of truth expressed. Ergo whatever it is that is expressed, or made manifest, is the truth to some degree. Ergo everything that is, is the truth. It is our responsibility to recognize it for what it is. It may be appropriate to review previous observations on dichotomies and gradient scale. Consistently we have contended that there are no dichotomies, which then properly puts each principle into the category of an absolute. To identify ‘truth’ as an absolute in that category then everything that is must have a measure of truth. It is a very simple and sensible approach to establish ‘what is’. It is the means of identifying a reality that must have truth as a base – whatever it is, and however nonsensical it may appear. All principles have an elementary gradient scale that we must use to identify knowledge that is honest. That gradation scale knows no dichotomies. Dichotomies are always the imaginary properties of pseudo subjective reasoning. Necessary factors establish that gradient scale where only objective realities exist to furnish a healthy subjective reason with truth, and so we learn to apply the conjunction to address reality for what it is. Truth comes in an abundant variety of ways in its commonality – and therein lays its overall ‘complete’ power, despite any denial to the contrary. Embedded knowledge as we see it is neither experience nor knowledge without principle content. A person may be well educated in all aspects of the geography of a beautiful South Sea island, but have no practical experience at all of its beauty. Being clever about a subject does not necessarily equate to an understanding of the subject. Nor should it lend itself to posing as adjudicators on a proposition preset we imagine by the same adjudicators, or essentially the same school of embedded thought processes. Long before human evolution, the principle of leverage has always existed in all Nature (as have all other principles, wherever there is space – time – energy – matter). Our adaptation to the existence of principles has added to the sum total of ‘knowledge’ as we know it, including the concept of knowledge itself. That form of ‘knowledge’, and our ‘knowing’ is natural and not any personal or esoteric acquisition. Just as a fish knows what its natural habitat is, or a bird to fly in the air, the human species uses all available principles it recognizes to add to its knowledge (already said). Any valid theory of knowledge must have as its base constructive definitive principles to support it, and it is evident that our accumulated common knowledge equates to our common experience. No matter how erudite or convoluted any argument may be, if in the end it is reduced to inane observations that have no factual basis in principle, then it is time to abandon them. Do try to consider the sort of ‘mind’ processes that offers us up a world that knows nothing but separation. How can we possibly evaluate what ‘wholesomeness’ really is? How can we possibly evaluate ‘who we are’? How can there be any theory of knowledge without addressing Nature or our innate and biological relationship with it? Any attempt to debate ‘who we are’ and the completeness of that concept must have some sense of reality on the real meaning of ‘completeness’, and some understanding of the principles that are the nucleus of human society. To wrap any argument around a non-existent concept that can never be realized is apropos to attaching oneself to a system of belief in things that do not exist. One can make ponderous and convoluted statements about those beliefs but in reality, they are morally and ethically misguided. The ability to correlate correct definitions to the reality of life offers up that direct link to the truths that are common to us all. It corrects the presumptive notion that there can be ‘different’ perspectives on the same reality. There can be ‘differences’ but there can not be ‘different’ measuring sticks for the same reality. No matter the multiplicity of perspectives, they can never alter the core principle of ‘what is’. Historically evolutionary progress can best be measured by the adoption of recognized principles. Reality at whatever level we find it can only be understood by addressing ‘what is’. Nothing can be understood by attempting to relegate it to a non-entity through questionable theories of ‘knowledge’, which in essence negate the very content of knowledge itself. The perpetuation of any theory of knowledge, which cannot recognize the principles that are its foundation, can only be a shadow of its own reality. Construct the ‘necessary factors’ around the skeleton and a body will take form. If any particular theory of knowledge cannot identify simple truths, how can we possibly question how anyone ‘knows’? A dichotomy is the human attempt to deny the existence of a whole reality of a principle. We have the principle of leverage and its necessary gradient scale. Mutual agreements of a consistent reality, at a communal level, are a passive form of the Eureka moment, which recognizes fundamental principles that relates to truth. In every social structure, there are varying degrees of recognition, which determine social use. The mosaics of differences, which make up the rich pattern of life, are a testament to human creativity. Principles offer up a form of predictability in which our brain forms knowledge through the process of interaction. The experience accumulated through each moment, forms exponentially in use, or becomes transparent immediately in a Eureka moment, in which we know. Real knowledge is through the constant interaction with natural principles, much more than the transference of divided embedded information. The problem we face is that ‘knowing’ or ‘how we know” is never a personal possession. Any theory of knowledge no matter how in that respect, is true interaction. Peeling a potato and ‘knowing’ it, is rewarding enough! All principles are the repository of pure erudite knowledge. We recognize Universal principles in play at all times in Nature and its by-product – human societies. The correlation between determined interdependence to objective reality requires our intellect to ‘honestly observe ‘what is’, and assimilate that subjectively. Then the equality of the external and the internal becomes a reality and we ‘know’. Knowledge is the process of natural action, reaction, and interaction. It is nonsensical to ask how do we ‘know’.Every moment in time is complete because it must contain all the principles that form its nucleus. It can only be like that to facilitate the immediate experience of Eternity, or the wholesomeness of any of its principles. On the gradient scale of experience, we all exist somewhere on that scale. It is called life. === Relativism: === Relativism can be consistent with interconnectedness and a gradient scale of knowledge. What it cannot do is confuse the relationship that correct gradient scales of principles have in reality. Hot and cold would be on a temperature scale – no dichotomy! Leverage could only be measured by its own scale (say a child’s sea-saw to a high-rise industrial crane) – no dichotomy! The human being is a human being whether it is a child or an adult – no dichotomy! The domestic cat is the same animal species as a wild lion – no dichotomy! The domestic dog is the same animal species as a wolf – no dichotomy! How can we manage to classify these as dichotomies? Gradient scales are the natural human mechanisms used to recognize constant principles. They ensure the human perspective is aligned correctly to identify ‘what is’. The distorted human perspective is the result of human thought processes unable to establish constants that must exist in each moment of time. The problem with embedded information is that it becomes stultified and it can stifle healthy reaction. The injection of recognizable principles invigorates and brings new life that offers countless avenues for human energy to be released. More importantly, those energies are used to enhance the evolutionary process. We are collectively gifted with the potential to elevate life itself. We can correctly use such information by transforming its content so that its inherent truth is made recognizable. It would be impossible for life to function if it was composed of ‘different’ opposite realities. All theories of knowledge are in essence interconnected and can only contain validity when the principles that are the coalescent mechanisms are recognized. The unification of the truth that must exist in any theory needs to be harvested and used to offer up a body of ‘knowledge’ that has commonality of meaning. The identification of principles, truth, knowledge, and their subsequent establishment can only be achieved through direct interaction with Nature and life. Gifted with life we have an obligation to demonstrate its capacity to use every resource to sustain and nourish its own environment. We all know through the constant natural process of action – re-action – interaction. Depending on the quality of that process, knowledge will take its appropriate place on the gradient scale. That we ‘know’ is natural. It is not some extraordinary esoteric attainment, posited by a body of theories that, by their very nature, look for difficulties where none exists. Universal belief systems based on mythologies can have an entrenched view of good principles being established because of their beliefs. Indeed the perpetuation of the beliefs throughout history offers a dynamic that is counterproductive to the ‘realization’ of principles that are necessarily true. Principles used in this approximate way, paradoxically hold no real meaning, and in fact, impose unhealthy dysfunctionality. When there is a critical change toward establishing correct principles, it is axiomatic that the diffusion of mythologies becomes an automatic process. True interaction lies in the knowledge that correct action is its own reward Any other interpretation is less than tangible. The accuracy in interpreting basic principles, and the alternate knowledge implicit in the interpretation, will always establish the primary principle sought. Archimedes et al. Truth can be found in the oddest places. (Archimedes bath image here please) Archimedes cognition on how to weigh metals in water through displacement. === Truth. === Language is the construct of human action and the word “truth” seems to hold pride of place by the power of its usage and the meanings it evokes. It is preferable if we could turn our attention to the unity of principles (including truth) that are the construct of every language we use. By uniting the principal terms we can elevate the meanings we desire. Reasonable constructs and the correct duality of established principles always lead toward meaning. It is the only form of meaning that leads to its own extension eg. how to weigh metals – how else could it be? All principles have reciprocal value one to the other. No foundation principle can stand alone. They can only exist in a union, one with the other, the source of reciprocity. All absolutes are universal. There is no hierarchy beyond the meanings they evoke in their joint construction. The binary connotations, however, one may express them, provide a constant reality beyond conventional consciousness. That experience is the immediate reward through disciplined application of their use. That discipline takes the form in all human action (such as the bathing scene above) disposed toward the correct functionality of basic principles. The daily connections we make always include the distinct possibility of their recognition, when we make those connections in a mindful state. From any common sense, or ‘more reasonable’ position, it would be more productive to view reality as possessing at every level the same innate values or principles consistent with our ability to measure, or recognize them. To view reality as having ‘different’, or antagonistic properties, is simply a misguided view of ‘what is’. That form of perspective is counter productive when it attempts to establish mythical dichotomies as realities in their own right. When the realities of principles are made transparent, we can then ‘more reasonably’ make use of them to further their basic existence. Here we use reason to exemplify their necessary function, and once established it becomes (if necessary), ‘more reasonable’ to locate them in all things. ‘More reasonable’ seems to have the particular philosophical motivation, not toward simple, sensible, and reasonable evidence, but more likely toward that ‘immaterialism’ ideology, and continually seeking for an elusive protracted answer is hardly ‘more reasonable'. Since we are apparently confined to a human perspective, we must settle with the latter position: the apparent state of representation of the world. The de-materializing of any object through the practice of ‘perceptual illusion’ is an attempt to deny the reality that exists. Where perceptual illusions are concerned, innate direct communion with that which is, suspends the effect of such illusions. All the properties in a chair are recognized as the reality that exists. That is materialism. A chair does have the principles of form, design, structure, colour, substance, etc. However it is analysed – it is a quantifiable reality. === Truth and Reason. === The reason could easily be defined, and validated, as the correct application of common sense. More expressions of common sense can only endorse the completeness of any concept. A true experience of reality does not require endless explanations as to its ‘wholeness’. It just is. Truth is in reality a network of implicit principles in which it is the predominant energy in each of them. They are identifiable by their interdependent nature (see network below) not the least of which is common sense. Dictionary given definitions of ‘truth’ place it in a very common sense acceptable category. One of which is ‘accuracy of representation’. Note how the two definitions in this paragraph coalesce. The human drive toward recognizing and understanding the place of principles (constants) correlates to the energy we expend on questioning ‘who we are". The constant principles of action, reaction, interaction, are the automated natural impulse toward ‘establishing’ a human reality, and human identity. The process of evolving within that process has an egalitarian dynamic that powers it. In essence, it is a natural gift that we must accept. Each life and its identity contains all its personal experiences which can never be known to anyone else. In a sense, we can never “know” another person. Their life is sacrosanct. We can know a lot about them, and there it ends. Truth is at the top of the gradient scale that measures the veracity of all things that are complete and related and paradoxically all reality is the truth. It gets back squarely to ‘who we are” and where we exist on that scale. To view gradient scales as having no truth to their structures is denying truth itself. For the entire interconnected, interdependent network of principles, each of them has a gradient scale whereby each measure expresses truth in its own manner. All forms of leverage, from the minuscule to the lever that will move the world, are in of themselves, true and exact at that point. It is the only way we can recognize their existence, and use that complete truth at that time, to move up the scale. Time is the relative measuring stick that determines the amount of knowledge we can absorb. Consider the advanced extensions to the Archimedes principle of leverage throughout time. Network scale example. Truth Knowledge Common sense Responsibility Reason Understanding Intelligence All interdependent, and interconnected with all other principles and absolutes. No ‘thought processes’ or ‘mind’ constructs can create reality. All we can ever do to gain knowledge is ‘act’ react’ and ‘interact’ within the confines of our immediate reality. The quality of that action is determined by the nature of available information. When there is freedom from embedded thought processes, there is a natural human ability to relate to the existence of truth as it is expressed in reality, and our brain records it accordingly. Thus, the principles of civilized societies evolve. Where there are predominant belief systems, the implicit energy will naturally direct itself toward human standards that blend all ethics together. That implicit energy will find its true home in the principles it seeks. The connected strength of those principles offers sanctity of experience that demands no sacrifice. Everything that is, must of necessity, have a true comparative value (not a distorted dichotomy value) for honest recognition to be realized. which is to ‘know’. All things are relative but only within their own true scale. It is the process by which we can identify reality, as it is. Principles cannot operate on any scale practicing negative discrimination. Thus a healthy individual can be at the top of the scale and someone with various health issues can be near the bottom of the scale. But that is how Healthy they are. There can be no relativity when ‘mind’ or ‘thought processes’ believe in mythical dichotomies. When human perception is distorted by such beliefs, they create a false reality and deny access to the true state. Where there is a network of connected basic standards that are universal, then it is possible to use them and be nourished accordingly. The scale of natural human progression provides recognizable evidence that we are constantly developing. Reality is the direct and conclusive evidence of possibilities realized. Therefore, the reality is always the source of all possibilities where truth exists. When the truth is used as the universal measure of ‘what is" there can be no discord as to its accuracy. It can only measure the principles that are implicit in everything there is, its natural milieu! Truth can only deal with ‘something’, it cannot measure an imaginative negative counterpart. Truth is the constant implicit property in every universal scale of principles. Thus, reality becomes transparent. Have writ large on the value of distorted comparative perception judgments. The standard of correct knowledge always carries with it, its own appraisal. Where principles are concerned there is an obvious scale of identification (e.g., leverage and the numerous references) that is all-inclusive and provides us with evidence of its existence. We could say with some truth, that the industrial crane has more leverage than a child’s see-saw, but we cannot deny the truth at the lower level or the reason applied. Where principles are concerned, truth is not a possibility, it is a constant reality (e.g., leverage). When mythological dichotomies are recognized and established for what they are, the process of ‘ironing’ them out and experiencing their constant reality will translate into the reality, which they are, and used accordingly. To evoke that new sense of reality, the mechanisms of ‘selective immaterialism’ need to be dismantled. Where human experience is presented with something it does not understand and is unable to appreciate the principles involved, the reaction can invoke a sense of fear. That condition can be a primary breeding ground to establish a language of dichotomies and put a selective name to something it does not deserve. The diffusion of a false singular dichotomy into the natural healthy state of the common good puts responsibility into its proper place. Within the process of diffusion, there is the natural and equal absorption of our true reality. The transition between separation and inclusion will be a seamless process because it is our natural state. Objectivity and Commonsense: Explore the plethora of principles - truths - constants that are the mark and phenomena of Homo sapiens. We cannot have any doubt about our existence in this present reality. The truth of reality is and can be experienced wholly and completely by anyone at any point in time. All human progress is the result of such experiences, manifest in principles throughout time and their subsequent ‘use’ evolves exponentially. The overwhelming evidence is our reality, now. A simple analogy of objectivity and commonsense. Somewhere in our early development, someone put the ingredients of a loaf of bread together, somehow baked it, and hey presto, the first experience of a loaf of bread. It is now a form of sustenance, which feeds billions of people. We no longer need to experience that ‘truth’ that ‘knowledge’. It is unnecessary because it is subjective assimilation and the act of external and internal activity. Because ‘our’ brain functions in a manner that can identify the natural elements it exists in –space, time, energy, matter, we learn to ‘know’ and recognize ‘completely’ a child’s see-saw. Knowing is a natural evolutionary function. The quality of ‘knowing’ has its own natural determinants which of necessity contain the measure of principles required for universal recognition. An Archimedes insight (or anyone else’s) could not become a universal reality unless those determinants were in play. I know the very same way we all know – by experience via action, reaction, and interaction. Truth persists and what Archimedes experienced was true and complete. Any experience of any truth, principle, or constant can be as ‘complete’ within a grain of sand, or knowledge of a pyramid. All experience of that nature is an experience in ‘time’, and when it is the truth, we use it accordingly. There is no mechanical translation, or opinion of ‘necessary factors’ as they are constant universals. That, which is ‘complete’, is transparent universal knowledge e.g., the principle of leverage. The precision of terms must include ‘necessary factors’. Necessary factors translate into a common universal language so there is no loss of meaning. All truth – principles-constants – absolutes, that stands the test of time we use accordingly. Thus, human societies evolve, and we evolve without the necessity of having to re-experience any of the principles we recognize and establish. That simply would not be a ‘natural’ commonsense proposition and an entire waste of unnecessary energy. The truth of any principle at any point in time, and at that point in time, can be experienced completely by anyone. Whatever it may be if the principle is established – from then on, it will evolve. There are some misguided notions that ‘truth’ ‘knowledge’ and ‘experience’ have some secret value that is unattainable by ordinary experience. It is a ‘natural’ evolutionary reality that what we ‘know’ becomes useful. We have a mutual responsibility to recognize, use and honor the principles, which are the common property we share. = Human consciousness. = [[File:Lane past Coombshead Farm - geograph.org.uk - 589598.jpg|center|thumb|369x369px|"The roads half travelled when you know the way"]] [[File:Sombrero Galaxy in infrared light (Hubble Space Telescope and Spitzer Space Telescope).jpg|center|300x300px]] === The Universe === ''<big>We do not own Space</big>'' ''<big>We do not own Time</big>'' ''<big>We do not own Energy</big>'' ''<big>We do not own Matter</big>'' The human capacity to understand the question of sovereignty or ownership of Space - Time - Energy - or Matter can only be accepted when any basis of dispute includes two dispositions - human and spiritual indigenous ties throughtout history. With the evolutionary appearance of indigenous peoples throughout this planet, their way of life should make it paramount that their existence be recognised as a natural law that has providence! Their culture and way of life has its own identity in which Space - Time - Energy - Matter is expressed as they experience it. That proof also lies in the existence of caves thousands of years old, and the existence their art and culture. So the constructs defined below are a new approach to understanding the concept of ‘knowledge’ and its proper place in an evolutionary expanding universe. Knowledge acquisition requires appropriate recognition through action, reaction, interaction, in which proper perception and comsciousnessvalues are applied. That form of construction requires the dismantling of previously embedded information. This requires a new direction to formulate a sound basis from which to extend. Construction of an analytical methodology to establish a form of ‘knowledge’ that is best suited to distinguish in a contemporary reality. A reality that adequately conforms to common notions of that which is true, and can only exist without any false relation to that which does not exist. All science needs the certainty that established absolutes provide. A pragmatic construction of real knowledge would propose that all reality expresses a form of evidence or proof and that the observer and the observed contain innate properties necessary to establish a foundational agreement. That form of agreement would necessarily function on the basis that everything that is – is truth. Any other interpretation would be a disconnect from reality, and the interdependent correspondence that must exist for true recognition of any absolute. Controversy will always emerge when the discourse on bifurcation and the introduction of non-existent dichotomies are used as arguments to be explored. Pragmatism would say that the human species would need basic properties to formulate any form of reliable epistemological analysis to explain and simplify the reality that forms their existence. That reality could well be recognized initially as the absolutes of space-time - energy-matter, through innate perceptual data that corresponds to an outward structure that is constantly evolving. That which is absolute is the determinative factor in establishing the existence of truth. When a chair, is a chair, is a chair, its recognition is established when we ‘commonly’ apply subjective and objective measures to that which is truly external to that which is truly internal. That which does exist becomes ‘common’ knowledge, and accepted as being true. Thus, the distinction between a priori knowledge and a posteriori knowledge no longer exists. Where there is consensus, everything is. The formulation of absolute criteria that offers ‘simplicity’ as a tool to measure all and everything, dispenses with the confusion of ‘difficulties’ historical philosophies engage in. It also offers an observable synthesis that clarifies the confusion. Within the combined properties of those absolutes then everything potentially would be. Everything that is by that definition is original, ipso facto, everything that has no false relationship, and therefore true (no dichotomies). Given contemporary human development, it would be ‘common’ pragmatism to accept the proposition that we exist within the absolutes of space-time – energy-matter. Within those absolutes and their innate properties, it would also be pragmatic to assert that ‘everything then is’ (whatever everything may be). Pragmatism would also dictate that ‘everything that is, is its own form of truth, and must contain available constructs of meaning. Therefore knowledge and understanding would be obtainable to that common experience, and at whatever level that experience is activated it is in interdependent unison with the source. The continuing establishment of basic absolute principles (their generalities and their specifics) forms our reason. 1+1= 2 is a ‘simple’ but true universal constant generality. That form of generality is used because we recognize the specific principles of a balanced equation that adds up and makes sense. Simple generalities with their inclusive specifics form the foundation of human reason and its constant evolution. Simplicity is the bane of a ‘mind’ that must have difficulties. = Availability. = The unifying feature that makes ‘knowledge’ available to us all, are the innate universal principles in all things. Archimedes established the principle of leverage. To paraphrase - ‘give me a lever, and I will move the world’. Once the law is established it can then be put to good use. The principle of leverage is manifest in countless ways, and put to good use! The principle of leverage is a constant available to us all, and always has been. Through his application Archimedes conveyed his insight in practical terms, and made aware of the principle and the laws governing it. We now use those laws. We no longer need to philosophise on its existence as a truth. Similarly whenever 2+2 = 4.  Whatever we use to make the equation – apples, oranges, bricks, the mathematical equation is a constant, and the principle of correctness applies. Here 2+2 =4 is empirical proof that the principle of correctness and agreement exist. Principle, or law: ‘a fundamental truth used as a basis of reasoning’.           If it were otherwise we could not measure anything. That which appears abstract then, requires no implausible argument as to its non-existence. Such is the nature of all universal principles, they exist whether the notion of a ‘mind’ can perceive their existence, or not. Because ‘thought’ does not create the reality of principles – universals- truth, it cannot from its mythical standpoint understand the simplicity of a Reality where ‘everything is’, nor the simple and factual conclusion – how could it be otherwise! Where ‘everything is’ evidently encompasses the whole evolutionary dimension. It is not an ideological enclosed static that stultifies expansion of an unfolding Reality. Any pure knowledge experience that ‘everything is’ ,(quite apart from the common-sense truth of the statement) is to experience the Absolute in any immediate part of anything that exists, which establishes its own truth forever. To examine a road code of law with that knowledge, and view the actions of drivers at traffic lights, it is more than reasonable to conclude with some conviction that there is  to some degree, Agreement – Knowledge – Understanding, and Conformity to that code of law. It becomes a ‘more reasonable’ proposition within Reality to understand that that code of law is multiplied exponentially, and the principles practised,  wherever drivers, motorcars, and traffic lights exist. It is the nature of the type of knowledge we are measuring that determines the measure of reason that can be applied to any given form of Reality. We can conclude that 2+2 = 4 is a reasonable mathematical calculation that contains the principles of Agreement – Understanding – Conformity. Given the accepted knowledge of these innate principles we can with more reason apply such a calculation Universally. Knowing is agreement with ‘what is’. Knowledge is not the attempted denial of any existing reality. That is a contradiction in terms. Mechanistic observation is akin to viewing from the outside, a straw in a glass of water. The straw always looks bent, but when removed from the glass we realise it is straight. To claim an experience of that which is Absolute, is not a claim of an experience from a higher domain, or an  isolated incident – it is common-place, numbered by just how many we are. Attempting to denigrate such experience is denying the everyday actions that contain the innate principles of a constant Reality. All life functions within the constraints of the laws that are the constructs of Nature and Reality. All life is an expression of the Absolute. It is when that expression is realised, not only in an instinctive sense, but in a real sense, that we penetrate reality beyond a comparative framework of mythology. To claim that you ‘know’ intrinsically what the principle of leverage is, or the principle behind the mathematical equation 2+2=4 is to claim experience of the Absolute. The Absolute is not some abstract esoteric truth – it is that which is immediate. Whether in awareness or not, we constantly comply, to some degree, with the laws of a constant reality. Therein lies the difficulty for a comparative framework mythology – the Absolute is everything! = Basic Equation. = However much the simplicity of the equation is, it contains the properties of correctness – balance – equality – mutual identity – meaning, which in its ‘simplicity’ presages all future mathematical equations. In that universal meaning, there is particular knowledge of consistent truth. That form of ‘simple’ consistency, creates its own natural equilibrium, and its ‘usefulness’ evolves exponentially up the reality scale. Here is where we need to give proper credence to ‘that which is’. All generalities have profound and specific principles as their ‘common’ identity, which are absolute. Unless those components are recognized, both objectively and subjectively, they are reduced to a comparative value spectrum (using dichotomies) as a misguided ‘simplistic’ factor. Deductive reasoning is then deprived of all value, and leads to the inevitable spurious question ‘how do we know? Given the above criteria to establish a correct basis for knowledge that is recognizable, and of a kind that can be used universally, ‘simplicity’ can be recognized as a tool that promotes its own established formula. That which we constantly use. Everything is the truth with regard to the methodology. How that truth or generality is expressed denotes the measure of the principle that is at its core, and forms that measure of reasoning we enjoy. Fortunately, although the truth is an innate property, it is not a ‘personal’ property per se, nor is the ‘experience’ of its reality. Its natural evolution is progressive. In that progression we are in common, the beneficent recipients that ‘evidently’ conform to its constant existence. Philosophical dissertations have become a monopolistic form of opinions that always seem to presume the ‘rightness’ of difficulties in establishing the source of our being, and are unable to put in ‘simple’ terms the question of ‘who we are’. There comes with that the denial of evidence that permeates human history, which establishes the principles of our ‘common’ reality. Those opinions carry with them a colossal library of questionable erudition that becomes embedded, using questionable values to support their argument. The most distinguished opponent of such arguments (Ludwig Wittgenstein 1889-1951<ref>{{Cite book|url=http://worldcat.org/oclc/1203018418|title=Tractatus logico-philosophicus|last=author.|first=Wittgenstein, Ludwig, 1889-1951,|isbn=978-1-78527-656-9|oclc=1203018418}}</ref>) proposed that language logic was a necessary tool to dismantle the convoluted ‘mind’ propositions that have permeated the philosophical hierarchy. Those ‘mind’ propositions only served to construct meaningless concepts as to ‘who we are’. Although he gained prominence in philosophical circles, his work was directed more toward academia. In his Tractatus Logico – Philosophies he quotes: “The limits of my language mean the limits of my world - What cannot be shown cannot be said”, and “There can be no representation of the logic of facts”. Having a belief in mystical truths that were inexpressible, his statements above are indicative of the embedded language of dichotomies. His form of ‘knowledge’ ‘philosophy’ gave (without question) the concept of ‘mind’ credence to formulate its own logic to clarify its own form of reasoning. Wittgenstein, by not recognizing within the language the distortion that dichotomies create, was unable to approach the evident constructs of true meaning that lie within the interdependent relationship of absolutes, and their constant existence. Evidential reality is all there ever is. The ‘meaning’ or ‘knowledge’ that becomes evident in reality is ‘commonly’ accepted and used accordingly. To repeat, the evidential reality is all there ever is. In that regard, the exponential drive toward ‘difficulties’ amassed a historical discourse of misinformation, which is used to address the very ‘difficulties’ created by spurious value systems. In effect, dealing with ‘nothingness’. We have managed to turn ‘truth’ into a problem, into a difficulty, when the truth is simplicity itself. An oxymoron of gargantuan proportions. This contemporary malady solidifies a diversion that discounts the reality of continuous progression. Progression in which ‘common sense’ is a motivating principle that promotes human evolution. The consistency of specific principles allows us to achieve correspondence. Real knowledge is not a fabrication of convoluted prescriptions. It is the ‘coalescence’ of ‘what is’ to internal reality. That established, evolutionary progress is assured, and reality factors are recognized for what they are. True meaning can best be attained by the interaction and interdependence of natural principles and so recognized as such. That meaning which contains all the specific components of reality is experienced as ‘true knowledge’, measure by measure by anyone. That form of ‘experience’ is not a ‘mind’ process, but a very natural state of realization consistent with our level of action, reaction, and interaction. A correct and pragmatically form of knowledge-seeking foundational answers to perennial questions would seek a direct passage to our ‘commonality’, the beacon that offers guidance. Consider the quantity and quality of knowledge we all pursue that has meaning and usefulness. To posit the notion that there are no dichotomies is a cataclysmic proposition that seems nonsensical to established embedded constructs of knowledge. Constructs of knowledge that offer only confusion, and continually pose impossible questions, whose absence would provide clarity. That absence of confusion would dispel and dismantle a reality of ‘mind’, which functions on its own selection of problems. To address reality as having only absolute constructs dispels the confusion of duality and its inability to ‘use’ relativism in its proper fashion. All general absolutes contain specific principles representing facts; the essential properties that confirm reality. This reference directs the observer to observe, and go beyond the restrictions of a ‘mind’ governed by dualism, monism, or any other spurious form of philosophy that distorts the very reality it exists in. Where there are no dichotomies, all we can deal with is ‘what is’, and the logic of ‘necessary factors’ thus destroying the possible inclusion of anything described as a “paradox”. = Pure Experience. = To design a chair our brain requires to exercise the qualities and properties necessary for its manifestation e.g., strength, balance , design, functionality etc, etc. A chair, is a chair, is a chair, the product of innate knowledge.   If we did address any tentative agreement that ‘I’  is a ‘fiction’, could not our ‘conscious experience’ of that ‘fiction’ be just as fictitious. It would follow that whatever perception of Reality we experience must also be fiction. Our contention, as always, has been that ’I think - therefore I am’ by Descartes is the greater fiction for reasons already explained. To consider to whatever degree that we can function on the basis of a fictional ’I’ precludes any attempt to honestly address ’who we are’. Saying that ‘conscious perspectives’ are limited and inconsistent with apparent reality are quite correct. It then brings into question the validity of ‘conscious perspectives’ to guide us toward ‘what is’. The entangled fictional relationship between ‘mind’ ‘I’ ‘thought’ ‘consciousness’ impose formidable barriers to that which is evident. Base observations on the construct and interpretation of what ‘knowledge’ is. Human experience is limited by its mechanical interpretation of Reality, especially ‘cogito ergo sum’. If everything that is, is its own measure of Reality (the differences) then everything must be measured, at whatever level, as being that part of the whole with all principles intact, making that measure available to be experienced as the Absolute. With absolutes there is no antagonism. We cannot exist or experience anything without a Universal complementary source of identification. It is notable that within the structure of Cartesian dualism, Descartes' personal address to innate knowledge he attributed to ‘thought’ which he identified as being distinct from his body. How different Western philosophy may have been if his attribution had been toward his brain and the existence and evidence of other physical entities that functioned every bit as efficiently as he did. The premise that Descartes operated from ‘never to accept anything as true’, was simply a wrong ended approach which brought him into conflict with his passing acceptance of innate knowledge, that the idea of God was innate to his being. To view the proposition that ‘everything is true’ allows reason to seek and identify that measure of truth. No quest can be productively based on cynicism or denial, nor adherence to belief systems that separate experience, knowledge, and Reality. We have the obligation to question whatever reality has placed before us , but if we constantly deny its existence and attempt to ‘disappear’ it from our experience, then we are in danger of never experiencing that reality. Not experiencing Reality as it is, is equivalent to not experiencing ‘who we are’, and is indeed the only human source and validity of truth, although Descartes held the erroneous belief that such knowledge was independent of any experience. That belief we suspect was some form of impetus toward his ’cogito ergo sum’.                   Knowledge and experience are co-existing ‘necessary factors’ So long as anyone believes that human experience is based solely on indirect conscious interpretation (mechanical disposition), therefore any ‘knowledge’ derived from experience will be incomplete. Clearly it is the quality of ‘knowledge’ that one experiences (e.g., Archimedes) that leads to a common certainty of evidence realised through direct experience. That quality of knowledge can be available when we observe directly the activity of drivers at traffic lights with the knowledge that it is a very common activity recognized internationally. In every case we can logically pronounce the premises to be true, therefore we have a conclusion that is also true - whether that conclusion is defined as Mutual Agreement, or Common Acceptance, it does not matter - they are mutual principles. It is this form of logical knowledge of innate principles that is the precursor to knowledge of the Absolute logically defined within all reason for ‘what it is’ . Where the basic premise is true that there is ‘Mutual Agreement’ between a multiplicity of drivers at traffic lights then we can with certainty conclude that the same principles exist Universally. We can also draw concrete conclusions, and establish knowledge, that it is not ‘absolutely necessary’ to experience by observation the multiplicity of drivers conforming to their particular road code. We have already established that knowledge. Knowledge and experience are not separate philosophical theories. One cannot be without the other.. Everyone has the potential to experience the Absolute paradoxically, in part or in whole. Everything that is, must contain the properties of the Absolute, otherwise nothing could be. To ask questions about human experience based solely and inevitably on our interpretation of ‘knowledge’, and co-existing with that, its particular meaning in human existence. So long as we can only deal with our conscious interpretation as representing Reality then we derive functionally less meaning than we are entitled to. When we see other humans consistently using levers to open crates then we can recognize a ‘social intelligence’ operating which equates to understanding that is not based on opinion, but is a clear expression of human activity that has correspondence. All of the principles involved in that experience can coalesce to provide that form of Reality that requires no interpretation. It becomes recognizable knowledge. How we understand that knowledge is through the realisation and identification of the principles involved, which become immediately transparent. The Absolute could be categorised as a knowledge experience that encompasses all and everything. Whatever is manifest is that measure (complete in itself) of the Whole with all its principles intact. Where there are at least two actions that are identical we can reach a common-sense conclusion that a definitive principle is operating. When that corresponds with innate knowledge then we have the complete cycle. The definition itself is language opening the door to an experience of Reality. No one can know in isolation. An imaginary ‘I’ restricts any experience of who “we are”,  and is not a necessary part of human experience.   Explaining experience beyond imaginary thought processes requires a definitive language that deals with the principles of Reality itself. Pure experience. The world-wide disposition that has no grounding in Nature and Reality becomes captive to any mythical fear that  offers a target to give some form of direction or stability. There is nothing more simple than to make Reality transparent - its evidence abounds. We can pronounce the principle properties that provide guidelines to its existence whereby the reality is made apparent. = Expansion. = To address concerns on ‘negativity’. Negativity is in essence the inability to establish a measure of Reality. Mechanistic processes of denial are the attempt to understand and make transparent that which is apparently unexplainable, and resolve a condition whose energy is driven toward finding that core of affirmation. The evolutionary principle from all available evidence is that human beings as a species progress. This seems a paradoxical contradiction to the embedded proposition that we can never know the ‘truth’. The consequence of such a traditional premise is that denial and negativity both hold paramount positions. We are conditioned to accept the premise that there is in fact no premise that will enable us to go beyond presently accepted norms of experience. We are conditioned to accept that the ‘truth’ is inexpressible. The evolution of the human species is constantly subject to contemporary ingrained social habits, which give some kind of credence to that particular point of existence. Indirect conscious interpretation classifies itself as a solid perspective to govern and justify human activity, which in many historical ways has proved disastrous.                   Our continued intention is to expose detrimental barriers to the realisation of ‘who we are’, and in that process establish a smoother, more realistic approach to ‘who we are’. The Archimedes legacy. When we establish knowledge of something that exists through a multiplicity of experience and evidence, then from every reasonable standard we can establish that it is true, ergo that which is true is Absolute. The principle of leverage is well grounded in social intelligence, and our natural knowledge of that does not need erudite explanations of its presence, nor any ‘conscious interpretation’ to realise its existence, or its practice. Evolution eventually removes restrictive passages to direct experience, the very purpose of evolution. The principle of leverage is not a matter of opinion, it is the realisation of actuality and our continued ‘more reasonable’ response each time the principle is applied. Children learn to speak their language primarily through experience without any direct, or indirect conscious interpretation, and so, universally we ‘know’ the most powerful means to communicate. Were we to move 50 miles in any compass direction from the town we live in, there is a certain predictability that we will meet others who speak the same English language that we do. If in that experience we find that these premises we have drawn about our travels were true, then the conclusion we would come to in particular, is that when we communicate we make known what we know. …………………………………………………………. '''Please note the date:''' '''Oct 2005.''' I am offering up this older material below to provide insight as to the progression of this work. There may well be some duplication to date. During this period my wife and I worked in collaboration to ensure an equality of experience. = Stepping Stones 1. = There is nothing other than what is – there is no hidden Reality that we need to seek, it embraces us at every turn. The Archimedes experience is the pure experience of Ultimate Reality, which provides indisputable knowledge. Reality is the source of complete knowledge, it is the constant source that has provided us with all human development, from the writings of William Shakespeare, to the computer development of Bill Gates. What they have produced is now an evident part of our reality that we can engage in. We can experience ‘mutual agreement’ through epiphanies, insights, enlightenment , understanding, Eureka moments etc, they are all one and the same. Implicit within the macrocosm is the microcosm – it cannot be otherwise. The more we conform within the microcosm the more we begin to appreciate that Reality contains everything, and that we can realise through experience its manifestation. Each Eureka moment is that personal point of experience that connects us with the Truth. The principle of leverage was always available, it took an Archimedes to explain it to us. Each Eureka moment necessarily engages with the reality of complete knowledge, and utilises its share at that time. When we have complete knowledge of who we are in that personal moment, then we understand that these, egalitarian properties, are rightfully shared by everyone and that we have experienced that which is infinite. It does not mean that the process of evolution is over – it has only just begun. It does mean that we can no longer continue coasting through this existence in a near comatose state. If there is a hypnotic fixation in holding the principles of Reality as being separate, and different, then the potential realisation of their immediate unity, and communion, becomes problematic. Knowledge, and experience are one and the same – they are not different!! Experience = Immediate knowledge of basic reality that is factually correct, and that we can reasonably use. Knowledge = Immediate experience of secure, and accurate information that is constantly stable, and sustains principles. Reality = Complete Knowledge. As the microcosmic part of the total macrocosm we are immersed in reality. The real question should be, ‘how can one not know Reality, or ‘who we are’. Mutual agreement is evident when we know we can go to the bank, and deal with money transactions. Mutual agreement is evident when we know we can go to the supermarket and exchange money for goods. Mutual agreement is evident when we know we can send our children to school to enhance their education. Evident proof is validation of what is – it is not a matter of anyone’s opinion, nor is it an assumption of ours. Neither do we assume, or offer any opinion, on the Universal Reality that there is ‘mutual agreement’ that we need air, food, and water to sustain us. Evident proof is also the basis for the mechanics toward realisation of ‘complete knowledge ‘ of who we are. Reality can be realised through concentration on its basic principles. We use language to express our understanding of who we are.It is relatively easy, it is reasonable, and it is responsible. We convey through language our measure of intelligence, and to the best of our ability conform to the basic rule of communication – ‘we make known’ Implicit within that exercise is ‘mutual agreement’. We may differ in some specifics, but we meet the basic obligation of communication – ‘we make known’, and always we progress to some degree. Simultaneity is one of the constant principles that we all share and they come from Here, Now, the Present, where they have always been. Everything is. Our being is always engaged in the present, and we each have an obligation to understand our relationship to what is. The present is the only point of contact we can ever have with Reality. To some degree or another, each one of us is directly connected to Reality (we do not have any choice in the matter), and we can potentially evaluate ‘what is’ through the utilisation, and examination of factual reality. We are the microcosmic part of that Universal Macrocosm, and because we already have that innate information it is a matching process when we have a Eureka moment, an epiphany, an understanding beyond question. Nothing enters our minds - we already know! Everyone has innate knowledge of the principle of leverage. It requires correct examination of ‘what is’ for realisation to occur. It is then a relief to have ‘mutual agreement’ on the things we would wish to make transparent to others. To use a traffic analogy, it is evident that there is ‘en masse’ mutual agreement when we know to drive off when the traffic light turns green. Mutual agreement is translated into people obeying traffic rules (otherwise chaos). Two cars, two drivers, sitting directly alongside each other at traffic lights, discuss their understanding of their Road Code in this particular position, and what they should do. When the light turns green there are a myriad of principles that apply when they drive off simultaneously. They have both demonstrated their ‘complete knowledge’ of the significance of the green light from this perspective. There is Mutual Agreement. There is Predictable Conformity. There is Common Ground. Each one complements the other. They are both right. One more remove: From an outsider’s point of view – they both know! The green light could be categorised as a Eureka moment, it sets in play all the above principles, whether the drivers are aware of it or not. From the perspective of two outside objective observers who know the traffic rules, if asked, did the two drivers at the lights obey the rules – the answer would be yes, there would be mutual agreement. They have complete knowledge of this particular circumstance concerning drivers, and green lights. Could it be that certain schools of thought are curtailed by a questionable refusal to recognize what is, and have a preference for creating a difficulty where none exists! No one can examine what isn’t! There is no such thing as ‘nothing’. Something is – what is it? A Scottish engineer functions on the same principles as an Italian Pope. Because Archimedes was prominent as a mathematician, his realisation of the principle of leverage, and his understanding of the difference in water displacement between silver and gold was widely reported. This does not mean that realisation of ‘what is’ is an exclusive experience. As said previously Archimedes did not realise something new – it has always existed, and all forms of life would have utilised the leverage principle to some degree or another (watch a bird build a nest). At that time there were probably many thousands of people who had some understanding of the principle, but Archimedes was the one who made statements about it. As in any Eureka moment, we can experience infinity and who we are. It is mutual agreement (an understanding) between the part, and the whole. It is when the principles are in unison Eureka! Reality is there to be examined, and experienced, it is not separate from us, nor should we try to make it so. = Stepping stones 2. = Knowledge is not conditional by the activities of what may be called ‘thought’ or ‘consciousness processes’. Real knowledge is that which is available to all, and to be shared by all. It cannot be contained by the ‘experiencer’ and then not ‘known’ by the accident of experience. It is the actual innate experience itself which conclusively establishes the truth. It can only deal in the truth which is its modus operandi of dissemination. For me to say that ‘everything is’, is a statement of fact which cannot be denied, and an intellectual dishonesty to attempt to deny the evidence by philosophical machinations. Hostility toward the truth leads inevitably toward attempted negation - looking for nothingness! ‘Being here’ demands its own recognition - attempting to deny it is simply perverse. Knowledge is the realisation of ‘what is’.. Rene Descartes ‘I think - therefore I am ‘did no service to human evolution, or education. It established in Western societies especially, the culture of individualism, with the precursor that so-called ‘thought’ was the inward evidence for existence, and for the following unfortunate claim that we have a ‘mind’, or to use the euphemism, a soul! Experience is true knowledge. When that experience marries up with its innate counterpart then recognition is realised (cognition). In simple terms, a light goes on in the brain. There can be no real knowledge without truth. All thought qualifies experience and attempts to reduce truth to near nothingness which is a widespread conditional activity. We cannot manufacture knowledge, or the principles which are its properties. No matter the amount of correct information anyone can ingest, it does not become knowledge until there is tripartite coalescence between inherent knowledge - ingested correct information - and ‘what is’. Then we truly recognize that which is Absolute. Within Nature we have the distinct privilege of evolving in a Universe that can only recognize the attributes of social cohesion. Knowledge is not anyone’s personal possession. Whatever measure of experience we may have of it, it is only available as a Universal sharing experience to be beneficially used. Human activity whereby we witness people using tools for leverage, or drivers at traffic lights obeying the rules of the road, are observable markers that contain the properties for understanding our own reality. Unless seen for what they are, they are only mechanical platitudes with an equally mechanical response. We could rightly claim that that at least is some response, but of no real value. The natural process that operates when we see that which is innate, overrides any erudite explanation from an academic base however intellectual its original source. ‘Thinking’ for oneself cannot make judgments about a ‘natural’ experience. When we see human duplicate functions in operation then we are in communion, and at another level we recognize who we are. When we actively see the activities of the human brain in action we are not dealing with any internal ‘will - o’ - the wisp’ that no one can ever experience. We exercise that prerogative (human activity) at every moment in time, but quite apparently without that focus of attention that denotes realistic recognition. To seek identity in sectarian, or secular belief systems to overcome the contemporary feeling of loss of identity leads to the acceptance of anything that offers some form of stability. That is then used to strengthen that which is euphemistically addressed as the ’self’. To retain that security the acceptance of information transmitted throughout generations, is absorbed into the culture, and defended to the death against those who would question that belief system. The greatest knowledge we can ever have is our own and it has the potential to transcend all else and provide insight into infinity. The most tragic human condition is the lack of experience of identity in a multiplicity of identities in which we all share. The real problem is not one of ‘identity’, but a lack of ’communion’. Whether we like it or not, whether we are aware of it or not, the principle of ’communion’ must always exist to some degree for evolution to proceed. It is within the experience of that principle that we understand the fallaciousness of that much heralded ‘self’ which draws down so much energy in an attempt to establish itself as a reality. Within positive language structure possibilities (no dichotomies), there should be the disposition toward the realisation that our relationships to cognize into ‘communion’ must be addressed as specifically dependent. Social attempts to be ‘independent’ are the very remove from reality and signify reduction attempts toward nothingness. Adherence to, and the cultivation of faith and belief systems give little elbow room for any factual occurrence to be anything other than a comparison to the myths that are held. The cultural and educational socialisation of generations of children must carry with it, its historical belief systems that overwhelm the natural instincts. Observe an animal out of its natural habitat and locked in a cage for its entire life. It would be a salutary exercise if we could dispense with the term ‘mind’ from our vocabulary and magnify the use of the word brain to promote a realistic discussion on ‘who we are’. My action of levering open a wooden crate and knowledge of it is one and the same. Our remarkable brain functions like that, the purpose of a brain, the natural repository of innate knowledge. The assertion of principles is critical to avoid all activity being submerged by questioning their very existence, and being unable to see directly. It would be a rarity today, if anyone using a lever to pry open a wooden crate would have the same enormity of experience that Archimedes had, nor the need to make pronouncements about it. It has all been done prior to our awareness of its value with the accompanying data attached. Our brain knows the value of a lever and activates our body accordingly when needed. It could be categorised as evolutionary transmission. The observance of someone prying open a crate with a lever, or drivers conforming to the road code at traffic lights, is a function of the brain in action, not a mythical entity in a singular locality that denies its own senses. When the brain is not burdened by distorted belief systems it then has the potential to experience ’that which is’, which is always constant. When we understand the function of a lever, or the presence of traffic lights, then we can activate the principles involved because we already know how! The negative impact in the use of dichotomies in language lies in their distraction from the truth, as our brain processes the words we use in relation to Reality. The tendency to attempt to separate inherent truths through the words we use disrupts that natural correspondence necessary for identification. A chair, is a chair, is a chair. = Stepping stones 3. = Where principles are concerned the constituent linkages in language are identity markers to that which is real - reference points. Without dichotomies there is no separation, or ambiguity between what we experience, and ‘what is’. Philosophy in its attempt to address something through denial is an elementary confusion. To say that that is a chair, and then attempt to deny it invoking philosophical theorems concerning the human ability to experience it, is a severe contradiction on the existence of the object , and the observer. When this form of contradiction is then taken as a constant, it then precludes any common-sense and definitive answer to the existence of a chair. '''For philosophers, George Orwell’s ’to see what is in front of one’s nose needs a constant struggle’ would be apt.''' Real concepts cannot exist in any mythology, therefore all that we experience is inevitably the truth that is there to beproperly categorised for what it is. The proper use of language in this context will identify whatever it is to correspond with present reality. Misuse of language (dichotomies and mythologies) leads only to the acceptance of a fractured state where nothing is whole and represents confusion. The dissipation of the supposed problem is never realised. Fiction has been elevated to the status of an accepted reality. Very early evolutionary physical dangers allowed the development of fictions that offered some form of imaginary protection beyond limited physical ability. That contemporary humanity endorses the mythology of ‘I’ is testament to the psychological fear that still exists and requires its proper recognition. Emphasis must be placed in the relationship between language and reality for understanding to proceed. The persistence of dichotomies has their own persistent confusion which then promotes a false reality through misleading information. Microcosm and macrocosm are one and the same in a Universe where ‘everything is’. Isolated viewpoints are exactly that, and are unable to view the expanse in which we are encompassed. We must learn to view reality through both ends of the same telescope. When we understand the extensive scope of ‘truth’, then we know that its values and properties do not change - which relates to ‘completeness’. Philosophical, ideological, and intellectual endeavour , try to shape the structure of ‘what is’ based on pre-dispositional knowledge, which can only ask the same questions, and look for the same answers. Not to experience that which is absolute or whole is the normal result of the confusion of language which has no correspondence to that which is real. To discuss with a philosopher the possibility that ’mind’ per se does not exist, and to dissolve it as a concept would place them in a realistic position, would indeed be a difficult proposition. The strength of that difficulty lies in another imaginary concept, that that ‘mind’ represents ’I’, and it is anathema to that fiction to consider its own demise! = Stepping stones 4. = There are no dichotomies. Everything is, and everything that is, is complete, everything is an Absolute complete Reality. You are experiencing your measure of that reality. It cannot be otherwise that you are experiencing that measure of completeness. When we come to terms with it we have the innate capacity to see the Absolute in a grain of sand. That is knowledge. Belief in dichotomies is the mythical barrier to that particular experience - which is only denial, supported by erudite protestations that human construct dichotomies exist. At a mechanical level Intelligence and Stupidity appear to be separate identifiable conditions, and they appear to be antagonistic. Stupidity is in Reality a measure of the Intelligence which is always constant. If someone was in a state of mythical utter and complete stupidity we would not attempt any form of emancipation from that condition. We know that that is misguided and proceed with techniques to advance intelligence. Consider the proposition that there are no dichotomies, and within that possibility all questions become irrelevant. Presuming that there are no dichotomies allows the process of establishing ‘necessary factors’ to proceed, and allows each measure of wholesomeness to be realised. Experience is the criteria for knowledge. Some Reality experiences were simply transposed into particular belief systems and elevated into a pseudo spiritual dimension, or a philosophical conundrum. Where there is a belief in a divisive fiction (dichotomies) there is automatic mechanistic restriction to that which is Real. There is a capacity beyond ego and intellect which can commune with ‘what is’, and recognize its properties. Reality is constant. Within the accepted comparative framework there is the view of principles as having different divisive categories e.g., as above, Intelligence and Stupidity, and classify them within ‘thought’ structure as dichotomies and give credence to them as being an antagonistic reality. The consequence of that, is, that one is always a remove from recognizing the structural properties of immediate existence. Any construct of knowledge necessary to evaluate ’what is’ will address the properties (principles) that are the constituent constant markers available in that which is the microcosm and the macrocosm. That identity (the Absolute) is found in any sphere of Reality.Everything is - and everything that is, must be experienced for what it is, and not for what anyone denies it to be. There is no mythical human construction that can deny ’what is”. Everything is - without dichotomies. To repeat, we do not have the ability to create ‘nothingness’ - ‘that which is’ has no imaginary comparative human construct. To attempt to deal with such constructs, and give credence to them is always the denial of ‘what is’, and adherence to ‘thought’ processes whose only purpose is to cement that activity. Indeed realising that the concepts of dichotomies are human mythical constructs, denying true perspective, is the beginning of insight. The dissipation of such processes through addressing the principles of Reality allows us the potential to experience directly ‘what is’, in simple terms -the truth! Intelligence is a ‘necessary factor’. Addressing stupidity is a denial of reality at whatever level we find it. Intelligence and Stupidity are not antagonistic, they are one and the same principle with measurable degrees of existence. Only from a comparative framework standpoint is credence given to any mythical form. The above observation is not negating the process, it is questioning the markers which evolve into imaginary separation (trapped in a comparative framework mythology). That particular process can and does create a false mythical reality that appears divisive. We cannot exist within a divisive reality! Reality must be complete for us to recognize its existence. Where there are no dichotomies within the premise that ‘everything is’, there exists no antagonistic position. The distinction between human constructs of positive and negative are matters of mythical perspective wherein no experience of the Absolute is available. It is because the human ’mind’ per se places its own construction on its immediate experience, and must have its particular interpretation based on what it considers ’knowledge’. There is a difference between ’mechanical knowledge’, and ’pure knowledge’. From the mechanical knowledge standpoint which can only deal ‘in indirect conscious interpretation’, it is quite correct to say that that form of knowledge is incomplete, and it always will be. Pure knowledge experienced via our brain knows no separation, nor antagonism, and is responsible for our ability to recognize the actions of others who may pry open wooden crates with a lever, or drive off uniformly at traffic lights. Within that cohesive activity it precludes ’a matter of opinion’ and by themselves can become subjects of a pure knowledge experience. To repeat, it is a form of ’communion’ with ’what is’, and available to all. Where drivers at traffic lights universally conform to their particular road code, and where universally there is a language which identifies their activity as Mutual Agreement, or any other logical definition, we can concur with the common-sense conclusion that we have universally established that within language and common activity, there is indeed a truth formed. The coalescence between universal language and universal activity are the logical constructs that create civilizations. There is a vast social network of common activity that solidifies the logic into an honest and persuasive conclusion that confirms innate common principles –knowledge. = Stepping stones 5. = The Art of making sense of everything. How to understand principles. # Principle. A fundamental truth or proposition that serves as the foundation for a system of belief or behaviour or for a chain of reasoning. # All principles are interdependent, interconnected, and infinite. # Each one is dependent on the other two. Examples of a principles template and how to define them without dichotomies. Communication. Truth. Standard. Proof. Express. Contribute. Mutual. Direction. Advance. Comfort. Organize. Certain. Immediate. Interest. Improve. Present. Constructive. Gain. Trust. Progress. Source. Knowledge. Basic. Original Reality. Awareness.Freedom. Purpose. Connect. Understand. Support. Peace. Cause. Unity. Ability. Rights. Honest. Discover. Positive. Energy. Balance. Good. Courage. Willing.  Control. Use. Association.  Observe. Reason. Easy. Wealth. Simple. Law. Increase. Order. Flow.Co-operation. Exact. Quality. Accuracy. Strength. Responsible. Operating. Creative. Measure. Recognition. Accept. Constant. Obligation. Include. Dependence. Relationship. Value. Success. Principle. Equality. Stable. Share. Love. Sustenance. Action. Identity. Intelligence. Education. Secure. Facts. Agreement. Information. For. Rules.Clear. Yield. Example: Success = Securing facts                = Responsible co-operation               = Constructive knowledge So success by definition is : Securing facts through constructive knowledge and cooperating  responsibly. All definitions of success from your template are infinite. You will find your own suitable definition. ==== There are no dichotomies! ==== Any principle is correctly defined by any two other principles. You create a new language of Absolutes. Using conjunctions you can write your own book. The man whose book is filled with quotations has been said to creep along the shore of authors as if he were afraid to trust himself to the free compass of reasoning. I would rather defend such authors by a different allusion and ask whether honey is the worse for being gathered from many flowers. Anonymous, quoted in Tryon Edwards (1853) The World’s Laconics: Or, The Best Thoughts of the Best Authors. p. 232 Amen to that! “One is not born, but rather becomes a woman” Simone de Beauvoir. “Time does not change us. It just unfolds us” Max Frisch. We experience ourselves our thoughts and feelings as something separate from the rest. A kind of optical delusion of consciousness. This delusion is a kind of prison for us, restricting us to our personal desires and to affection for a few persons nearest to us. Albert Einstein, in One Home, One Family, One Future. = Who we are. =   It is notable that within the structure of Cartesian dualism, Descartes' personal address to innate knowledge he attributed to ‘thought’ which he identified as being distinct from his body. How different Western philosophy may have been if his attribution had been toward his brain and the existence and evidence of other physical entities that functioned every bit as efficiently as he did. The premise that Descartes operated from ‘never to accept anything as true’, was simply a wrong ended approach which brought him into conflict with his passing acceptance of innate knowledge, that the idea of God was innate to his being. To view the proposition that ‘everything is true’ allows reason to seek and identify that measure of truth. No quest can be productively based on cynicism or denial, nor adherence to belief systems that separate experience, knowledge, and Reality. We have the obligation to question whatever reality has placed before us , but if we constantly deny its existence and attempt to ‘disappear’ it from our experience, then we are in danger of never experiencing that reality for what it is..   For anyone to say that ‘everything is’ is a simple linguistic absolute that no amount of ‘more reasonable’ requests (above) can deny. Those requests only appear to be governed by the difficulties of ‘mental complexities’, and embedded ideologies. To accept that ‘everything is’ as an absolute, is a realistic basis to establish any reality, and comprises the basis for reason to be activated. If there is ‘nothing’, nothing can be achieved. Within that which is Absolute there are no dichotomies. Therefore there are no antagonistic positions available. Everything that is, is a measure of the Absolute.  We are always in the present, everyone and everything. Instant elementary ‘knowledge’ which we all share, and must admit to. In being alive, we do not have the ability to not be here, and we do not have the ability to not know! To have a problem in addressing what Truth and Knowledge are, to the point of denying their existence, then that problem exists well below the scale of Reality. To repeat knowledge is not the proprietary right of any individual, it is enshrined in the principle of agreement that we mutually exercise to establish its own reality. The reality of experience is not, nor ever will be, a personal possession that we can have and hold. Its reality becomes more alive when we see the same activity being practised by others. Then we know we are sharing that reality, and that experience. We cannot "have" the principles that exist, but when we undrestand them then we are obliged to use them with integrity. That form of integrity in any language, is an added foundation stone to any belief system Mechanistic ‘I’ has no concept or understanding of ‘pure knowledge’. Only when we break free of the myth that some clarity becomes apparent, and we have the opportunity to engage with what is real. Philosophy it appears to me is constrained by individual ‘thought’ processes, which (without experience) cannot escape from that individuality. Those ‘thought’ processes conjure up a human history of inflexible, and impossible propositions which only serve to protect that individuality. Descartes ‘cogito ergo sum’ has compounded the difficulties by strengthening the incorrect premise of a false individuality. ‘I’ is a phantom consciousness much like a phantom pain experienced after a limb is amputated. The brain registers the pain signifying that something should be there. Likewise our brain has that same relationship with Nature and Reality. It is analogous to our brain dealing with a ‘phantom reality’ knowing that something is missing but is continuing to evolve to establish the whole. There exists a ‘phantom chasm’ between our brain and Reality and an understanding of its properties. We are robbed of real meaning. Evidence, recognition, and the truth are the principles it uses to reform. Within their structure is the meaning of reforms. Nature does not impose any morality on us, the principles implied in morality are there for us to understand and use. Our brain has the capacity, once reality is correctly examined, to recognize ‘that which is’. Once realised it becomes embedded. That ‘phantom consciousness’ is an experience removed from its proper environment. It takes its proper place when we experience reality for ‘what it is’, which provides the totality of meaning. True experience allows us entry to the quality of knowledge that is a continuous reality. So long as anyone believes that human experience is based solely on indirect conscious interpretation (mechanical disposition), any ‘knowledge’ derived from that experience will be incomplete. That form of philosophical negative conclusion can come down to not believing that Reality exists (a chair is not a chair, is not a chair etc,), or that our experience of ourselves and others is real, and discount any other form of knowledge that threatens that belief. There is an intellectual dishonesty in denying the existence of principles. Hostility towards the truth leads inevitably to negativity. Being here demands, not denial, but the right to be recognized. We are the recipients of a ubiquitous communication system – making known. We can only understand that which we know. We make known all the time. Knowledge of Reality – Truth – the Absolute is a collective inclusive experience of the principles we share, and never the property of any individual. To ‘know’ ‘who we are’ is an inclusive experience of the principles involved. Never ‘cogito ergo sum’.   Philosophers in investigating the nature of knowledge and the Universe, firmly established for themselves that the source of reason and logic was located in a mythical concept ‘the mind’. From the wrong basis evolved elaborate and metaphysical constructions which removed the investigations further, and further, from the truth. To comprehend the material world, and give it credibility, the recognition of implicit principles is paramount. We need to construct a language that provides that form of recognition. Any philosophical theory of ‘mind’ that will deny the evident structure of solid objects is misguided by the injection of a mythical entity (mind) that determines that seeing solid objects is a ‘perceptual illusion’. That form of determination is singularly narcissistic, empowered by the self-induced threat that venturing into a ‘materialistic’ world is a loss of that illusory self, and all the belief systems it has constructed to protect it. That erroneous established view that not addressing ‘materialism’ as a profound Reality, and as only a ‘perceptual illusion, is compounded by the belief that that form of illusion is implicit in every human view available. We cannot manufacture knowledge that leads to a mechanistic understanding of ‘what is’ , nor the principles which are its properties – however much dogma is practised. We can only aspire to relate to ‘necessary factors’ that are the implicit fundamentals of existence. For me to use a lever to open a crate is a form of communion with Archimedes through the principle he enunciated. It is now not ‘necessary’ for me to go through the same experience as Archimedes to establish that ‘necessary factor’ or ‘principle’. It is now common-place, and common-sense to utilise the principle. The extract below provides some explanation of the brain processes in action Universally, and coincides with any reasoning on the observance of the leverage principle, and the actions of motorists conforming to the Road Code wherever traffic lights exist. ''The right-to-left shift of mental control looked increasingly like a universal phenomenon, capturing the essence of every learning process on every time scale, from hours to years. An individual faced with a truly novel situation or problem tackles it mostly with the right hemisphere. But once the situation becomes familiar and is mastered, the dominant role of the left hemisphere becomes evident. It looked like the empowering patterns capturing the essence of the situations (or rather the whole class of similar situations) were, once formed, stored in the left hemisphere. (The Wisdom Paradox. Professor Elkhonon Goldberg. P202)<ref>{{Cite journal|last=Vandermeulen|first=Jo|date=2008-08|title=Verstand komt met de jaren|url=http://dx.doi.org/10.1007/bf03077135|journal=Neuropraxis|volume=12|issue=4|pages=137–139|doi=10.1007/bf03077135|issn=1387-5817}}</ref>'' = Limitations. = However limited our view of connectedness is, or however tenuous the reality our experience is, ‘everything is’, and everything is connected. Innate knowledge and the fundamental nature of Man is the prior source of knowledge that seeks and identifies that connectedness. Attempting to address what we don’t know is that mythical infinite regress toward that parallel mythical ‘nothingness’. To always address what we know establishes Reality. To establish knowledge of principles, start from ‘we are here’. The natural principles within the diversity of human culture and activity when recognized as mutual ‘necessary factors’ will have the effect of enhancing and directing vital energy toward the very process of  the communion we seek, and the gradual penetration of a reality that always exists. Knowledge of Reality is not ‘different’ in other locales. The fundamental principles are the same. Addressing ‘what is’ instead of denying ‘what is’ is the basic construction of real knowledge. Within the structure of the Absolute we are all the same with a magnificent differential in our expression of the principles of necessity. That expression is our ongoing effort toward its own experience which gives it life and meaning. That experience in turn exposes us to an immediate Reality that is in communion with the fundamental structure of our being. All that we can contribute toward that is 50%, the other half is in our momentary relationship with Reality – then we know! That form of knowledge is always available through that form of experience, and it always comes in the form of confirmation which reforms. Until that experience our prior condition appears mechanistic, without direction, or understanding. Reality, life, is not mechanistic. We are the recipients of innate principles with the constant potential to experience those principles in action (Archimedes et al). Dogmas, ideologies, are the restrictive practices used to blur the recognition of principles operating to a level that understanding of that common and constant activity is virtually denied. Our natural capital (principles) is degraded to the point that their factual evidence is reduced even to the point that they are categorised as a ‘perceptual illusion”. We can trust facts 2+2=4. Simplicity has its own majesty. Anything circumscribed by reason requires control of our emotions. To comprehend the material world, bring it alive, and give it credibility, the recognition of its implicit principles is paramount. We need to construct the language that provides evidence of that Reality. Any philosophical theory of ‘mind’ that will deny the evident structure of solid objects, is misinformed by the injection of that mythical entity (mind) that determines that seeing solid objects is a ‘perceptual illusion’. That form of determination is singularly narcissistic, empowered by the self-induced threat that venturing into a ‘materialistic’ world is a loss of that illusory self, and all the belief systems it has constructed to protect it. The erroneous established view that not addressing ‘materialism’ as a profound reality, and as only a ‘perceptual illusion’, is paradoxically compounded by the belief that that form of illusion is implicit in every other human view available, thus it then makes its own sense, form and justification to the illusion! The evident question we must ask, ‘how does a ‘mind’ conclude that ‘immaterialism’ exists universally? Surely it is a simple but massive contradiction in terms. If there is nothing there but ‘perceptual illusion’, how can you attribute it to other ‘minds’. The oxymoronic effect of narcissism is that it is the very denial of ‘who we are’. Man is not composed of an overwhelming self-love. That mythical embedded belief cannot consider the possibility of underlying principles that are the real life force of Man in his relationship with Reality. The truth of that, is that humankind (in spite of itself), evolves towards its own Reality. The only human values that exist, lie in Man’s recognition of the principles involved that provide human direction. Our ‘material brain’ is a product of Nature's evolutionary process, and has innate within it the same principles that exist in all matter. That ‘which is’, is the truth, and our brain evolves to process that at every level, and we constantly manifest that in every action we take – whether we like it or not. The fundamental similarities between human beings is that we are not only evidently human, but that we also function and construct societies that we recognize as beneficial to our immediate well-being. All social function is determined by our brain capacity and its ability to postulate the relationship it has with Universal principles. = Illusion. = Considering that we can contradict things is an illusion. We can never contradict the truth. We do not have the ability to create proprietary constructs of reality. That ‘which is’, can only make its basic properties transparent to us through direct experience. Imaginary concepts must in the end conform to a measurable construct that we can identify. Within the structure of any philosophical theory of ‘knowledge’ it must contain the basic elements of truth at all times, or there is nothing!! To say that ‘everything is’ is motivated by pure reason experience as an objective, and subjective reality and as an axiomatic grammatical premise that no amount of mental acrobatics can deny. We can only deal with ‘something’, whatever it may be. There is no metaphysical construct that can provide evidence that ‘nothing’ exists, outside a mythical mind. Explaining experience beyond ‘thought’ processes requires a definitive language that deals with the reality itself. We all Know. It is innate. The ‘difference’ between us is only the measure of the knowledge that is made manifest, and that knowledge continually proliferates. The ‘individual perspective’, and the illusory ‘I’ which dominates, is the barrier to any relation to ‘what is’, and the malady of never experiencing the truth directly!   Truth, knowledge, agreement are the abundant and embedded Absolutes that form the structure of human evolution. That we constantly utilise and improve on their use is evidence of their reality, and the material transparency within every social structure. The survival and proliferation of such realities should be the evidence to establish that ‘that which is’ is Absolute. When we focus our ‘perspective’, opinion, or a hypothetical consideration of a space, time, or identity to question a Universally accepted fact, it is hardly a categorical argument to dismiss that which is true as nonsensical. Any denial that 2+2=4 is a fundamental truth hardly takes into account that the reality of such basics are vital to the success of higher mathematics. Unless the basics are continually correct, and evidently so, then no correct solutions could evolve. We know that within any basic structural ‘use’ that the calculation is correct. We commonly accept its correctness as an embedded reality. All forms of lower or higher mathematics would have the axiomatic principle of ‘correctness’ as their basis to extend from. Also, they would have as an axiom that the reverse is true. The 2+2=4 is, in its reality, the epitome of balance and construction. The 2+2 reality forms its correct conclusion when the principles of mathematics are propounded and they conform to transparent truth and arrive at 4. Only when it ‘adds up’, does it become a truth that we all recognize. Our greatest ignorance is taking for granted the proliferation of such truths through an ideological blinkered perspective. Because truth takes a commonplace form it is no less fundamental. Unless there is correct knowledge as to the existence of fundamental truth, that ‘which is’, goes unrecognised. That form of truth must be applicable to all. Truth exists in everything – it is an evidential reality. Searching for an esoteric truth is chasing shadows. Every truth is a ‘necessary factor’, and fundamental to our existence. Because of the imposed limited perspectives (via education, ideology, beliefs) that which is evidently true, and transparent, is delegated to a position of simple practicality with conditions placed on it which further deletes its substance, and we have the awful predilection of conforming to the attempted destruction of that which is true. Do we have a problem with seeing something, which is correct, as also being true? All truths are fundamental. They are not subject to attempted denial because of any diminished realisation at any point in time. Where there is reasonable evidence of balance, equity, and agreement we can conclude that a truth exists. Once innate information of that truth becomes transparent, it becomes an embedded useful human utility that must have some measure of fundamental truth as their starting point. From any common-sense, or ‘more reasonable’ position, it would be more productive to view reality as possessing at every level the same innate values or principles consistent with our ability to measure, or recognize them. To view reality as having ‘different’, or antagonistic properties, is simply a misguided view of ‘what is’.  That form of perspective is counter productive when it attempts to establish mythical dichotomies as realities in their own right. When the reality of principles are made transparent, we can then ‘more reasonably’ make use of them to further their basic existence. Here we use reason to exemplify their necessary function, and once established it becomes (if necessary), ‘more reasonable’ to locate them in all things. The dematerialization of any object through the practice of ‘perceptual illusion’ is an attempt to deny the reality that exists. Where perceptual illusions are concerned, innate direct communion with that which is, suspends the effect of such illusions. All the properties in a chair are recognized as the reality that exists. That is materialism. A chair does have the principles of form, design, structure, colour, substance etc.  However it is analysed – it is a quantifiable reality.   = Human representation. When we understand the validity and existence of principles in all things, it is easy to understand that ideological dogmas are never the foundation for real knowledge, or that direct experience of ‘what is’. Our real perspective is not some individualistic experience that confines us, it is that expanse in which we exist that offers us the view of that expanse. Everyone has the potential to go beyond their ‘apparent’ human perspective limitations. Shifting our sense of perception toward that which is basic, paradoxically extends the experience of that which is true. Let general knowledge be directed toward the performance that identifies the measure of principles that are enacted. Therein lies the production of knowledge that offers a sustainable growth of that vital universal aspect of knowledge, where, reason and truth, can prevail. Any correct definition is language itself, opening the door to that reality experience which is critical. Only when we know and experience that the same reality (with all its principles intact) exists for all of us can we then recognize the mythical distinctions that are taken as being real. The majestic experience of that reality goes well beyond historical beliefs. Exploring simple ‘necessities’ is not based on any sacred text, but the privilege of recognizing a sensible evolutionary path through life. Whatever may be in the future, is implicit in the material world now, and it has always been so. k8o2ezz45uu2h4p37plq0rzsvlsido4 2 281683 2416279 2411960 2022-08-18T20:01:53Z Atcovi 276019 /* Kapital 5 - 2/6/2022 */ wikitext text/x-wiki == Kapital 5 - 2/6/2022 == - '''Weak verbs''': ge- + verb stem + -t/-et (gearbeitet, geregnet, gehört)<br> --verbs ending with "-ieren" don't have the "ge-" * Review "strong", "weak" and "mixed" verbs * https://chatterbug.com/grammar/german/perfect-tense-questions - "Perfekt" Good quizzes: https://www.germanzone.org/accusative-case-personal-pronouns-1/ === "Strong"/"Weak"/"Mixed" Verbs === - https://www.vocabulix.com/german/strong-weak-verbs.shtml === Accusative Pronouns === -accusacative verbs: https://en.wikibooks.org/wiki/German/Grammar/Transitive_verbs#The_accusative_pronouns {| class="wikitable" |+ !English !Deutsch |- |me |'''mich''' |- |you (informal) |'''dich''' |- |you (formal) |'''Sie''' |- |him |'''ihn''' |- |her |'''sie''' |- |it |'''es''' |- |us |'''uns''' |- |you all (informal) |'''euch''' |- |you all (formal) |'''Sie''' |- |them |'''sie''' |} ====Examples==== {| class="wikitable" |+ !Deutsch !Deutsch (accusativ) |- |Ich kaufe den Apfel |Ich kaufe '''''ihn''''' [masculine "it"] |- |Ich höre das Mäuschen |Ich höre '''''es''''' [neutral "it"] |- |Ich sehe die Blume |Ich sehe '''''sie''''' [feminine "it"] |} === Dative Pronouns === accusative pronoun > dative pronoun ==== Review of what dative means ==== ''What makes a dative case''? Indirect object/"to whom?" (Ich gebe '''dem alten Mann''' eine Fahrkarte; "dem alten Mann" is the indirect object, while the "Fahrkarte" is the direct job that is being acted on. You'd ask yourself, in this case: "Who was the direct object (eine Fahrkarte) given [verb] to?" - The man) -dative pronouns: https://en.wikibooks.org/wiki/German/Grammar/Ditransitive_verbs#The_dative_pronouns {| class="wikitable" |+ !English !Deutsch |- |I |'''mir''' |- |You (informal) |'''dir''' |- |You (formal) |'''Ihnen''' |- |Him |'''ihm''' |- |Her |'''ihr''' |- |It |'''ihm''' |- |We |'''uns''' |- |You (plural) |'''euch''' |- |Them |'''ihnen''' |} ====== Overview ====== - https://german.net/exercises/cases/accusative-dative/ === Vocabulary (Feiern & Party)=== Es ist die Zeit Feste zu feiern! *Ich '''feiere''' *Du '''feierst''' *Er/Sie/Es '''feiert''' *Wir '''feieren''' *Ihr '''feiert''' *Sie '''feieren''' ==== Feste ==== {| class="wikitable" |+ !English !Deutsch |- |The holiday |Der Feiertag |- |The card |Die Karte |- |The party |Die Party |- |The room |Das Zimmer |- |To toast |anstoßen (stößt... an) |- |To recieve |bekommen |- |To invite |einladen (lädt... an) |- |To celebrate |feiern |- |To throw a party |eine Party geben |- |To [not] have fun |(keinen) Spaß haben |- |To give [a gift] |schenken |- |To surprise |Überraschen |- |Congratulations! |Herlizchen Glückwunsch |} ==== Besondere Anlässe ==== {| class="wikitable" |+ !English !Deutsch |- |The marriage |Die Ehe |- |The newlywed |Der/Die Frischvermählte |- |The birth |Die Geburt |- |The birthday |Der Geburtstag |- |The wedding |Die Hochzeit |- |The wedding anniversary |Der Hochzeitstag |- |The anniversary |Der Jahrestag |- |The New Year's Eve |Das/Der Silvester |- |The Christmas |Das Weihnachten |- |To [go into] retirement |in Rente gehen |- |To [make] graduate [happen] |einen Abschluss machen |} ==== Ausdrücke ==== {| class="wikitable" |+ !English !Deutsch |- |The friendship |Die Freündschaft |- |The happiness |Das Glück |- |The kiss |Der Kuss |- |The love |Die Liebe |} === Vocabulary (das Kleidung) === {| class="wikitable" |+ !Deutsch !Englisch |- |Der Hut, -e |hat |- |Der Badeanzug, -e |bathing suit |- |Teuer |expensive |- |Die Krawatte |tie |- |Der Gürtel |belt |- |Das Kleid |dress |- |Das Hemd |shirt |- |Die kurze Hose |shorts |- |Der Turnschuh |sneakers |- |Er trägt einen Anzug |He's wearing a suit |- |Die Handtasche |purse |- |gelb |yellow |- |grün |green |- |lila |purple |- |rosa |pink |- |grau |gray |- |schwarz |black |- |weiß |white |- |braun |brown |- |blau |blue |- |rot |red |- |orange |orange |- |die Sonnenbrille, -n |sunglasses |- |die Mütze, -n |cap |- |die Halskette, -n |necklace |- |der Schal, -s |scarf |- |Es ist günstig |It's cheap |- |die Jacke, -n |jacket |- |der Rock, -e |skirt |- |die Hose, -n |pants |- |der Schuh, -e |shoe |} == Wissen vs. Kennen == * '''Wissen''' - To know as a fact (I know that her name is... I know her address is...) * '''Kennen''' - To be familiar with/acquainted with (Familiar with people or places) - https://languageposters.com/pages/german-verbs-wissen-conjugation -https://www.vocabulix.com/conjugation/query_german.html?conjugation3,kennen,kennen,_german_present ''Are you familiar with the subject? If not, then use wissen.'' === Examples === ==== Wissen ==== * die Adresse von Amy * ihren Familiennamen * die Antwort * viel ueber Mode * die Uhrzeit ==== Kennen ==== * das Anne Frank-Haus in Amsterdam * deinen Freundin nicht * Maries Eltern * ein gutes chinesisches Restaurant * Wien gut * eine Stadt == Das Perfekt: The conversational past == Ich kaufe ein Brot in der Bäckerei. * Decide: sein or haben? '''Sein''' is a change of a physical condition/location (Point A --> Point B), while '''haben''' is for everything else. * Conguate based on decision. * Pay attention to if the verb is weak, strong, or mixed. == Two way prepositions == Either destination/movement towards a destination (accusative) or location/no movement towards a destination (dative). TIP: Use '''wohin''' ''(where to)'' with accusative as it talks about a movement towards a destination. Use '''wo''' with dative as it talks about a location/the item in question already being there. * '''An''' [at, on (vertical)] ''(Ich haenge das Bild an die Wand [accusative]) or (Das Bild haengt an der Wand)'' * '''Auf''' [on, on top of (horizontal)] ''(Stell deine Schuhe nicht auf den Tisch)'' * '''Zwischen''' [between] * '''Neben''' [next] * '''Hinter''' [behind] * '''Über''' [above, over] * '''Unter''' [under] ''(Ich trage ein T-Shirt unter dem Pullover)'' * '''In''' [in, into] ''(William geht in die Stadt; Was macht er in der Stadt?) or (Ich laufe jeden Tag in die Schule; Ich lerne in der Bibliothek)'' * '''Vor''' [in front of] {{info| Common ''Wo''/''Wohin'' verbs, use dative/accusative after these verbs *'''liegen''' --> to lie *'''sitzen''' --> to sit *'''stehen''' --> to stand *'''hängen''' --> to hang}} '''Tips''' * A movement verb = accusative === Contractions === * an das --> '''ans''' * auf das --> '''aufs''' * in das --> '''ins''' * an dem --> '''am''' * in dem --> '''im''' [[Category:Atcovi's Work]] 0yptcpq6egel56rs5vsndyhmbww9pu8 0 281704 2416310 2389328 2022-08-18T21:51:49Z Saltrabook 1417466 wikitext text/x-wiki Not updated # Self-perceived safety and psychosocial work environment amongst Danish dockworkers IMH 2019;70(3):171–179. # Mental and Physical Health and Working Environment of Icelandic Fishermen Published in Research Gate, June 2021. # Global Occupational Health Research Methods Education Published July 2021 MethodsX 8 # Well-being among French fisheries and maritime students questionnaire-based study GHQ12 data 2022 # Le bien-être des étudiants pêcheurs et maritimes GHQ12 data 2020 # Early diagnosis of T2DM using high sensitive tests in the mandatory medical examinations for fishers, seafarers and other transport workers. Primary Care Diabetes. 2022 Feb;16(1):211-213. https://doi.org/10.1016/j.pcd.2021.12.018 # Early diagnosis of diabetes-2 for transport workers with routine medical examination - submitted 9 Dec to Primary Care Diabetes Elsevier # [[/Systematic Review on Psychological Distress among maritime students 2022-23 /]] # [[/Psychological Distress in Venezuelan, Thai, French, and Spanish maritime students GHQ12 2022-23/]] # [[/Well-being among medical students in Panamá study using GHQ12 data 2021-22 with Diploma 1/|Well-being among medical students in Panamá study using GHQ12 data 2021-22]] # [[/Health services among port workers in Costa Rica: analysis of data from a container terminal clinic/]] [[/Well-being among medical students in Panamá study using GHQ12 data 2021-22 with Diploma 1/|2021-22]] # [[/Well-being among Japanese maritime students questionnaire-based study GHQ12 data 2022/]] # Early Diagnosis of T2DM Using High Sensitive Tests in the Mandatory Medical Examinations for Fishers, Seafarers and Other Transport Workers”. Primary Care Diabetes, January 2022, https://doi.org/10.1016/j.pcd.2021.12.018. # Rethinking the Use of Urine Dipstick for Early Diagnosis of Type 2 Diabetes Mellitus”. Diabetes Research and Clinical Practice, 31. januar 2022, 109222. https://doi.org/10.1016/j.diabres.2022.109222. # Koirala S, Jensen OC, Lucas D, Thapa A. Systematic Review on Psychological Distress among seafarers and its Assessment on Venezuelan, Thai, French, and Spanish maritime students. International Journal of Environmental Research and Public Health. 2023 Mar 1. # Martinez A, Jensen OC. Port workers use of medical service in a big Maritime Container Terminal. International Maritime Health. 2024 Jun 6. # Thapa A, Jensen OC, Lucas D, Koirala S. Psychological Distress among seafarers and its Assessment on Venezuelan, Thai, French, and Spanish maritime students by using General Health Questionnaire- 12 items. International Journal of Environmental Research and Public Health. 2024 Feb 1. # Jensen OC, Canals L, Estopà Pujol H. Prevalence of pre-diabetes, diabetes type 2 and obesity in seamen from a national data base: how to improve the data from the medical examinations. International Maritime Health. 2025 Feb 1. # Corman V, Lucas D, Jensen OC. Evaluating the relationship between musculoskeletal disorders and working environment among Panamanian port workers. International Journal of Environmental Research and Public Health. 2023 Feb 1. # Lucero-Prisno DEIII, Jensen et al. OC. Top Ten Public Health Challenges to Track in 2022. Public Health Challenges. 2023 Jan 1. # Jensen OCEA, International Diabetes and Hypertension Research group in seafaring, fishing and transport workers, Mahre-Net. Screening for diabetes type 2 and hypertension in seafarers´ routine medical examinations: Subtitle Revision of ILO Guidelines for medical examination of seafarers. International Maritime Health. 2022 Jun 22. # Lucero-Prisno DEIII, Jensen OC, Lucas D, Denisenko I, Canals ML. Maritime Workers and their Global Health: need to improve scientific knowledge and prevention. Journal of Global Health. 2022 May 25. # Lucas D, Lucero-Prisno DEIII, Luisa Canals M, Jensen OC. Important topics on seafarer’s health are Mental health and wellbeing. Journal of Global Health. 2022 Apr 5. # Shrestha S, Shrestha B, Bygvraa DA, Jensen OC. Risk Assessment in Artisanal Fisheries in Developing Countries: A Systematic Review. American Journal of Preventive Medicine. 2022 Apr;62(4):e255-e264. https://doi.org/10.1016/j.amepre.2021.08.031 # Jensen OC, Flores A, Corman V, Luisa Canals M, Lucas D, Denisenko I et al. Rethinking the use of urine dipstick for early diagnosis of Type 2 diabetes mellitus. Diabetes Research and Clinical Practice. 2022 Feb;184. 109222. https://doi.org/10.1016/j.diabres.2022.109222 # Lucero-Prisno DEIII, Jensen OC. The Need to Strengthen Maritime Safety in Africa: The Boat Sinking Tragedy in Madagascar Don Eliseo Lucero-Prisno III1, 2, Yusuf Babatunde3, Alhaji Umar Sow4, Mohamed Hoosen Suleman5, Olaf Jensen6. International Maritime Health. 2022 Jan 2. # Lucas D, Jensen OC, Loddé B, Pougnet R, Adam B, Lux H et al. Occupational allergic diseases in the fishing industry harvesting workers: a systematic review. international Maritime Health. 2022 Jan 1. # Siame S, Andrioti Bygvraa D, Jensen OC. Facilitators and barriers to implementing occupational safety interventions in the fishing industry: A scoping review. Safety Science. 2022 Jan;145. 105512. https://doi.org/10.1016/j.ssci.2021.105512 # Babatunde Y, Lucero-Prisno DE, Adegbite MA, Adediji N, Oladipo H, Sampson-Oladipupo E et al. The role of pharmacists in global maritime health. international Maritime Health. 2022;73(2):96-97. https://doi.org/10.5603/IMH.2022.0016 # Jensen OC. Global open-access curriculum for education in research methodology. 2021. Abstract from European Society of Medicine ESMED Congress 2021 , Vienna, Austria. # Lucas D, Jego C, Chresten Jensen O, Loddé B, Pougnet R, Dewitte JD et al. Santé mentale chez les gens de mer: connaissances actuelles et impact de la pandémie COVID 19. Archives des Maladies Professionnelles et de l'Environnement. 2021 Nov;82(6):619-623. https://doi.org/10.1016/j.admp.2021.05.001 # an de Sijpe P, Canals ML, Jensen O, Lucas D. Acute occupational phosphine intoxications in the maritime shipping sector: a scoping review. Archives of Environmental and Occupational Health. 2021 Oct 16. https://doi.org/10.1080/19338244.2021.1990829 # Baygi F, Mohammadi-Nasrabadi F, Zyriax B-C, Jensen OC, Bygvraa DA, Oldenburg M et al. Global overview of dietary outcomes and dietary intake assessment methods in maritime settings: a systematic review. BMC Public Health. 2021 Aug 21;21. 1579. https://doi.org/10.1186/s12889-021-11593-z # Jensen OC, Flores A, Canals ML, Lucas D, Denisenko I, Lucero-Prisno DEIII et al. Global occupational health research methods education. MethodsX. 2021 Jul;8. 101436. https://doi.org/10.1016/j.mex.2021.101436 # Lucas D, Jego C, Jensen OC, Loddé B, Pougnet R, Dewitte J-D et al. Seafarers’ mental health in the COVID-19 era: lost at sea? International Maritime Health. 2021 Jun 28;72(2):138-141. https://doi.org/10.5603/IMH.2021.0023 # Jensen OC. Norwegian textbook 8.6.6 Maritime Occupational Health (OH) practitioners: Purpose and tasks, Skills and competency. In Norwegian Textbook of Maritime Medicine. 2021 # Fotteler ML, Andrioti Bygvraa D, Jensen OC. The impact of the Maritime Labor Convention on seafarers’ working and living conditions: an analysis of port state control statistics. B M C Public Health. 2020 Dec 1;20(1). 1586. https://doi.org/10.1186/s12889-020-09682-6 # Van de Sijpe P, Lucas D, Canals ML, Jensen O. Acute occupational phosphine intoxications in the maritime shipping sector: Belgian and French reported cases. international Maritime Health. 2020 Sep 28;71(3):151-159. https://doi.org/10.5603/IMH.2020.0028 # Pesel G, Canals ML, Sandrin M, Jensen O. Wellbeing of a selection of seafarers in Eastern Adriatic Sea during the COVID-19 pandemic 2020. international Maritime Health. 2020 Sep 28;71(3):184-190. https://doi.org/10.5603/IMH.2020.0033 # Jensen OC. Social-strategisk, evidensbaseret Covid-19 forebyggelse. Ugeskrift for Laeger. 2020 Sep 14;182(18). q0ybq405r6ve4x2489us8jh1h8ar8ej 2416311 2416310 2022-08-18T21:53:03Z Saltrabook 1417466 wikitext text/x-wiki Not updated # Self-perceived safety and psychosocial work environment amongst Danish dockworkers IMH 2019;70(3):171–179. # Mental and Physical Health and Working Environment of Icelandic Fishermen Published in Research Gate, June 2021. # Global Occupational Health Research Methods Education Published July 2021 MethodsX 8 # Well-being among French fisheries and maritime students questionnaire-based study GHQ12 data 2022 # Le bien-être des étudiants pêcheurs et maritimes GHQ12 data 2020 # Early diagnosis of T2DM using high sensitive tests in the mandatory medical examinations for fishers, seafarers and other transport workers. Primary Care Diabetes. 2022 Feb;16(1):211-213. https://doi.org/10.1016/j.pcd.2021.12.018 # Early diagnosis of diabetes-2 for transport workers with routine medical examination - submitted 9 Dec to Primary Care Diabetes Elsevier # [[/Systematic Review on Psychological Distress among maritime students 2022-23 /]] # [[/Psychological Distress in Venezuelan, Thai, French, and Spanish maritime students GHQ12 2022-23/]] # [[/Well-being among medical students in Panamá study using GHQ12 data 2021-22 with Diploma 1/|Well-being among medical students in Panamá study using GHQ12 data 2021-22]] # [[/Health services among port workers in Costa Rica: analysis of data from a container terminal clinic/]] [[/Well-being among medical students in Panamá study using GHQ12 data 2021-22 with Diploma 1/|2021-22]] # [[/Well-being among Japanese maritime students questionnaire-based study GHQ12 data 2022/]] # Early Diagnosis of T2DM Using High Sensitive Tests in the Mandatory Medical Examinations for Fishers, Seafarers and Other Transport Workers”. Primary Care Diabetes, January 2022, https://doi.org/10.1016/j.pcd.2021.12.018. # Rethinking the Use of Urine Dipstick for Early Diagnosis of Type 2 Diabetes Mellitus”. Diabetes Research and Clinical Practice, 31. januar 2022, 109222. https://doi.org/10.1016/j.diabres.2022.109222. # Koirala S, Jensen OC, Lucas D, Thapa A. Systematic Review on Psychological Distress among seafarers and its Assessment on Venezuelan, Thai, French, and Spanish maritime students. International Journal of Environmental Research and Public Health. 2023 Mar 1. # Martinez A, Jensen OC. Port workers use of medical service in a big Maritime Container Terminal. International Maritime Health. 2024 Jun 6. # Thapa A, Jensen OC, Lucas D, Koirala S. Psychological Distress among seafarers and its Assessment on Venezuelan, Thai, French, and Spanish maritime students by using General Health Questionnaire- 12 items. International Journal of Environmental Research and Public Health. 2024 Feb 1. # Jensen OC, Canals L, Estopà Pujol H. Prevalence of pre-diabetes, diabetes type 2 and obesity in seamen from a national data base: how to improve the data from the medical examinations. International Maritime Health. 2025 Feb 1. # Corman V, Lucas D, Jensen OC. Evaluating the relationship between musculoskeletal disorders and working environment among Panamanian port workers. International Journal of Environmental Research and Public Health. 2023 Feb 1. # Lucero-Prisno DEIII, Jensen et al. OC. Top Ten Public Health Challenges to Track in 2022. Public Health Challenges. 2023 Jan 1. # Jensen OCEA, International Diabetes and Hypertension Research group in seafaring, fishing and transport workers, Mahre-Net. Screening for diabetes type 2 and hypertension in seafarers´ routine medical examinations: Subtitle Revision of ILO Guidelines for medical examination of seafarers. International Maritime Health. 2022 Jun 22. # Lucero-Prisno DEIII, Jensen OC, Lucas D, Denisenko I, Canals ML. Maritime Workers and their Global Health: need to improve scientific knowledge and prevention. Journal of Global Health. 2022 May 25. # Lucas D, Lucero-Prisno DEIII, Luisa Canals M, Jensen OC. Important topics on seafarer’s health are Mental health and wellbeing. Journal of Global Health. 2022 Apr 5. # Shrestha S, Shrestha B, Bygvraa DA, Jensen OC. Risk Assessment in Artisanal Fisheries in Developing Countries: A Systematic Review. American Journal of Preventive Medicine. 2022 Apr;62(4):e255-e264. https://doi.org/10.1016/j.amepre.2021.08.031 # Jensen OC, Flores A, Corman V, Luisa Canals M, Lucas D, Denisenko I et al. Rethinking the use of urine dipstick for early diagnosis of Type 2 diabetes mellitus. Diabetes Research and Clinical Practice. 2022 Feb;184. 109222. https://doi.org/10.1016/j.diabres.2022.109222 # Lucero-Prisno DEIII, Jensen OC. The Need to Strengthen Maritime Safety in Africa: The Boat Sinking Tragedy in Madagascar Don Eliseo Lucero-Prisno III1, 2, Yusuf Babatunde3, Alhaji Umar Sow4, Mohamed Hoosen Suleman5, Olaf Jensen6. International Maritime Health. 2022 Jan 2. # Lucas D, Jensen OC, Loddé B, Pougnet R, Adam B, Lux H et al. Occupational allergic diseases in the fishing industry harvesting workers: a systematic review. international Maritime Health. 2022 Jan 1. # Siame S, Andrioti Bygvraa D, Jensen OC. Facilitators and barriers to implementing occupational safety interventions in the fishing industry: A scoping review. Safety Science. 2022 Jan;145. 105512. https://doi.org/10.1016/j.ssci.2021.105512 # Babatunde Y, Lucero-Prisno DE, Adegbite MA, Adediji N, Oladipo H, Sampson-Oladipupo E et al. The role of pharmacists in global maritime health. international Maritime Health. 2022;73(2):96-97. https://doi.org/10.5603/IMH.2022.0016 # Jensen OC. Global open-access curriculum for education in research methodology. 2021. Abstract from European Society of Medicine ESMED Congress 2021 , Vienna, Austria. # Lucas D, Jego C, Chresten Jensen O, Loddé B, Pougnet R, Dewitte JD et al. Santé mentale chez les gens de mer: connaissances actuelles et impact de la pandémie COVID 19. Archives des Maladies Professionnelles et de l'Environnement. 2021 Nov;82(6):619-623. https://doi.org/10.1016/j.admp.2021.05.001 # Van de Sijpe P, Canals ML, Jensen O, Lucas D. Acute occupational phosphine intoxications in the maritime shipping sector: a scoping review. Archives of Environmental and Occupational Health. 2021 Oct 16. https://doi.org/10.1080/19338244.2021.1990829 # Baygi F, Mohammadi-Nasrabadi F, Zyriax B-C, Jensen OC, Bygvraa DA, Oldenburg M et al. Global overview of dietary outcomes and dietary intake assessment methods in maritime settings: a systematic review. BMC Public Health. 2021 Aug 21;21. 1579. https://doi.org/10.1186/s12889-021-11593-z # Jensen OC, Flores A, Canals ML, Lucas D, Denisenko I, Lucero-Prisno DEIII et al. Global occupational health research methods education. MethodsX. 2021 Jul;8. 101436. https://doi.org/10.1016/j.mex.2021.101436 # Lucas D, Jego C, Jensen OC, Loddé B, Pougnet R, Dewitte J-D et al. Seafarers’ mental health in the COVID-19 era: lost at sea? International Maritime Health. 2021 Jun 28;72(2):138-141. https://doi.org/10.5603/IMH.2021.0023 # Jensen OC. Norwegian textbook 8.6.6 Maritime Occupational Health (OH) practitioners: Purpose and tasks, Skills and competency. In Norwegian Textbook of Maritime Medicine. 2021 # Fotteler ML, Andrioti Bygvraa D, Jensen OC. The impact of the Maritime Labor Convention on seafarers’ working and living conditions: an analysis of port state control statistics. B M C Public Health. 2020 Dec 1;20(1). 1586. https://doi.org/10.1186/s12889-020-09682-6 # Van de Sijpe P, Lucas D, Canals ML, Jensen O. Acute occupational phosphine intoxications in the maritime shipping sector: Belgian and French reported cases. international Maritime Health. 2020 Sep 28;71(3):151-159. https://doi.org/10.5603/IMH.2020.0028 # Pesel G, Canals ML, Sandrin M, Jensen O. Wellbeing of a selection of seafarers in Eastern Adriatic Sea during the COVID-19 pandemic 2020. international Maritime Health. 2020 Sep 28;71(3):184-190. https://doi.org/10.5603/IMH.2020.0033 # Jensen OC. Social-strategisk, evidensbaseret Covid-19 forebyggelse. Ugeskrift for Laeger. 2020 Sep 14;182(18). 2em32yd1i1iuyfg22qr00km5bzloxje 2416312 2416311 2022-08-18T21:55:05Z Saltrabook 1417466 wikitext text/x-wiki Not updated # Self-perceived safety and psychosocial work environment amongst Danish dockworkers IMH 2019;70(3):171–179. # Mental and Physical Health and Working Environment of Icelandic Fishermen Published in Research Gate, June 2021. # Global Occupational Health Research Methods Education Published July 2021 MethodsX 8 # Well-being among French fisheries and maritime students questionnaire-based study GHQ12 data 2022 # Le bien-être des étudiants pêcheurs et maritimes GHQ12 data 2020 # Early diagnosis of T2DM using high sensitive tests in the mandatory medical examinations for fishers, seafarers and other transport workers. Primary Care Diabetes. 2022 Feb;16(1):211-213. https://doi.org/10.1016/j.pcd.2021.12.018 # Early diagnosis of diabetes-2 for transport workers with routine medical examination - submitted 9 Dec to Primary Care Diabetes Elsevier # [[/Systematic Review on Psychological Distress among maritime students 2022-23 /]] # [[/Psychological Distress in Venezuelan, Thai, French, and Spanish maritime students GHQ12 2022-23/]] # [[/Well-being among medical students in Panamá study using GHQ12 data 2021-22 with Diploma 1/|Well-being among medical students in Panamá study using GHQ12 data 2021-22]] # [[/Health services among port workers in Costa Rica: analysis of data from a container terminal clinic/]] [[/Well-being among medical students in Panamá study using GHQ12 data 2021-22 with Diploma 1/|2021-22]] # [[/Well-being among Japanese maritime students questionnaire-based study GHQ12 data 2022/]] # Early Diagnosis of T2DM Using High Sensitive Tests in the Mandatory Medical Examinations for Fishers, Seafarers and Other Transport Workers”. Primary Care Diabetes, January 2022, https://doi.org/10.1016/j.pcd.2021.12.018. # Rethinking the Use of Urine Dipstick for Early Diagnosis of Type 2 Diabetes Mellitus”. Diabetes Research and Clinical Practice, 31. januar 2022, 109222. https://doi.org/10.1016/j.diabres.2022.109222. # Koirala S, Jensen OC, Lucas D, Thapa A. Systematic Review on Psychological Distress among seafarers and its Assessment on Venezuelan, Thai, French, and Spanish maritime students. International Journal of Environmental Research and Public Health. 2023 Mar 1. # Martinez A, Jensen OC. Port workers use of medical service in a big Maritime Container Terminal. International Maritime Health. 2024 Jun 6. # Thapa A, Jensen OC, Lucas D, Koirala S. Psychological Distress among seafarers and its Assessment on Venezuelan, Thai, French, and Spanish maritime students by using General Health Questionnaire- 12 items. International Journal of Environmental Research and Public Health. 2024 Feb 1. # Jensen OC, Canals L, Estopà Pujol H. Prevalence of pre-diabetes, diabetes type 2 and obesity in seamen from a national data base: how to improve the data from the medical examinations. International Maritime Health. 2025 Feb 1. # Corman V, Lucas D, Jensen OC. Evaluating the relationship between musculoskeletal disorders and working environment among Panamanian port workers. International Journal of Environmental Research and Public Health. 2023 Feb 1. # Lucero-Prisno DEIII, Jensen et al. OC. Top Ten Public Health Challenges to Track in 2022. Public Health Challenges. 2023 Jan 1. # Jensen OCEA, International Diabetes and Hypertension Research group in seafaring, fishing and transport workers, Mahre-Net. Screening for diabetes type 2 and hypertension in seafarers´ routine medical examinations: Subtitle Revision of ILO Guidelines for medical examination of seafarers. International Maritime Health. 2022 Jun 22. # Lucero-Prisno DEIII, Jensen OC, Lucas D, Denisenko I, Canals ML. Maritime Workers and their Global Health: need to improve scientific knowledge and prevention. Journal of Global Health. 2022 May 25. # Lucas D, Lucero-Prisno DEIII, Luisa Canals M, Jensen OC. Important topics on seafarer’s health are Mental health and wellbeing. Journal of Global Health. 2022 Apr 5. # Shrestha S, Shrestha B, Bygvraa DA, Jensen OC. Risk Assessment in Artisanal Fisheries in Developing Countries: A Systematic Review. American Journal of Preventive Medicine. 2022 Apr;62(4):e255-e264. https://doi.org/10.1016/j.amepre.2021.08.031 # Jensen OC, Flores A, Corman V, Luisa Canals M, Lucas D, Denisenko I et al. Rethinking the use of urine dipstick for early diagnosis of Type 2 diabetes mellitus. Diabetes Research and Clinical Practice. 2022 Feb;184. 109222. https://doi.org/10.1016/j.diabres.2022.109222 # Lucero-Prisno DEIII, Jensen OC. The Need to Strengthen Maritime Safety in Africa: The Boat Sinking Tragedy in Madagascar Don Eliseo Lucero-Prisno III1, 2, Yusuf Babatunde3, Alhaji Umar Sow4, Mohamed Hoosen Suleman5, Olaf Jensen6. International Maritime Health. 2022 Jan 2. # Lucas D, Jensen OC, Loddé B, Pougnet R, Adam B, Lux H et al. Occupational allergic diseases in the fishing industry harvesting workers: a systematic review. international Maritime Health. 2022 Jan 1 in preparation # Siame S, Andrioti Bygvraa D, Jensen OC. Facilitators and barriers to implementing occupational safety interventions in the fishing industry: A scoping review. Safety Science. 2022 Jan;145. 105512. https://doi.org/10.1016/j.ssci.2021.105512 # Babatunde Y, Lucero-Prisno DE, Adegbite MA, Adediji N, Oladipo H, Sampson-Oladipupo E et al. The role of pharmacists in global maritime health. international Maritime Health. 2022;73(2):96-97. https://doi.org/10.5603/IMH.2022.0016 # Jensen OC. Global open-access curriculum for education in research methodology. 2021. Abstract from European Society of Medicine ESMED Congress 2021 , Vienna, Austria. # Lucas D, Jego C, Chresten Jensen O, Loddé B, Pougnet R, Dewitte JD et al. Santé mentale chez les gens de mer: connaissances actuelles et impact de la pandémie COVID 19. Archives des Maladies Professionnelles et de l'Environnement. 2021 Nov;82(6):619-623. https://doi.org/10.1016/j.admp.2021.05.001 # Van de Sijpe P, Canals ML, Jensen O, Lucas D. Acute occupational phosphine intoxications in the maritime shipping sector: a scoping review. Archives of Environmental and Occupational Health. 2021 Oct 16. https://doi.org/10.1080/19338244.2021.1990829 # Baygi F, Mohammadi-Nasrabadi F, Zyriax B-C, Jensen OC, Bygvraa DA, Oldenburg M et al. Global overview of dietary outcomes and dietary intake assessment methods in maritime settings: a systematic review. BMC Public Health. 2021 Aug 21;21. 1579. https://doi.org/10.1186/s12889-021-11593-z # Jensen OC, Flores A, Canals ML, Lucas D, Denisenko I, Lucero-Prisno DEIII et al. Global occupational health research methods education. MethodsX. 2021 Jul;8. 101436. https://doi.org/10.1016/j.mex.2021.101436 # Lucas D, Jego C, Jensen OC, Loddé B, Pougnet R, Dewitte J-D et al. Seafarers’ mental health in the COVID-19 era: lost at sea? International Maritime Health. 2021 Jun 28;72(2):138-141. https://doi.org/10.5603/IMH.2021.0023 # Jensen OC. Norwegian textbook 8.6.6 Maritime Occupational Health (OH) practitioners: Purpose and tasks, Skills and competency. In Norwegian Textbook of Maritime Medicine. 2021 # Fotteler ML, Andrioti Bygvraa D, Jensen OC. The impact of the Maritime Labor Convention on seafarers’ working and living conditions: an analysis of port state control statistics. B M C Public Health. 2020 Dec 1;20(1). 1586. https://doi.org/10.1186/s12889-020-09682-6 # Van de Sijpe P, Lucas D, Canals ML, Jensen O. Acute occupational phosphine intoxications in the maritime shipping sector: Belgian and French reported cases. international Maritime Health. 2020 Sep 28;71(3):151-159. https://doi.org/10.5603/IMH.2020.0028 # Pesel G, Canals ML, Sandrin M, Jensen O. Wellbeing of a selection of seafarers in Eastern Adriatic Sea during the COVID-19 pandemic 2020. international Maritime Health. 2020 Sep 28;71(3):184-190. https://doi.org/10.5603/IMH.2020.0033 # Jensen OC. Social-strategisk, evidensbaseret Covid-19 forebyggelse. Ugeskrift for Laeger. 2020 Sep 14;182(18). ce2h06kh3yzzu8d6y7micbtovarvgs3 2416313 2416312 2022-08-18T21:55:55Z Saltrabook 1417466 wikitext text/x-wiki Not updated # Self-perceived safety and psychosocial work environment amongst Danish dockworkers IMH 2019;70(3):171–179. # Mental and Physical Health and Working Environment of Icelandic Fishermen Published in Research Gate, June 2021. # Global Occupational Health Research Methods Education Published July 2021 MethodsX 8 # Well-being among French fisheries and maritime students questionnaire-based study GHQ12 data 2022 # Le bien-être des étudiants pêcheurs et maritimes GHQ12 data 2020 # Early diagnosis of T2DM using high sensitive tests in the mandatory medical examinations for fishers, seafarers and other transport workers. Primary Care Diabetes. 2022 Feb;16(1):211-213. https://doi.org/10.1016/j.pcd.2021.12.018 # [[/Systematic Review on Psychological Distress among maritime students 2022-23 /]] # [[/Psychological Distress in Venezuelan, Thai, French, and Spanish maritime students GHQ12 2022-23/]] # [[/Well-being among medical students in Panamá study using GHQ12 data 2021-22 with Diploma 1/|Well-being among medical students in Panamá study using GHQ12 data 2021-22]] # [[/Health services among port workers in Costa Rica: analysis of data from a container terminal clinic/]] [[/Well-being among medical students in Panamá study using GHQ12 data 2021-22 with Diploma 1/|2021-22]] # [[/Well-being among Japanese maritime students questionnaire-based study GHQ12 data 2022/]] # Early Diagnosis of T2DM Using High Sensitive Tests in the Mandatory Medical Examinations for Fishers, Seafarers and Other Transport Workers”. Primary Care Diabetes, January 2022, https://doi.org/10.1016/j.pcd.2021.12.018. # Rethinking the Use of Urine Dipstick for Early Diagnosis of Type 2 Diabetes Mellitus”. Diabetes Research and Clinical Practice, 31. januar 2022, 109222. https://doi.org/10.1016/j.diabres.2022.109222. # Koirala S, Jensen OC, Lucas D, Thapa A. Systematic Review on Psychological Distress among seafarers and its Assessment on Venezuelan, Thai, French, and Spanish maritime students. International Journal of Environmental Research and Public Health. 2023 Mar 1. # Martinez A, Jensen OC. Port workers use of medical service in a big Maritime Container Terminal. International Maritime Health. 2024 Jun 6. # Thapa A, Jensen OC, Lucas D, Koirala S. Psychological Distress among seafarers and its Assessment on Venezuelan, Thai, French, and Spanish maritime students by using General Health Questionnaire- 12 items. International Journal of Environmental Research and Public Health. 2024 Feb 1. # Jensen OC, Canals L, Estopà Pujol H. Prevalence of pre-diabetes, diabetes type 2 and obesity in seamen from a national data base: how to improve the data from the medical examinations. International Maritime Health. 2025 Feb 1. # Corman V, Lucas D, Jensen OC. Evaluating the relationship between musculoskeletal disorders and working environment among Panamanian port workers. International Journal of Environmental Research and Public Health. 2023 Feb 1. # Lucero-Prisno DEIII, Jensen et al. OC. Top Ten Public Health Challenges to Track in 2022. Public Health Challenges. 2023 Jan 1. # Jensen OCEA, International Diabetes and Hypertension Research group in seafaring, fishing and transport workers, Mahre-Net. Screening for diabetes type 2 and hypertension in seafarers´ routine medical examinations: Subtitle Revision of ILO Guidelines for medical examination of seafarers. International Maritime Health. 2022 Jun 22. # Lucero-Prisno DEIII, Jensen OC, Lucas D, Denisenko I, Canals ML. Maritime Workers and their Global Health: need to improve scientific knowledge and prevention. Journal of Global Health. 2022 May 25. # Lucas D, Lucero-Prisno DEIII, Luisa Canals M, Jensen OC. Important topics on seafarer’s health are Mental health and wellbeing. Journal of Global Health. 2022 Apr 5. # Shrestha S, Shrestha B, Bygvraa DA, Jensen OC. Risk Assessment in Artisanal Fisheries in Developing Countries: A Systematic Review. American Journal of Preventive Medicine. 2022 Apr;62(4):e255-e264. https://doi.org/10.1016/j.amepre.2021.08.031 # Jensen OC, Flores A, Corman V, Luisa Canals M, Lucas D, Denisenko I et al. Rethinking the use of urine dipstick for early diagnosis of Type 2 diabetes mellitus. Diabetes Research and Clinical Practice. 2022 Feb;184. 109222. https://doi.org/10.1016/j.diabres.2022.109222 # Lucero-Prisno DEIII, Jensen OC. The Need to Strengthen Maritime Safety in Africa: The Boat Sinking Tragedy in Madagascar Don Eliseo Lucero-Prisno III1, 2, Yusuf Babatunde3, Alhaji Umar Sow4, Mohamed Hoosen Suleman5, Olaf Jensen6. International Maritime Health. 2022 Jan 2. # Lucas D, Jensen OC, Loddé B, Pougnet R, Adam B, Lux H et al. Occupational allergic diseases in the fishing industry harvesting workers: a systematic review. international Maritime Health. 2022 Jan 1 in preparation # Siame S, Andrioti Bygvraa D, Jensen OC. Facilitators and barriers to implementing occupational safety interventions in the fishing industry: A scoping review. Safety Science. 2022 Jan;145. 105512. https://doi.org/10.1016/j.ssci.2021.105512 # Babatunde Y, Lucero-Prisno DE, Adegbite MA, Adediji N, Oladipo H, Sampson-Oladipupo E et al. The role of pharmacists in global maritime health. international Maritime Health. 2022;73(2):96-97. https://doi.org/10.5603/IMH.2022.0016 # Jensen OC. Global open-access curriculum for education in research methodology. 2021. Abstract from European Society of Medicine ESMED Congress 2021 , Vienna, Austria. # Lucas D, Jego C, Chresten Jensen O, Loddé B, Pougnet R, Dewitte JD et al. Santé mentale chez les gens de mer: connaissances actuelles et impact de la pandémie COVID 19. Archives des Maladies Professionnelles et de l'Environnement. 2021 Nov;82(6):619-623. https://doi.org/10.1016/j.admp.2021.05.001 # Van de Sijpe P, Canals ML, Jensen O, Lucas D. Acute occupational phosphine intoxications in the maritime shipping sector: a scoping review. Archives of Environmental and Occupational Health. 2021 Oct 16. https://doi.org/10.1080/19338244.2021.1990829 # Baygi F, Mohammadi-Nasrabadi F, Zyriax B-C, Jensen OC, Bygvraa DA, Oldenburg M et al. Global overview of dietary outcomes and dietary intake assessment methods in maritime settings: a systematic review. BMC Public Health. 2021 Aug 21;21. 1579. https://doi.org/10.1186/s12889-021-11593-z # Jensen OC, Flores A, Canals ML, Lucas D, Denisenko I, Lucero-Prisno DEIII et al. Global occupational health research methods education. MethodsX. 2021 Jul;8. 101436. https://doi.org/10.1016/j.mex.2021.101436 # Lucas D, Jego C, Jensen OC, Loddé B, Pougnet R, Dewitte J-D et al. Seafarers’ mental health in the COVID-19 era: lost at sea? International Maritime Health. 2021 Jun 28;72(2):138-141. https://doi.org/10.5603/IMH.2021.0023 # Jensen OC. Norwegian textbook 8.6.6 Maritime Occupational Health (OH) practitioners: Purpose and tasks, Skills and competency. In Norwegian Textbook of Maritime Medicine. 2021 # Fotteler ML, Andrioti Bygvraa D, Jensen OC. The impact of the Maritime Labor Convention on seafarers’ working and living conditions: an analysis of port state control statistics. B M C Public Health. 2020 Dec 1;20(1). 1586. https://doi.org/10.1186/s12889-020-09682-6 # Van de Sijpe P, Lucas D, Canals ML, Jensen O. Acute occupational phosphine intoxications in the maritime shipping sector: Belgian and French reported cases. international Maritime Health. 2020 Sep 28;71(3):151-159. https://doi.org/10.5603/IMH.2020.0028 # Pesel G, Canals ML, Sandrin M, Jensen O. Wellbeing of a selection of seafarers in Eastern Adriatic Sea during the COVID-19 pandemic 2020. international Maritime Health. 2020 Sep 28;71(3):184-190. https://doi.org/10.5603/IMH.2020.0033 # Jensen OC. Social-strategisk, evidensbaseret Covid-19 forebyggelse. Ugeskrift for Laeger. 2020 Sep 14;182(18). gg6nw1z8nvtm392fpsewl2k9j6hwcas 0 281790 2416261 2416036 2022-08-18T19:17:32Z Professorbrendan 2917898 wikitext text/x-wiki <noinclude> Hi there! <strong>If you reached this page after clicking the join button, you can head back to the [[Eventmath/Participants|Participants page]] to see the result.</strong> If you just happened across this page and are looking to browse the list of participants, you'll want to head over to the Participants page as well. The current page only exists for technical reasons. ==How to edit an entry== To edit your entry in the participants list, just edit it directly on this page, by clicking the ''Edit source'' tab near the top of the page. The entry on the Participants page will be updated automatically. ==How to add an entry== Use the Join button on the Participants page. ==What's the purpose of this page?== The current page exists only for technical reasons... Still reading? Okay, here's the technical stuff if you care to know. Entries to the participants list get added to the current page via the Join button of the Participants page, and then the current page is [https://www.mediawiki.org/wiki/Transclusion transcluded] into the Participants page in the appropriate place. Why? Otherwise, the [https://www.mediawiki.org/wiki/Extension:InputBox InputBox extension] that's used to implement the Join button will simply place each entry at the very bottom of the wikitext on the Participants page. When that happens, the entries are placed outside of the main container of the page, so that they appear outside of the Eventmath content. ==The list== [[Category:Eventmath project pages]] </noinclude> ===Greg Stanton (he/him) === * ''Twitter:'' [https://twitter.com/HigherMathNotes @HigherMathNotes] * ''User:'' [[User:Greg at Higher Math Help|Greg at Higher Math Help]] * ''Bio:'' Hi there! I'm a professional tutor of high-school through graduate-level math. I was inspired by a [https://twitter.com/professorbrenda/status/1334257531271712768 conversation on Twitter] to propose Eventmath and am [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working to get this project off the ground. I’m excited to see what we can make together! ===Brendan W. Sullivan (he/him)=== * ''Twitter:'' [https://twitter.com/professorbrenda @professorbrenda] (no final N!) * ''User:'' [[User:Professorbrendan|Professorbrendan]] * ''Bio:'' I am also [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working on this project. I've been teaching undergraduate mathematics for 7+ years (12+ including graduate school) and have experience using current events articles in the classroom to teach quantitative reasoning and problem-solving skills. I hope this project will make it easier for teachers and students to explore how mathematical thinking is useful in our modern world. ===Jyotika Wadhwa=== * ''User:'' [[User:jodubdub|jodubdub]] * ''Bio:'' I am a software engineer turned finance professional - both professions that rely heavily on quantitative reasoning. I am excited about participating in a project that uses real-life examples to further mathematical literacy. ===Heather Charles=== * ''User:'' [[User:Heathercharles|Heathercharles]] * ''Bio:'' I'm a middle school science and math teacher in Tucson, AZ, with a bachelor's degree in physics from Northern Arizona University. I've been teaching secondary math for 5 years and middle school math for 4! '''Ozlem Ugurlu (she/her)''' * ''Twitter:'' [https://twitter.com/ozziugurlu @Ozziugurlu] * ''User:'' [[User:Wikiversity-username|Ougurlu]] * ''Bio:'' I received my Bs and MSc in Mathematics in Turkey. In 2013, I moved to the USA to pursue my doctorate in Mathematics at Tulane University, New Orleans, LA. Since then, I have been teaching undergraduate/graduate level courses. I have no editing experience on Wiki; however, it is important to me that I can provide my students with accessible and tangible learning experiences. So, I am excited to be part of this project. ===Suzie Garfield=== * ''User:'' [[User:Suzie_Garfield|Suzie Garfield]] * ''Bio:'' I’m interested in informal education, particularly in on-the-job and more structured business training and in promoting quantitative literacy by embedding it into the stories I tell as a professional and business storyteller. My experience has been in business, technology, education, and performance. In the past, I’ve worked as a computer programmer and classroom teacher. Now I am mostly a consultant, trainer, storyteller, coach, and mentor. ===Linda McGuire=== * ''User:'' [[Lindamc29]] * ''Bio:'' I have been teaching the full spectrum of undergraduate mathematics courses for over 20 years. I also teach interdisciplinary courses in other programs such as writing, women and gender studies, and social justice. I try my best to cultivate collaborative learning spaces and I am hopeful that the Eventmath wiki will be a site of collaboration with other teaching-focused colleagues around the globe. ===Asia Matthews=== * ''User:'' [[User:Asia.Matthews|Asia.Matthews]] * ''Bio:'' I teach undergraduates how to think mathematically and to use this knowledge to figure out how to be in this world as one of many. I’ve been teaching undergraduate mathematics since my Master’s degree in 2005 and I like to push my students to be mathematically creative. As of 2022 I have been teaching at Quest University Canada for +/- 5 years. ===Jessie Oehrlein (she/her)=== * ''Twitter:'' [https://twitter.com/numberdance @numberdance * ''User:'' [[User:Jessoehrlein|Jessoehrlein]] * ''Bio:'' I teach undergraduate statistics and mathematics and have been at Fitchburg State University since 2021. By research area, I'm at atmospheric/climate scientist. dzk07qe4s1n0rfh5arah8nugzi39rrv 2416262 2416261 2022-08-18T19:18:43Z Professorbrendan 2917898 wikitext text/x-wiki <noinclude> Hi there! <strong>If you reached this page after clicking the join button, you can head back to the [[Eventmath/Participants|Participants page]] to see the result.</strong> If you just happened across this page and are looking to browse the list of participants, you'll want to head over to the Participants page as well. The current page only exists for technical reasons. ==How to edit an entry== To edit your entry in the participants list, just edit it directly on this page, by clicking the ''Edit source'' tab near the top of the page. The entry on the Participants page will be updated automatically. ==How to add an entry== Use the Join button on the Participants page. ==What's the purpose of this page?== The current page exists only for technical reasons... Still reading? Okay, here's the technical stuff if you care to know. Entries to the participants list get added to the current page via the Join button of the Participants page, and then the current page is [https://www.mediawiki.org/wiki/Transclusion transcluded] into the Participants page in the appropriate place. Why? Otherwise, the [https://www.mediawiki.org/wiki/Extension:InputBox InputBox extension] that's used to implement the Join button will simply place each entry at the very bottom of the wikitext on the Participants page. When that happens, the entries are placed outside of the main container of the page, so that they appear outside of the Eventmath content. ==The list== [[Category:Eventmath project pages]] </noinclude> ===Greg Stanton (he/him) === * ''Twitter:'' [https://twitter.com/HigherMathNotes @HigherMathNotes] * ''User:'' [[User:Greg at Higher Math Help|Greg at Higher Math Help]] * ''Bio:'' Hi there! I'm a professional tutor of high-school through graduate-level math. I was inspired by a [https://twitter.com/professorbrenda/status/1334257531271712768 conversation on Twitter] to propose Eventmath and am [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working to get this project off the ground. I’m excited to see what we can make together! ===Brendan W. Sullivan (he/him)=== * ''Twitter:'' [https://twitter.com/professorbrenda @professorbrenda] (no final N!) * ''User:'' [[User:Professorbrendan|Professorbrendan]] * ''Bio:'' I am also [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working on this project. I've been teaching undergraduate mathematics for 7+ years (12+ including graduate school) and have experience using current events articles in the classroom to teach quantitative reasoning and problem-solving skills. I hope this project will make it easier for teachers and students to explore how mathematical thinking is useful in our modern world. ===Jyotika Wadhwa=== * ''User:'' [[User:jodubdub|jodubdub]] * ''Bio:'' I am a software engineer turned finance professional - both professions that rely heavily on quantitative reasoning. I am excited about participating in a project that uses real-life examples to further mathematical literacy. ===Heather Charles=== * ''User:'' [[User:Heathercharles|Heathercharles]] * ''Bio:'' I'm a middle school science and math teacher in Tucson, AZ, with a bachelor's degree in physics from Northern Arizona University. I've been teaching secondary math for 5 years and middle school math for 4! '''Ozlem Ugurlu (she/her)''' * ''Twitter:'' [https://twitter.com/ozziugurlu @Ozziugurlu] * ''User:'' [[User:Wikiversity-username|Ougurlu]] * ''Bio:'' I received my Bs and MSc in Mathematics in Turkey. In 2013, I moved to the USA to pursue my doctorate in Mathematics at Tulane University, New Orleans, LA. Since then, I have been teaching undergraduate/graduate level courses. I have no editing experience on Wiki; however, it is important to me that I can provide my students with accessible and tangible learning experiences. So, I am excited to be part of this project. ===Suzie Garfield=== * ''User:'' [[User:Suzie_Garfield|Suzie Garfield]] * ''Bio:'' I’m interested in informal education, particularly in on-the-job and more structured business training and in promoting quantitative literacy by embedding it into the stories I tell as a professional and business storyteller. My experience has been in business, technology, education, and performance. In the past, I’ve worked as a computer programmer and classroom teacher. Now I am mostly a consultant, trainer, storyteller, coach, and mentor. ===Linda McGuire=== * ''User:'' [[Lindamc29]] * ''Bio:'' I have been teaching the full spectrum of undergraduate mathematics courses for over 20 years. I also teach interdisciplinary courses in other programs such as writing, women and gender studies, and social justice. I try my best to cultivate collaborative learning spaces and I am hopeful that the Eventmath wiki will be a site of collaboration with other teaching-focused colleagues around the globe. ===Asia Matthews=== * ''User:'' [[User:Asia.Matthews|Asia.Matthews]] * ''Bio:'' I teach undergraduates how to think mathematically and to use this knowledge to figure out how to be in this world as one of many. I’ve been teaching undergraduate mathematics since my Master’s degree in 2005 and I like to push my students to be mathematically creative. As of 2022 I have been teaching at Quest University Canada for +/- 5 years. ===Jessie Oehrlein (she/her)=== * ''Twitter:'' [https://twitter.com/numberdance @numberdance * ''User:'' [[User:Jessoehrlein|Jessoehrlein]] * ''Bio:'' I teach undergraduate statistics and mathematics and have been at Fitchburg State University since 2021. By research area, I'm at atmospheric/climate scientist. ===Debasmita Basu=== * ''User:'' [[User:basud2|basud2]] * ''Bio:'' I teach Quantitative Reasoning and Mathematics at Eugene Lang College of Liberal Arts at The New School, and I am the Director of the Quantitative Reasoning program at my institution. Before pursuing my doctoral studies, I was a high school mathematics teacher in India for 4 years. As a cisgender woman of color, I aim to design mathematical activities that cultivate students’ critical consciousness towards various social and environmental justice issues and help them realize the power of mathematics. ssb7k3yw8qt6fdrvczdufzyqev52tx9 2416263 2416262 2022-08-18T19:19:00Z Professorbrendan 2917898 /* Jessie Oehrlein (she/her) */ wikitext text/x-wiki <noinclude> Hi there! <strong>If you reached this page after clicking the join button, you can head back to the [[Eventmath/Participants|Participants page]] to see the result.</strong> If you just happened across this page and are looking to browse the list of participants, you'll want to head over to the Participants page as well. The current page only exists for technical reasons. ==How to edit an entry== To edit your entry in the participants list, just edit it directly on this page, by clicking the ''Edit source'' tab near the top of the page. The entry on the Participants page will be updated automatically. ==How to add an entry== Use the Join button on the Participants page. ==What's the purpose of this page?== The current page exists only for technical reasons... Still reading? Okay, here's the technical stuff if you care to know. Entries to the participants list get added to the current page via the Join button of the Participants page, and then the current page is [https://www.mediawiki.org/wiki/Transclusion transcluded] into the Participants page in the appropriate place. Why? Otherwise, the [https://www.mediawiki.org/wiki/Extension:InputBox InputBox extension] that's used to implement the Join button will simply place each entry at the very bottom of the wikitext on the Participants page. When that happens, the entries are placed outside of the main container of the page, so that they appear outside of the Eventmath content. ==The list== [[Category:Eventmath project pages]] </noinclude> ===Greg Stanton (he/him) === * ''Twitter:'' [https://twitter.com/HigherMathNotes @HigherMathNotes] * ''User:'' [[User:Greg at Higher Math Help|Greg at Higher Math Help]] * ''Bio:'' Hi there! I'm a professional tutor of high-school through graduate-level math. I was inspired by a [https://twitter.com/professorbrenda/status/1334257531271712768 conversation on Twitter] to propose Eventmath and am [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working to get this project off the ground. I’m excited to see what we can make together! ===Brendan W. Sullivan (he/him)=== * ''Twitter:'' [https://twitter.com/professorbrenda @professorbrenda] (no final N!) * ''User:'' [[User:Professorbrendan|Professorbrendan]] * ''Bio:'' I am also [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working on this project. I've been teaching undergraduate mathematics for 7+ years (12+ including graduate school) and have experience using current events articles in the classroom to teach quantitative reasoning and problem-solving skills. I hope this project will make it easier for teachers and students to explore how mathematical thinking is useful in our modern world. ===Jyotika Wadhwa=== * ''User:'' [[User:jodubdub|jodubdub]] * ''Bio:'' I am a software engineer turned finance professional - both professions that rely heavily on quantitative reasoning. I am excited about participating in a project that uses real-life examples to further mathematical literacy. ===Heather Charles=== * ''User:'' [[User:Heathercharles|Heathercharles]] * ''Bio:'' I'm a middle school science and math teacher in Tucson, AZ, with a bachelor's degree in physics from Northern Arizona University. I've been teaching secondary math for 5 years and middle school math for 4! '''Ozlem Ugurlu (she/her)''' * ''Twitter:'' [https://twitter.com/ozziugurlu @Ozziugurlu] * ''User:'' [[User:Wikiversity-username|Ougurlu]] * ''Bio:'' I received my Bs and MSc in Mathematics in Turkey. In 2013, I moved to the USA to pursue my doctorate in Mathematics at Tulane University, New Orleans, LA. Since then, I have been teaching undergraduate/graduate level courses. I have no editing experience on Wiki; however, it is important to me that I can provide my students with accessible and tangible learning experiences. So, I am excited to be part of this project. ===Suzie Garfield=== * ''User:'' [[User:Suzie_Garfield|Suzie Garfield]] * ''Bio:'' I’m interested in informal education, particularly in on-the-job and more structured business training and in promoting quantitative literacy by embedding it into the stories I tell as a professional and business storyteller. My experience has been in business, technology, education, and performance. In the past, I’ve worked as a computer programmer and classroom teacher. Now I am mostly a consultant, trainer, storyteller, coach, and mentor. ===Linda McGuire=== * ''User:'' [[Lindamc29]] * ''Bio:'' I have been teaching the full spectrum of undergraduate mathematics courses for over 20 years. I also teach interdisciplinary courses in other programs such as writing, women and gender studies, and social justice. I try my best to cultivate collaborative learning spaces and I am hopeful that the Eventmath wiki will be a site of collaboration with other teaching-focused colleagues around the globe. ===Asia Matthews=== * ''User:'' [[User:Asia.Matthews|Asia.Matthews]] * ''Bio:'' I teach undergraduates how to think mathematically and to use this knowledge to figure out how to be in this world as one of many. I’ve been teaching undergraduate mathematics since my Master’s degree in 2005 and I like to push my students to be mathematically creative. As of 2022 I have been teaching at Quest University Canada for +/- 5 years. ===Jessie Oehrlein (she/her)=== * ''Twitter:'' [https://twitter.com/numberdance @numberdance] * ''User:'' [[User:Jessoehrlein|Jessoehrlein]] * ''Bio:'' I teach undergraduate statistics and mathematics and have been at Fitchburg State University since 2021. By research area, I'm at atmospheric/climate scientist. ===Debasmita Basu=== * ''User:'' [[User:basud2|basud2]] * ''Bio:'' I teach Quantitative Reasoning and Mathematics at Eugene Lang College of Liberal Arts at The New School, and I am the Director of the Quantitative Reasoning program at my institution. Before pursuing my doctoral studies, I was a high school mathematics teacher in India for 4 years. As a cisgender woman of color, I aim to design mathematical activities that cultivate students’ critical consciousness towards various social and environmental justice issues and help them realize the power of mathematics. 14s6o3bs1a6ig4awn37wqxch7bzvfdj 2416264 2416263 2022-08-18T19:20:53Z Professorbrendan 2917898 wikitext text/x-wiki <noinclude> Hi there! <strong>If you reached this page after clicking the join button, you can head back to the [[Eventmath/Participants|Participants page]] to see the result.</strong> If you just happened across this page and are looking to browse the list of participants, you'll want to head over to the Participants page as well. The current page only exists for technical reasons. ==How to edit an entry== To edit your entry in the participants list, just edit it directly on this page, by clicking the ''Edit source'' tab near the top of the page. The entry on the Participants page will be updated automatically. ==How to add an entry== Use the Join button on the Participants page. ==What's the purpose of this page?== The current page exists only for technical reasons... Still reading? Okay, here's the technical stuff if you care to know. Entries to the participants list get added to the current page via the Join button of the Participants page, and then the current page is [https://www.mediawiki.org/wiki/Transclusion transcluded] into the Participants page in the appropriate place. Why? Otherwise, the [https://www.mediawiki.org/wiki/Extension:InputBox InputBox extension] that's used to implement the Join button will simply place each entry at the very bottom of the wikitext on the Participants page. When that happens, the entries are placed outside of the main container of the page, so that they appear outside of the Eventmath content. ==The list== [[Category:Eventmath project pages]] </noinclude> ===Greg Stanton (he/him) === * ''Twitter:'' [https://twitter.com/HigherMathNotes @HigherMathNotes] * ''User:'' [[User:Greg at Higher Math Help|Greg at Higher Math Help]] * ''Bio:'' Hi there! I'm a professional tutor of high-school through graduate-level math. I was inspired by a [https://twitter.com/professorbrenda/status/1334257531271712768 conversation on Twitter] to propose Eventmath and am [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working to get this project off the ground. I’m excited to see what we can make together! ===Brendan W. Sullivan (he/him)=== * ''Twitter:'' [https://twitter.com/professorbrenda @professorbrenda] (no final N!) * ''User:'' [[User:Professorbrendan|Professorbrendan]] * ''Bio:'' I am also [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working on this project. I've been teaching undergraduate mathematics for 7+ years (12+ including graduate school) and have experience using current events articles in the classroom to teach quantitative reasoning and problem-solving skills. I hope this project will make it easier for teachers and students to explore how mathematical thinking is useful in our modern world. ===Jyotika Wadhwa=== * ''User:'' [[User:jodubdub|jodubdub]] * ''Bio:'' I am a software engineer turned finance professional - both professions that rely heavily on quantitative reasoning. I am excited about participating in a project that uses real-life examples to further mathematical literacy. ===Heather Charles=== * ''User:'' [[User:Heathercharles|Heathercharles]] * ''Bio:'' I'm a middle school science and math teacher in Tucson, AZ, with a bachelor's degree in physics from Northern Arizona University. I've been teaching secondary math for 5 years and middle school math for 4! '''Ozlem Ugurlu (she/her)''' * ''Twitter:'' [https://twitter.com/ozziugurlu @Ozziugurlu] * ''User:'' [[User:Wikiversity-username|Ougurlu]] * ''Bio:'' I received my Bs and MSc in Mathematics in Turkey. In 2013, I moved to the USA to pursue my doctorate in Mathematics at Tulane University, New Orleans, LA. Since then, I have been teaching undergraduate/graduate level courses. I have no editing experience on Wiki; however, it is important to me that I can provide my students with accessible and tangible learning experiences. So, I am excited to be part of this project. ===Suzie Garfield=== * ''User:'' [[User:Suzie_Garfield|Suzie Garfield]] * ''Bio:'' I’m interested in informal education, particularly in on-the-job and more structured business training and in promoting quantitative literacy by embedding it into the stories I tell as a professional and business storyteller. My experience has been in business, technology, education, and performance. In the past, I’ve worked as a computer programmer and classroom teacher. Now I am mostly a consultant, trainer, storyteller, coach, and mentor. ===Linda McGuire=== * ''User:'' [[Lindamc29]] * ''Bio:'' I have been teaching the full spectrum of undergraduate mathematics courses for over 20 years. I also teach interdisciplinary courses in other programs such as writing, women and gender studies, and social justice. I try my best to cultivate collaborative learning spaces and I am hopeful that the Eventmath wiki will be a site of collaboration with other teaching-focused colleagues around the globe. ===Asia Matthews=== * ''User:'' [[User:Asia.Matthews|Asia.Matthews]] * ''Bio:'' I teach undergraduates how to think mathematically and to use this knowledge to figure out how to be in this world as one of many. I’ve been teaching undergraduate mathematics since my Master’s degree in 2005 and I like to push my students to be mathematically creative. As of 2022 I have been teaching at Quest University Canada for +/- 5 years. ===Jessie Oehrlein (she/her)=== * ''Twitter:'' [https://twitter.com/numberdance @numberdance] * ''User:'' [[User:Jessoehrlein|Jessoehrlein]] * ''Bio:'' I teach undergraduate statistics and mathematics and have been at Fitchburg State University since 2021. By research area, I'm at atmospheric/climate scientist. ===Debasmita Basu=== * ''User:'' [[User:basud2|basud2]] * ''Bio:'' I teach Quantitative Reasoning and Mathematics at Eugene Lang College of Liberal Arts at The New School, and I am the Director of the Quantitative Reasoning program at my institution. Before pursuing my doctoral studies, I was a high school mathematics teacher in India for 4 years. As a cisgender woman of color, I aim to design mathematical activities that cultivate students’ critical consciousness towards various social and environmental justice issues and help them realize the power of mathematics. ===Carrie Muir=== * ''User:'' [[User:Carrie Muir|Carrie Muir]] * ''Bio:'' I'm a community college math professor, and have been teaching undergrad mathematics since before Y2K. I frequently teach quantitative literacy and statistics courses. d7xix7u5vu22tfplej53mnb1vt9zfih 2416265 2416264 2022-08-18T19:21:46Z Professorbrendan 2917898 wikitext text/x-wiki <noinclude> Hi there! <strong>If you reached this page after clicking the join button, you can head back to the [[Eventmath/Participants|Participants page]] to see the result.</strong> If you just happened across this page and are looking to browse the list of participants, you'll want to head over to the Participants page as well. The current page only exists for technical reasons. ==How to edit an entry== To edit your entry in the participants list, just edit it directly on this page, by clicking the ''Edit source'' tab near the top of the page. The entry on the Participants page will be updated automatically. ==How to add an entry== Use the Join button on the Participants page. ==What's the purpose of this page?== The current page exists only for technical reasons... Still reading? Okay, here's the technical stuff if you care to know. Entries to the participants list get added to the current page via the Join button of the Participants page, and then the current page is [https://www.mediawiki.org/wiki/Transclusion transcluded] into the Participants page in the appropriate place. Why? Otherwise, the [https://www.mediawiki.org/wiki/Extension:InputBox InputBox extension] that's used to implement the Join button will simply place each entry at the very bottom of the wikitext on the Participants page. When that happens, the entries are placed outside of the main container of the page, so that they appear outside of the Eventmath content. ==The list== [[Category:Eventmath project pages]] </noinclude> ===Greg Stanton (he/him) === * ''Twitter:'' [https://twitter.com/HigherMathNotes @HigherMathNotes] * ''User:'' [[User:Greg at Higher Math Help|Greg at Higher Math Help]] * ''Bio:'' Hi there! I'm a professional tutor of high-school through graduate-level math. I was inspired by a [https://twitter.com/professorbrenda/status/1334257531271712768 conversation on Twitter] to propose Eventmath and am [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working to get this project off the ground. I’m excited to see what we can make together! ===Brendan W. Sullivan (he/him)=== * ''Twitter:'' [https://twitter.com/professorbrenda @professorbrenda] (no final N!) * ''User:'' [[User:Professorbrendan|Professorbrendan]] * ''Bio:'' I am also [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working on this project. I've been teaching undergraduate mathematics for 7+ years (12+ including graduate school) and have experience using current events articles in the classroom to teach quantitative reasoning and problem-solving skills. I hope this project will make it easier for teachers and students to explore how mathematical thinking is useful in our modern world. ===Jyotika Wadhwa=== * ''User:'' [[User:jodubdub|jodubdub]] * ''Bio:'' I am a software engineer turned finance professional - both professions that rely heavily on quantitative reasoning. I am excited about participating in a project that uses real-life examples to further mathematical literacy. ===Heather Charles=== * ''User:'' [[User:Heathercharles|Heathercharles]] * ''Bio:'' I'm a middle school science and math teacher in Tucson, AZ, with a bachelor's degree in physics from Northern Arizona University. I've been teaching secondary math for 5 years and middle school math for 4! '''Ozlem Ugurlu (she/her)''' * ''Twitter:'' [https://twitter.com/ozziugurlu @Ozziugurlu] * ''User:'' [[User:Wikiversity-username|Ougurlu]] * ''Bio:'' I received my Bs and MSc in Mathematics in Turkey. In 2013, I moved to the USA to pursue my doctorate in Mathematics at Tulane University, New Orleans, LA. Since then, I have been teaching undergraduate/graduate level courses. I have no editing experience on Wiki; however, it is important to me that I can provide my students with accessible and tangible learning experiences. So, I am excited to be part of this project. ===Suzie Garfield=== * ''User:'' [[User:Suzie_Garfield|Suzie Garfield]] * ''Bio:'' I’m interested in informal education, particularly in on-the-job and more structured business training and in promoting quantitative literacy by embedding it into the stories I tell as a professional and business storyteller. My experience has been in business, technology, education, and performance. In the past, I’ve worked as a computer programmer and classroom teacher. Now I am mostly a consultant, trainer, storyteller, coach, and mentor. ===Linda McGuire=== * ''User:'' [[Lindamc29]] * ''Bio:'' I have been teaching the full spectrum of undergraduate mathematics courses for over 20 years. I also teach interdisciplinary courses in other programs such as writing, women and gender studies, and social justice. I try my best to cultivate collaborative learning spaces and I am hopeful that the Eventmath wiki will be a site of collaboration with other teaching-focused colleagues around the globe. ===Asia Matthews=== * ''User:'' [[User:Asia.Matthews|Asia.Matthews]] * ''Bio:'' I teach undergraduates how to think mathematically and to use this knowledge to figure out how to be in this world as one of many. I’ve been teaching undergraduate mathematics since my Master’s degree in 2005 and I like to push my students to be mathematically creative. As of 2022 I have been teaching at Quest University Canada for +/- 5 years. ===Jessie Oehrlein (she/her)=== * ''Twitter:'' [https://twitter.com/numberdance @numberdance] * ''User:'' [[User:Jessoehrlein|Jessoehrlein]] * ''Bio:'' I teach undergraduate statistics and mathematics and have been at Fitchburg State University since 2021. By research area, I'm at atmospheric/climate scientist. ===Debasmita Basu=== * ''User:'' [[User:basud2|basud2]] * ''Bio:'' I teach Quantitative Reasoning and Mathematics at Eugene Lang College of Liberal Arts at The New School, and I am the Director of the Quantitative Reasoning program at my institution. Before pursuing my doctoral studies, I was a high school mathematics teacher in India for 4 years. As a cisgender woman of color, I aim to design mathematical activities that cultivate students’ critical consciousness towards various social and environmental justice issues and help them realize the power of mathematics. ===Carrie Muir=== * ''User:'' [[User:Carrie Muir|Carrie Muir]] * ''Bio:'' I'm a community college math professor, and have been teaching undergrad mathematics since before Y2K. I frequently teach quantitative literacy and statistics courses. ===Dave Jehle=== * ''User:'' [[User:davidejehle|davidejehle]] * ''Bio:'' BS in Math (RPI), MA in Math (SUNY at Buffalo) and MBA (RIT). Worked as a manufacturing systems consultant for 30+ years. As consulting engagements wound down, began adjunct teaching over the last seventeen years. Have taught a wide range of undergraduate classes from Basic Mathematics through Calculus, Statistics and Discrete Mathematics. ffyd1rma9piyqr3zlj893lxruiedlo6 2416267 2416265 2022-08-18T19:22:47Z Professorbrendan 2917898 wikitext text/x-wiki <noinclude> Hi there! <strong>If you reached this page after clicking the join button, you can head back to the [[Eventmath/Participants|Participants page]] to see the result.</strong> If you just happened across this page and are looking to browse the list of participants, you'll want to head over to the Participants page as well. The current page only exists for technical reasons. ==How to edit an entry== To edit your entry in the participants list, just edit it directly on this page, by clicking the ''Edit source'' tab near the top of the page. The entry on the Participants page will be updated automatically. ==How to add an entry== Use the Join button on the Participants page. ==What's the purpose of this page?== The current page exists only for technical reasons... Still reading? Okay, here's the technical stuff if you care to know. Entries to the participants list get added to the current page via the Join button of the Participants page, and then the current page is [https://www.mediawiki.org/wiki/Transclusion transcluded] into the Participants page in the appropriate place. Why? Otherwise, the [https://www.mediawiki.org/wiki/Extension:InputBox InputBox extension] that's used to implement the Join button will simply place each entry at the very bottom of the wikitext on the Participants page. When that happens, the entries are placed outside of the main container of the page, so that they appear outside of the Eventmath content. ==The list== [[Category:Eventmath project pages]] </noinclude> ===Greg Stanton (he/him) === * ''Twitter:'' [https://twitter.com/HigherMathNotes @HigherMathNotes] * ''User:'' [[User:Greg at Higher Math Help|Greg at Higher Math Help]] * ''Bio:'' Hi there! I'm a professional tutor of high-school through graduate-level math. I was inspired by a [https://twitter.com/professorbrenda/status/1334257531271712768 conversation on Twitter] to propose Eventmath and am [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working to get this project off the ground. I’m excited to see what we can make together! ===Brendan W. Sullivan (he/him)=== * ''Twitter:'' [https://twitter.com/professorbrenda @professorbrenda] (no final N!) * ''User:'' [[User:Professorbrendan|Professorbrendan]] * ''Bio:'' I am also [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working on this project. I've been teaching undergraduate mathematics for 7+ years (12+ including graduate school) and have experience using current events articles in the classroom to teach quantitative reasoning and problem-solving skills. I hope this project will make it easier for teachers and students to explore how mathematical thinking is useful in our modern world. ===Jyotika Wadhwa=== * ''User:'' [[User:jodubdub|jodubdub]] * ''Bio:'' I am a software engineer turned finance professional - both professions that rely heavily on quantitative reasoning. I am excited about participating in a project that uses real-life examples to further mathematical literacy. ===Heather Charles=== * ''User:'' [[User:Heathercharles|Heathercharles]] * ''Bio:'' I'm a middle school science and math teacher in Tucson, AZ, with a bachelor's degree in physics from Northern Arizona University. I've been teaching secondary math for 5 years and middle school math for 4! '''Ozlem Ugurlu (she/her)''' * ''Twitter:'' [https://twitter.com/ozziugurlu @Ozziugurlu] * ''User:'' [[User:Wikiversity-username|Ougurlu]] * ''Bio:'' I received my Bs and MSc in Mathematics in Turkey. In 2013, I moved to the USA to pursue my doctorate in Mathematics at Tulane University, New Orleans, LA. Since then, I have been teaching undergraduate/graduate level courses. I have no editing experience on Wiki; however, it is important to me that I can provide my students with accessible and tangible learning experiences. So, I am excited to be part of this project. ===Suzie Garfield=== * ''User:'' [[User:Suzie_Garfield|Suzie Garfield]] * ''Bio:'' I’m interested in informal education, particularly in on-the-job and more structured business training and in promoting quantitative literacy by embedding it into the stories I tell as a professional and business storyteller. My experience has been in business, technology, education, and performance. In the past, I’ve worked as a computer programmer and classroom teacher. Now I am mostly a consultant, trainer, storyteller, coach, and mentor. ===Linda McGuire=== * ''User:'' [[Lindamc29]] * ''Bio:'' I have been teaching the full spectrum of undergraduate mathematics courses for over 20 years. I also teach interdisciplinary courses in other programs such as writing, women and gender studies, and social justice. I try my best to cultivate collaborative learning spaces and I am hopeful that the Eventmath wiki will be a site of collaboration with other teaching-focused colleagues around the globe. ===Asia Matthews=== * ''User:'' [[User:Asia.Matthews|Asia.Matthews]] * ''Bio:'' I teach undergraduates how to think mathematically and to use this knowledge to figure out how to be in this world as one of many. I’ve been teaching undergraduate mathematics since my Master’s degree in 2005 and I like to push my students to be mathematically creative. As of 2022 I have been teaching at Quest University Canada for +/- 5 years. ===Jessie Oehrlein (she/her)=== * ''Twitter:'' [https://twitter.com/numberdance @numberdance] * ''User:'' [[User:Jessoehrlein|Jessoehrlein]] * ''Bio:'' I teach undergraduate statistics and mathematics and have been at Fitchburg State University since 2021. By research area, I'm at atmospheric/climate scientist. ===Debasmita Basu=== * ''User:'' [[User:basud2|basud2]] * ''Bio:'' I teach Quantitative Reasoning and Mathematics at Eugene Lang College of Liberal Arts at The New School, and I am the Director of the Quantitative Reasoning program at my institution. Before pursuing my doctoral studies, I was a high school mathematics teacher in India for 4 years. As a cisgender woman of color, I aim to design mathematical activities that cultivate students’ critical consciousness towards various social and environmental justice issues and help them realize the power of mathematics. ===Carrie Muir=== * ''User:'' [[User:Carrie Muir|Carrie Muir]] * ''Bio:'' I'm a community college math professor, and have been teaching undergrad mathematics since before Y2K. I frequently teach quantitative literacy and statistics courses. ===Dave Jehle=== * ''User:'' [[User:davidejehle|davidejehle]] * ''Bio:'' BS in Math (RPI), MA in Math (SUNY at Buffalo) and MBA (RIT). Worked as a manufacturing systems consultant for 30+ years. As consulting engagements wound down, began adjunct teaching over the last seventeen years. Have taught a wide range of undergraduate classes from Basic Mathematics through Calculus, Statistics and Discrete Mathematics. ===Phillip Andreae=== * ''User:'' [[User:Phillip Andreae|Phillip Andreae] * ''Bio:'' I teach courses including statistics, calculus, proofs, and geometry at Meredith College in Raleigh, NC. My background is in differential geometry, but recently I have gotten interested in math and democracy, especially gerrymandering and voting methods. I enjoy math when it makes sense, and I want to help students make their own sense out of math, including math in the news i19e810owk9x3wjf63tr6zbfhn6xiqa 2416268 2416267 2022-08-18T19:23:06Z Professorbrendan 2917898 /* Phillip Andreae */ wikitext text/x-wiki <noinclude> Hi there! <strong>If you reached this page after clicking the join button, you can head back to the [[Eventmath/Participants|Participants page]] to see the result.</strong> If you just happened across this page and are looking to browse the list of participants, you'll want to head over to the Participants page as well. The current page only exists for technical reasons. ==How to edit an entry== To edit your entry in the participants list, just edit it directly on this page, by clicking the ''Edit source'' tab near the top of the page. The entry on the Participants page will be updated automatically. ==How to add an entry== Use the Join button on the Participants page. ==What's the purpose of this page?== The current page exists only for technical reasons... Still reading? Okay, here's the technical stuff if you care to know. Entries to the participants list get added to the current page via the Join button of the Participants page, and then the current page is [https://www.mediawiki.org/wiki/Transclusion transcluded] into the Participants page in the appropriate place. Why? Otherwise, the [https://www.mediawiki.org/wiki/Extension:InputBox InputBox extension] that's used to implement the Join button will simply place each entry at the very bottom of the wikitext on the Participants page. When that happens, the entries are placed outside of the main container of the page, so that they appear outside of the Eventmath content. ==The list== [[Category:Eventmath project pages]] </noinclude> ===Greg Stanton (he/him) === * ''Twitter:'' [https://twitter.com/HigherMathNotes @HigherMathNotes] * ''User:'' [[User:Greg at Higher Math Help|Greg at Higher Math Help]] * ''Bio:'' Hi there! I'm a professional tutor of high-school through graduate-level math. I was inspired by a [https://twitter.com/professorbrenda/status/1334257531271712768 conversation on Twitter] to propose Eventmath and am [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working to get this project off the ground. I’m excited to see what we can make together! ===Brendan W. Sullivan (he/him)=== * ''Twitter:'' [https://twitter.com/professorbrenda @professorbrenda] (no final N!) * ''User:'' [[User:Professorbrendan|Professorbrendan]] * ''Bio:'' I am also [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working on this project. I've been teaching undergraduate mathematics for 7+ years (12+ including graduate school) and have experience using current events articles in the classroom to teach quantitative reasoning and problem-solving skills. I hope this project will make it easier for teachers and students to explore how mathematical thinking is useful in our modern world. ===Jyotika Wadhwa=== * ''User:'' [[User:jodubdub|jodubdub]] * ''Bio:'' I am a software engineer turned finance professional - both professions that rely heavily on quantitative reasoning. I am excited about participating in a project that uses real-life examples to further mathematical literacy. ===Heather Charles=== * ''User:'' [[User:Heathercharles|Heathercharles]] * ''Bio:'' I'm a middle school science and math teacher in Tucson, AZ, with a bachelor's degree in physics from Northern Arizona University. I've been teaching secondary math for 5 years and middle school math for 4! '''Ozlem Ugurlu (she/her)''' * ''Twitter:'' [https://twitter.com/ozziugurlu @Ozziugurlu] * ''User:'' [[User:Wikiversity-username|Ougurlu]] * ''Bio:'' I received my Bs and MSc in Mathematics in Turkey. In 2013, I moved to the USA to pursue my doctorate in Mathematics at Tulane University, New Orleans, LA. Since then, I have been teaching undergraduate/graduate level courses. I have no editing experience on Wiki; however, it is important to me that I can provide my students with accessible and tangible learning experiences. So, I am excited to be part of this project. ===Suzie Garfield=== * ''User:'' [[User:Suzie_Garfield|Suzie Garfield]] * ''Bio:'' I’m interested in informal education, particularly in on-the-job and more structured business training and in promoting quantitative literacy by embedding it into the stories I tell as a professional and business storyteller. My experience has been in business, technology, education, and performance. In the past, I’ve worked as a computer programmer and classroom teacher. Now I am mostly a consultant, trainer, storyteller, coach, and mentor. ===Linda McGuire=== * ''User:'' [[Lindamc29]] * ''Bio:'' I have been teaching the full spectrum of undergraduate mathematics courses for over 20 years. I also teach interdisciplinary courses in other programs such as writing, women and gender studies, and social justice. I try my best to cultivate collaborative learning spaces and I am hopeful that the Eventmath wiki will be a site of collaboration with other teaching-focused colleagues around the globe. ===Asia Matthews=== * ''User:'' [[User:Asia.Matthews|Asia.Matthews]] * ''Bio:'' I teach undergraduates how to think mathematically and to use this knowledge to figure out how to be in this world as one of many. I’ve been teaching undergraduate mathematics since my Master’s degree in 2005 and I like to push my students to be mathematically creative. As of 2022 I have been teaching at Quest University Canada for +/- 5 years. ===Jessie Oehrlein (she/her)=== * ''Twitter:'' [https://twitter.com/numberdance @numberdance] * ''User:'' [[User:Jessoehrlein|Jessoehrlein]] * ''Bio:'' I teach undergraduate statistics and mathematics and have been at Fitchburg State University since 2021. By research area, I'm at atmospheric/climate scientist. ===Debasmita Basu=== * ''User:'' [[User:basud2|basud2]] * ''Bio:'' I teach Quantitative Reasoning and Mathematics at Eugene Lang College of Liberal Arts at The New School, and I am the Director of the Quantitative Reasoning program at my institution. Before pursuing my doctoral studies, I was a high school mathematics teacher in India for 4 years. As a cisgender woman of color, I aim to design mathematical activities that cultivate students’ critical consciousness towards various social and environmental justice issues and help them realize the power of mathematics. ===Carrie Muir=== * ''User:'' [[User:Carrie Muir|Carrie Muir]] * ''Bio:'' I'm a community college math professor, and have been teaching undergrad mathematics since before Y2K. I frequently teach quantitative literacy and statistics courses. ===Dave Jehle=== * ''User:'' [[User:davidejehle|davidejehle]] * ''Bio:'' BS in Math (RPI), MA in Math (SUNY at Buffalo) and MBA (RIT). Worked as a manufacturing systems consultant for 30+ years. As consulting engagements wound down, began adjunct teaching over the last seventeen years. Have taught a wide range of undergraduate classes from Basic Mathematics through Calculus, Statistics and Discrete Mathematics. ===Phillip Andreae=== * ''User:'' [[User:Phillip Andreae|Phillip Andreae]] * ''Bio:'' I teach courses including statistics, calculus, proofs, and geometry at Meredith College in Raleigh, NC. My background is in differential geometry, but recently I have gotten interested in math and democracy, especially gerrymandering and voting methods. I enjoy math when it makes sense, and I want to help students make their own sense out of math, including math in the news 61hdlnm9tsj8kf3kfo7kwt1m8h7zzpo 2416269 2416268 2022-08-18T19:28:05Z Professorbrendan 2917898 /* Heather Charles */ wikitext text/x-wiki <noinclude> Hi there! <strong>If you reached this page after clicking the join button, you can head back to the [[Eventmath/Participants|Participants page]] to see the result.</strong> If you just happened across this page and are looking to browse the list of participants, you'll want to head over to the Participants page as well. The current page only exists for technical reasons. ==How to edit an entry== To edit your entry in the participants list, just edit it directly on this page, by clicking the ''Edit source'' tab near the top of the page. The entry on the Participants page will be updated automatically. ==How to add an entry== Use the Join button on the Participants page. ==What's the purpose of this page?== The current page exists only for technical reasons... Still reading? Okay, here's the technical stuff if you care to know. Entries to the participants list get added to the current page via the Join button of the Participants page, and then the current page is [https://www.mediawiki.org/wiki/Transclusion transcluded] into the Participants page in the appropriate place. Why? Otherwise, the [https://www.mediawiki.org/wiki/Extension:InputBox InputBox extension] that's used to implement the Join button will simply place each entry at the very bottom of the wikitext on the Participants page. When that happens, the entries are placed outside of the main container of the page, so that they appear outside of the Eventmath content. ==The list== [[Category:Eventmath project pages]] </noinclude> ===Greg Stanton (he/him) === * ''Twitter:'' [https://twitter.com/HigherMathNotes @HigherMathNotes] * ''User:'' [[User:Greg at Higher Math Help|Greg at Higher Math Help]] * ''Bio:'' Hi there! I'm a professional tutor of high-school through graduate-level math. I was inspired by a [https://twitter.com/professorbrenda/status/1334257531271712768 conversation on Twitter] to propose Eventmath and am [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working to get this project off the ground. I’m excited to see what we can make together! ===Brendan W. Sullivan (he/him)=== * ''Twitter:'' [https://twitter.com/professorbrenda @professorbrenda] (no final N!) * ''User:'' [[User:Professorbrendan|Professorbrendan]] * ''Bio:'' I am also [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working on this project. I've been teaching undergraduate mathematics for 7+ years (12+ including graduate school) and have experience using current events articles in the classroom to teach quantitative reasoning and problem-solving skills. I hope this project will make it easier for teachers and students to explore how mathematical thinking is useful in our modern world. ===Jyotika Wadhwa=== * ''User:'' [[User:jodubdub|jodubdub]] * ''Bio:'' I am a software engineer turned finance professional - both professions that rely heavily on quantitative reasoning. I am excited about participating in a project that uses real-life examples to further mathematical literacy. ===Heather Charles=== * ''User:'' [[User:Heathercharles|Heathercharles]] * ''Bio:'' I'm a middle school science and math teacher in Tucson, AZ, with a bachelor's degree in physics from Northern Arizona University. I've been teaching secondary math for 5 years and middle school math for 4! ===Suzie Garfield=== * ''User:'' [[User:Suzie_Garfield|Suzie Garfield]] * ''Bio:'' I’m interested in informal education, particularly in on-the-job and more structured business training and in promoting quantitative literacy by embedding it into the stories I tell as a professional and business storyteller. My experience has been in business, technology, education, and performance. In the past, I’ve worked as a computer programmer and classroom teacher. Now I am mostly a consultant, trainer, storyteller, coach, and mentor. ===Linda McGuire=== * ''User:'' [[Lindamc29]] * ''Bio:'' I have been teaching the full spectrum of undergraduate mathematics courses for over 20 years. I also teach interdisciplinary courses in other programs such as writing, women and gender studies, and social justice. I try my best to cultivate collaborative learning spaces and I am hopeful that the Eventmath wiki will be a site of collaboration with other teaching-focused colleagues around the globe. ===Asia Matthews=== * ''User:'' [[User:Asia.Matthews|Asia.Matthews]] * ''Bio:'' I teach undergraduates how to think mathematically and to use this knowledge to figure out how to be in this world as one of many. I’ve been teaching undergraduate mathematics since my Master’s degree in 2005 and I like to push my students to be mathematically creative. As of 2022 I have been teaching at Quest University Canada for +/- 5 years. ===Jessie Oehrlein (she/her)=== * ''Twitter:'' [https://twitter.com/numberdance @numberdance] * ''User:'' [[User:Jessoehrlein|Jessoehrlein]] * ''Bio:'' I teach undergraduate statistics and mathematics and have been at Fitchburg State University since 2021. By research area, I'm at atmospheric/climate scientist. ===Debasmita Basu=== * ''User:'' [[User:basud2|basud2]] * ''Bio:'' I teach Quantitative Reasoning and Mathematics at Eugene Lang College of Liberal Arts at The New School, and I am the Director of the Quantitative Reasoning program at my institution. Before pursuing my doctoral studies, I was a high school mathematics teacher in India for 4 years. As a cisgender woman of color, I aim to design mathematical activities that cultivate students’ critical consciousness towards various social and environmental justice issues and help them realize the power of mathematics. ===Carrie Muir=== * ''User:'' [[User:Carrie Muir|Carrie Muir]] * ''Bio:'' I'm a community college math professor, and have been teaching undergrad mathematics since before Y2K. I frequently teach quantitative literacy and statistics courses. ===Dave Jehle=== * ''User:'' [[User:davidejehle|davidejehle]] * ''Bio:'' BS in Math (RPI), MA in Math (SUNY at Buffalo) and MBA (RIT). Worked as a manufacturing systems consultant for 30+ years. As consulting engagements wound down, began adjunct teaching over the last seventeen years. Have taught a wide range of undergraduate classes from Basic Mathematics through Calculus, Statistics and Discrete Mathematics. ===Phillip Andreae=== * ''User:'' [[User:Phillip Andreae|Phillip Andreae]] * ''Bio:'' I teach courses including statistics, calculus, proofs, and geometry at Meredith College in Raleigh, NC. My background is in differential geometry, but recently I have gotten interested in math and democracy, especially gerrymandering and voting methods. I enjoy math when it makes sense, and I want to help students make their own sense out of math, including math in the news qk4ff9kak7zo192o79cxz0rt0d0spx3 2416271 2416269 2022-08-18T19:28:40Z Professorbrendan 2917898 wikitext text/x-wiki <noinclude> Hi there! <strong>If you reached this page after clicking the join button, you can head back to the [[Eventmath/Participants|Participants page]] to see the result.</strong> If you just happened across this page and are looking to browse the list of participants, you'll want to head over to the Participants page as well. The current page only exists for technical reasons. ==How to edit an entry== To edit your entry in the participants list, just edit it directly on this page, by clicking the ''Edit source'' tab near the top of the page. The entry on the Participants page will be updated automatically. ==How to add an entry== Use the Join button on the Participants page. ==What's the purpose of this page?== The current page exists only for technical reasons... Still reading? Okay, here's the technical stuff if you care to know. Entries to the participants list get added to the current page via the Join button of the Participants page, and then the current page is [https://www.mediawiki.org/wiki/Transclusion transcluded] into the Participants page in the appropriate place. Why? Otherwise, the [https://www.mediawiki.org/wiki/Extension:InputBox InputBox extension] that's used to implement the Join button will simply place each entry at the very bottom of the wikitext on the Participants page. When that happens, the entries are placed outside of the main container of the page, so that they appear outside of the Eventmath content. ==The list== [[Category:Eventmath project pages]] </noinclude> ===Greg Stanton (he/him) === * ''Twitter:'' [https://twitter.com/HigherMathNotes @HigherMathNotes] * ''User:'' [[User:Greg at Higher Math Help|Greg at Higher Math Help]] * ''Bio:'' Hi there! I'm a professional tutor of high-school through graduate-level math. I was inspired by a [https://twitter.com/professorbrenda/status/1334257531271712768 conversation on Twitter] to propose Eventmath and am [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working to get this project off the ground. I’m excited to see what we can make together! ===Brendan W. Sullivan (he/him)=== * ''Twitter:'' [https://twitter.com/professorbrenda @professorbrenda] (no final N!) * ''User:'' [[User:Professorbrendan|Professorbrendan]] * ''Bio:'' I am also [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working on this project. I've been teaching undergraduate mathematics for 7+ years (12+ including graduate school) and have experience using current events articles in the classroom to teach quantitative reasoning and problem-solving skills. I hope this project will make it easier for teachers and students to explore how mathematical thinking is useful in our modern world. ===Jyotika Wadhwa=== * ''User:'' [[User:jodubdub|jodubdub]] * ''Bio:'' I am a software engineer turned finance professional - both professions that rely heavily on quantitative reasoning. I am excited about participating in a project that uses real-life examples to further mathematical literacy. ===Heather Charles=== * ''User:'' [[User:Heathercharles|Heathercharles]] * ''Bio:'' I'm a middle school science and math teacher in Tucson, AZ, with a bachelor's degree in physics from Northern Arizona University. I've been teaching secondary math for 5 years and middle school math for 4! ===Suzie Garfield=== * ''User:'' [[User:Suzie_Garfield|Suzie Garfield]] * ''Bio:'' I’m interested in informal education, particularly in on-the-job and more structured business training and in promoting quantitative literacy by embedding it into the stories I tell as a professional and business storyteller. My experience has been in business, technology, education, and performance. In the past, I’ve worked as a computer programmer and classroom teacher. Now I am mostly a consultant, trainer, storyteller, coach, and mentor. ===Linda McGuire=== * ''User:'' [[Lindamc29]] * ''Bio:'' I have been teaching the full spectrum of undergraduate mathematics courses for over 20 years. I also teach interdisciplinary courses in other programs such as writing, women and gender studies, and social justice. I try my best to cultivate collaborative learning spaces and I am hopeful that the Eventmath wiki will be a site of collaboration with other teaching-focused colleagues around the globe. ===Asia Matthews=== * ''User:'' [[User:Asia.Matthews|Asia.Matthews]] * ''Bio:'' I teach undergraduates how to think mathematically and to use this knowledge to figure out how to be in this world as one of many. I’ve been teaching undergraduate mathematics since my Master’s degree in 2005 and I like to push my students to be mathematically creative. As of 2022 I have been teaching at Quest University Canada for +/- 5 years. ===Jessie Oehrlein (she/her)=== * ''Twitter:'' [https://twitter.com/numberdance @numberdance] * ''User:'' [[User:Jessoehrlein|Jessoehrlein]] * ''Bio:'' I teach undergraduate statistics and mathematics and have been at Fitchburg State University since 2021. By research area, I'm at atmospheric/climate scientist. ===Debasmita Basu=== * ''User:'' [[User:basud2|basud2]] * ''Bio:'' I teach Quantitative Reasoning and Mathematics at Eugene Lang College of Liberal Arts at The New School, and I am the Director of the Quantitative Reasoning program at my institution. Before pursuing my doctoral studies, I was a high school mathematics teacher in India for 4 years. As a cisgender woman of color, I aim to design mathematical activities that cultivate students’ critical consciousness towards various social and environmental justice issues and help them realize the power of mathematics. ===Carrie Muir=== * ''User:'' [[User:Carrie Muir|Carrie Muir]] * ''Bio:'' I'm a community college math professor, and have been teaching undergrad mathematics since before Y2K. I frequently teach quantitative literacy and statistics courses. ===Dave Jehle=== * ''User:'' [[User:davidejehle|davidejehle]] * ''Bio:'' BS in Math (RPI), MA in Math (SUNY at Buffalo) and MBA (RIT). Worked as a manufacturing systems consultant for 30+ years. As consulting engagements wound down, began adjunct teaching over the last seventeen years. Have taught a wide range of undergraduate classes from Basic Mathematics through Calculus, Statistics and Discrete Mathematics. ===Phillip Andreae=== * ''User:'' [[User:Phillip Andreae|Phillip Andreae]] * ''Bio:'' I teach courses including statistics, calculus, proofs, and geometry at Meredith College in Raleigh, NC. My background is in differential geometry, but recently I have gotten interested in math and democracy, especially gerrymandering and voting methods. I enjoy math when it makes sense, and I want to help students make their own sense out of math, including math in the news ===Ozlem Ugurlu (she/her)=== * ''Twitter:'' [https://twitter.com/ozziugurlu @Ozziugurlu] * ''User:'' [[User:Wikiversity-username|Ougurlu]] * ''Bio:'' I received my Bs and MSc in Mathematics in Turkey. In 2013, I moved to the USA to pursue my doctorate in Mathematics at Tulane University, New Orleans, LA. Since then, I have been teaching undergraduate/graduate level courses. I have no editing experience on Wiki; however, it is important to me that I can provide my students with accessible and tangible learning experiences. So, I am excited to be part of this project. rhs0egryr7ufw451jbu70m56ru8040l 2416384 2416271 2022-08-19T01:12:53Z Professorbrendan 2917898 wikitext text/x-wiki <noinclude> Hi there! <strong>If you reached this page after clicking the join button, you can head back to the [[Eventmath/Participants|Participants page]] to see the result.</strong> If you just happened across this page and are looking to browse the list of participants, you'll want to head over to the Participants page as well. The current page only exists for technical reasons. ==How to edit an entry== To edit your entry in the participants list, just edit it directly on this page, by clicking the ''Edit source'' tab near the top of the page. The entry on the Participants page will be updated automatically. ==How to add an entry== Use the Join button on the Participants page. ==What's the purpose of this page?== The current page exists only for technical reasons... Still reading? Okay, here's the technical stuff if you care to know. Entries to the participants list get added to the current page via the Join button of the Participants page, and then the current page is [https://www.mediawiki.org/wiki/Transclusion transcluded] into the Participants page in the appropriate place. Why? Otherwise, the [https://www.mediawiki.org/wiki/Extension:InputBox InputBox extension] that's used to implement the Join button will simply place each entry at the very bottom of the wikitext on the Participants page. When that happens, the entries are placed outside of the main container of the page, so that they appear outside of the Eventmath content. ==The list== [[Category:Eventmath project pages]] </noinclude> ===Greg Stanton (he/him) === * ''Twitter:'' [https://twitter.com/HigherMathNotes @HigherMathNotes] * ''User:'' [[User:Greg at Higher Math Help|Greg at Higher Math Help]] * ''Bio:'' Hi there! I'm a professional tutor of high-school through graduate-level math. I was inspired by a [https://twitter.com/professorbrenda/status/1334257531271712768 conversation on Twitter] to propose Eventmath and am [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working to get this project off the ground. I’m excited to see what we can make together! ===Brendan W. Sullivan (he/him)=== * ''Twitter:'' [https://twitter.com/professorbrenda @professorbrenda] (no final N!) * ''User:'' [[User:Professorbrendan|Professorbrendan]] * ''Bio:'' I am also [https://meta.wikimedia.org/wiki/Grants:Project/Eventmath#Grantees_and_project_leads one of two Wikimedia grantees] working on this project. I've been teaching undergraduate mathematics for 7+ years (12+ including graduate school) and have experience using current events articles in the classroom to teach quantitative reasoning and problem-solving skills. I hope this project will make it easier for teachers and students to explore how mathematical thinking is useful in our modern world. ===Jyotika Wadhwa=== * ''User:'' [[User:jodubdub|jodubdub]] * ''Bio:'' I am a software engineer turned finance professional - both professions that rely heavily on quantitative reasoning. I am excited about participating in a project that uses real-life examples to further mathematical literacy. ===Heather Charles=== * ''User:'' [[User:Heathercharles|Heathercharles]] * ''Bio:'' I'm a middle school science and math teacher in Tucson, AZ, with a bachelor's degree in physics from Northern Arizona University. I've been teaching secondary math for 5 years and middle school math for 4! ===Suzie Garfield=== * ''User:'' [[User:Suzie_Garfield|Suzie Garfield]] * ''Bio:'' I’m interested in informal education, particularly in on-the-job and more structured business training and in promoting quantitative literacy by embedding it into the stories I tell as a professional and business storyteller. My experience has been in business, technology, education, and performance. In the past, I’ve worked as a computer programmer and classroom teacher. Now I am mostly a consultant, trainer, storyteller, coach, and mentor. ===Linda McGuire=== * ''User:'' [[Lindamc29]] * ''Bio:'' I have been teaching the full spectrum of undergraduate mathematics courses for over 20 years. I also teach interdisciplinary courses in other programs such as writing, women and gender studies, and social justice. I try my best to cultivate collaborative learning spaces and I am hopeful that the Eventmath wiki will be a site of collaboration with other teaching-focused colleagues around the globe. ===Asia Matthews=== * ''User:'' [[User:Asia.Matthews|Asia.Matthews]] * ''Bio:'' I teach undergraduates how to think mathematically and to use this knowledge to figure out how to be in this world as one of many. I’ve been teaching undergraduate mathematics since my Master’s degree in 2005 and I like to push my students to be mathematically creative. As of 2022 I have been teaching at Quest University Canada for +/- 5 years. ===Jessie Oehrlein (she/her)=== * ''Twitter:'' [https://twitter.com/numberdance @numberdance] * ''User:'' [[User:Jessoehrlein|Jessoehrlein]] * ''Bio:'' I teach undergraduate statistics and mathematics and have been at Fitchburg State University since 2021. By research area, I'm at atmospheric/climate scientist. ===Debasmita Basu=== * ''User:'' [[User:basud2|basud2]] * ''Bio:'' I teach Quantitative Reasoning and Mathematics at Eugene Lang College of Liberal Arts at The New School, and I am the Director of the Quantitative Reasoning program at my institution. Before pursuing my doctoral studies, I was a high school mathematics teacher in India for 4 years. As a cisgender woman of color, I aim to design mathematical activities that cultivate students’ critical consciousness towards various social and environmental justice issues and help them realize the power of mathematics. ===Carrie Muir=== * ''User:'' [[User:Carrie Muir|Carrie Muir]] * ''Bio:'' I'm a community college math professor, and have been teaching undergrad mathematics since before Y2K. I frequently teach quantitative literacy and statistics courses. ===Dave Jehle=== * ''User:'' [[User:davidejehle|davidejehle]] * ''Bio:'' BS in Math (RPI), MA in Math (SUNY at Buffalo) and MBA (RIT). Worked as a manufacturing systems consultant for 30+ years. As consulting engagements wound down, began adjunct teaching over the last seventeen years. Have taught a wide range of undergraduate classes from Basic Mathematics through Calculus, Statistics and Discrete Mathematics. ===Phillip Andreae=== * ''User:'' [[User:Phillip Andreae|Phillip Andreae]] * ''Bio:'' I teach courses including statistics, calculus, proofs, and geometry at Meredith College in Raleigh, NC. My background is in differential geometry, but recently I have gotten interested in math and democracy, especially gerrymandering and voting methods. I enjoy math when it makes sense, and I want to help students make their own sense out of math, including math in the news ===Ozlem Ugurlu (she/her)=== * ''Twitter:'' [https://twitter.com/ozziugurlu @Ozziugurlu] * ''User:'' [[User:Wikiversity-username|Ougurlu]] * ''Bio:'' I received my Bs and MSc in Mathematics in Turkey. In 2013, I moved to the USA to pursue my doctorate in Mathematics at Tulane University, New Orleans, LA. Since then, I have been teaching undergraduate/graduate level courses. I have no editing experience on Wiki; however, it is important to me that I can provide my students with accessible and tangible learning experiences. So, I am excited to be part of this project. ===Erik Wallace=== * ''User:'' [[User:ErikWallace|ErikWallace]] * ''Bio:'' I am currently an Assistant Math professor at Hartwick College. I have been teaching since the beginning of my time as a graduate student at Indiana University in 2008, and have taught a wide range of undergraduate classes to both majors and non-majors. My current teaching style emphasizes hands-on approaches, such as guided activities, games, and problem solving. 46qpnej1fnp5j5rmiwj2vsya66dzwit 2 281930 2416367 2409334 2022-08-19T00:09:57Z WikiJournalBot 2938943 Automated Bot List Update wikitext text/x-wiki <noinclude>{{WikiJMed top menu}} {{col-begin}} {{col-break}} </noinclude> {{Issue header|year = 2022 |volume = 9 |issue = 1}} {{WikiJournalBotList |Volume=6 |Issue=1 |Journal=WikiJournal of Medicine }} {{Article info|Q=Q99676829|image=Components of Parenting Stress.png}} {{Article info|Q=Q100400590|image=Signs and symptoms of leptospirosis.svg}} {{Article info|Q=Q105411509|image=DRC cases south and north kivu , ituri.jpg}} {{ListEnd}} <noinclude>{{col-break}} {{WikiJMed right menu}} {{col-end}} [[category:{{ROOTPAGENAME}} issues]] </noinclude> 7hxmz63gwrcxyvm284vuy5cq07uug5e 1 281986 2416455 2382888 2022-08-19T07:51:32Z Rwatson1955 2831054 Ran plagiarism check wikitext text/x-wiki {{#section-h:{{ARTICLEPAGENAMEE}}}} == Plagiarism check == {{Pass}} Report from WMF copyvios tool flagged some false positives (not regarded as plagiarism) due to common stock phrases. Totalled 23.1% with 'violation unlikely' with regard to plagiarism. Please see the results here [https://copyvios.toolforge.org/?lang=en&project=wikiversity&action=search&turnitin=1&title=WikiJournal_Preprints/Body_image_disturbance_in_eating_disorders] [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) ==Editorial note== Plagiarism check indicates 23.1% similarity to other sources meaning 'Violation unlikely': https://copyvios.toolforge.org/?lang=en&project=wikiversity&action=search&turnitin=1&title=WikiJournal_Preprints/Body_image_disturbance_in_eating_disorders [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) 07:51, 19 August 2022 (UTC) ej39x57hge2x30ucwn8dnzrrtsalo6f 2416456 2416455 2022-08-19T07:52:19Z Rwatson1955 2831054 wikitext text/x-wiki {{#section-h:{{ARTICLEPAGENAMEE}}}} == Plagiarism check == {{Pass}} Report from WMF copyvios tool flagged some false positives (not regarded as plagiarism) due to common stock phrases. Totalled 23.1% with 'violation unlikely' with regard to plagiarism. Please see the results here [https://copyvios.toolforge.org/?lang=en&project=wikiversity&action=search&turnitin=1&title=WikiJournal_Preprints/Body_image_disturbance_in_eating_disorders] [[User:Rwatson1955|Rwatson1955]] ([[User talk:Rwatson1955|discuss]] • [[Special:Contributions/Rwatson1955|contribs]]) f28lwzh0omblii37fy0yu9byjuw57w8 3 282077 2416277 2384227 2022-08-18T19:53:02Z Atcovi 276019 /* The Präteritum (preterite) */ new section wikitext text/x-wiki == Quizlet == Quizlet, including these verbs, prepositions from K5, and verbs that I need to be familiar with, coming soon... —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:44, 26 February 2022 (UTC) == Kapitel 6: Writing Excersise == In a couple sentences, begin by briefly describing your kitchen and living room using the present tense. You do not have to follow what is in the pictures, you may describe your own home. The rest of the paragraph will be written in the Perfekt tense. You recently had a party, and needed to clean it up. What was the party about and what did you do at the party? Please describe what you did to clean your kitchen and living room after the party. Please read further instructions below: *Be sure to use vocabulary and verbs from 6A & 6B for this writing assignment. *Write a minimum of 10 sentences in German, but no more than 15. *Be sure to conjugate your verbs and follow correct word order for your sentences. *You should also be able to use coordinating conjunctions in your paragraph. *Modal verbs should be used in the Präteritum for this writing assignment. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 02:50, 16 March 2022 (UTC) :Meine Küche ist etwas klein und mein Wohnzimmer ist groß. Am Montag habe ich eine Party gegeben. Ich habe eine Party gegeben, weil Montag Maxs Geburtstag war. Auf dieser Party habe ich mit süße Mädchen getanzt. Max hat manche Küchen gegessen. Unsere Freunde haben manche Küchen gebackt. Nach der Party war meine Küche & mein Wohzimmer schmutzig! Max habe Geschirr gespült, denn er spült Geschirr gern. Ich habe die Zimmer mit dem Besen gefegt. Ich habe die gelbe Lampe in das Wohnzimmer geputzt. Obwohl wir haben das ganze Haus nach geputzt, haben wir viele Spaß! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 03:16, 16 March 2022 (UTC) :<small>(correction)</small> Meine Küche ist etwas klein und mein Wohnzimmer ist groß. Am Montag habe ich eine Party gegeben. Ich habe eine Party gegeben, weil Montag Maxs Geburtstag war. Auf dieser Party habe ich mit süß'''en''' Mädchen getanzt. Max hat '''viel Kuchen''' gegessen. Unsere Freunde haben '''viele Kuchen''' gebackt. Nach der Party war meine Küche & mein Wohzimmer schmutzig! Max '''hat''' Geschirr gespült, denn er spült Geschirr gern. Ich habe die Zimmer mit dem Besen gefegt. Ich habe die gelbe Lampe in '''dem''' Wohnzimmer geputzt. Obwohl wir das ganze Haus '''danach''' geputzt '''haben''', '''hatten''' wir viel Spaß! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:17, 18 March 2022 (UTC) == The Präteritum (preterite) == *blieben --> bl'''ei'''b *schreiben --> schr'''ie'''b in these cases, the "ie" or "ei" switched. *nehmen --> n'''a'''hm *gehen --> g'''a'''b in these cases, the "e" was changed to an "a" (e --> a). *fahren --> f'''u'''hr in this case, the "a" was changed to a "u" (a --> u). *sein --> '''war''' in this case, this was an irregular verb. *essen --> '''ass''' *gehen --> '''gig''' *stehen --> '''stand ''' in these cases, the verbs are irregular. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:53, 18 August 2022 (UTC) jp4qcv5cpuhmn2scwn7e5cmb5gynwns 0 282260 2416282 2403227 2022-08-18T20:58:04Z Greg at Higher Math Help 2911873 Made wording in the overview box more concise, based on survey feedback received during a workshop. wikitext text/x-wiki <!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DAF3EF; padding: 1em; overflow:hidden;"> <div style="overflow:hidden;"> <!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --> <div style="width:50%; margin-right:4em; float:left;"> {{Eventmath lesson plan overview <!--EDIT BELOW THIS LINE ONLY--> <!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--> |assumed-knowledge = <!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--> Proportions and basic spreadsheet calculations. Basic mechanics of the Electoral College in the United States (see [[Eventmath/Lesson_plans/Proportions_and_voting_power_under_the_Electoral_College#Resources|Resources]]). |activities = <!--Below, please briefly summarize what students will do during this lesson and what they will learn.--> Students will propose different measures of the "individual voter power" of residents in a state, based on its population and its voting representation in the Electoral College. They will calculate these measures in a spreadsheet and compare their findings to the article's main assertion. |class-time = 60-90 minutes <!--Below, please type one of the following, exactly as it appears: 00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes. Your best guess is fine! It can be changed later.--> <!--Next, you'll provide all citation data for the source (article/social media post/video). Article example: | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2021-10-28 Post example: | source-website = Twitter | source-title = Remember: Germany is supposed to be a SUCCESS STORY. | source-date = 2020-12-22 | source-url = https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-url = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-date = 2021-02-16 Notes: 1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD, including leading zeros: January is 01, not 1, for example. 2. If you cannot find a date, use n.d. as the date (no date). 3. To ensure that the article/post will remain accessible for educators and students... a. Archive it at https://archive.org/web/ (use the "Save Page Now" form to generate a permanent URL). b. Paste that URL here as the source-archive-url. c. Enter the date you archived the source as the source-archive date. 4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may avoid spreading disinformation by omitting the url of the original site entirely. In that case, use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date. --> | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It's Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2020-12-10 }} </div> <div style="float:left; width:20%;"> [[File:Eventmath_logo_bitmap_graphic.png|250px]] </div> </div> <div style="width:70%; float:left;"> <!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN 1. Note that "==My heading==" is a level 2 section heading, "===My subheading===" is level 3, etc. Level 1 is reserved for the title. 2. After you fill in a section, please delete the ''italicized instructions.'' 3. Check out the "Advanced" and "Help" menus at the top of the editor if you don't know how to format lists, etc. --> ==Activities== ===Discussion of articles=== # '''Preparation:''' Read the following article before class: [https://thehill.com/opinion/judiciary/519826-let-us-abolish-the-electoral-college "Let us abolish the Electoral College,"] October 6, 2020, ''The Hill''. Discuss with students whether the stated opinions are supported by evidence, especially numerical evidence. # '''Preparation:''' Read the main article from ''Huffington Post'', although it's so short that perhaps this could be done together during class. Discuss with students whether this provides more contextual information than the first article. # '''Discussion:''' Ask students whether the value of 3.6 in the title of the ''Huffington Post'' article refers to the ratio of a Wyomingite voter's power to a Californian voter's power or whether it refers to the reciprocal. # '''Discussion:''' Ask students to define a quantified measure of individual voter power by state. Encourage them to think of more than one way to do this, emphasizing that there may not necessarily be "one correct way." However, for the sake of the rest of the lesson, make sure that students eventually decide to include at least these two measures: ## Number of residents / number of Electoral College votes. ## Number of Electoral College votes / number of residents. (This is the reciprocal of the first one.) ===Data collection and analysis=== [[File:Faenza-x-office-spreadsheet.svg|right|350px|spreadsheet]] The main activity now is to create a spreadsheet containing the most recent census data showing each state's population, the number of Electoral College votes each state has, and the individual voter power in each state according to each of the proposed measures stated above. (For the instructor's convenience, [https://docs.google.com/spreadsheets/d/1HOca2MKYheQ6gJirmaSazNGvAlYOj_OLnr8rYaRiDvI/edit?usp=sharing here is a Google Sheets file] with each state's population -- both 2010 and 2020 census data -- and their # of Electoral College votes.) For the data collection, you may consider having students work on this in groups so they can continue the task as homework later. Or, you may consider working together as a class to create one spreadsheet, perhaps by assigning individual tasks to groups of students (e.g. "This group will find recent census data while this group sets up the spreadsheet calculations.") For the data analysis, you may consider having some prepared discussion questions or a worksheet of problems for groups of students to solve using the spreadsheet. For example, you might ask: * Which state's individuals have the smallest voting power by measure #1? What about measure by #2? * Which state's individuals have the largest voting power by measure #1, and by measure #2? Could there theoretically be a state with a larger value than that? How large do you think it could be? * Which states are traditionally considered "influential" in presidential elections? What are the values of their individual voting power, and what does that tell us, if anything? ===Concluding discussion=== [[File:California Presidential Election Results 2020.svg|right|300px|California Presidential Election Results 2020]] Ask students some probing questions to inspire further discussion, such as: * Which of these measures is more helpful for quantifying what we actually want to measure? * Which of these measures is more helpful for understanding the real world situation in question? * Which of these measures do the articles actually use? * And do the articles make all of this information clear? Finally, use all of the data and any points raised in discussion to compare Wyoming and California, as the article does: * Ask students whether they agree with the article’s headline, or whether they might have written it differently. * Discuss with the students why they think Wyoming and California were chosen, out of all the states. How might the headline have been different if other states were chosen as comparison points? ===Instructor notes=== ====Comparison of the proposed measures==== <!--Table copied from https://en.wikipedia.org/wiki/Scientific_notation and modified because the Val template on Wikiversity needs to be updated --> {| class="wikitable" style="float:right; margin:5px;" |- !Decimal notation !Scientific notation |- |<math>2</math> |<math>2\times 10^0</math> |- |<math>300</math> |<math>3\times 10^2</math> |- |<math>4321.768</math> |<math>4.321768\times 10^3</math> |- |<math>-53000</math> ||<math>-5.3\times 10^4</math> |- |<math>6720000000</math> |<math>6.72\times 10^9</math> |- |<math>0.2</math> |<math>2\times 10^{-1}</math> |- |<math>987</math> |<math>9.87\times 10^2</math> |- |<math>0.00000000751</math> |<math>7.51\times 10^{-9}</math> |} Discussing these finer points with students might help them better understand how we make conventional choices about definitions in mathematics. * If you define individual voter power in a state as the number of residents per Electoral College vote (calculation: number of residents / number of Electoral College votes), then you will find that Wyoming has the absolute minimum and California has the absolute maximum. However, this is arguably "incorrect" for two reasons: ** The ratio of individual voter power in Wyoming to individual voter power in California would be 1/3.6 and not 3.6, as the headline states. So, this measure is at least inconsistent with the article's claims. ** Moreover, this measure is inconsistent with our intuition about how individual voter power might change. For example, if the number of Electoral College votes allotted to a given state were to instantly increase (say, due to a legislative action), then we would expect the power of each individual voter in that state to increase, as well. However, the number of residents per Electoral College vote would decrease. * If you define individual voter power in a state as the amount of an Electoral College vote that each voter represents (calculation: number of Electoral College votes / number of residents), then you will find that Wyoming has the absolute maximum and California has the absolute minimum, and the ratio of those ratios is 3.6. ** This supports the article’s claim. Since this definition is consistent with intuition, as well, it better captures the quantity we are trying to measure. ** However, this measure leads to tiny numbers. This may prompt a mini-lesson on scientific notation or other ways of expressing and working with large/small numbers. Moreover, you may have to help students format the spreadsheet to display the values correctly and helpfully. ====Limitations of the proposed measures==== Each of the measures is subject to limitations. For example, Wyoming and California are currently [https://en.wikipedia.org/wiki/United_States_Electoral_College#Summary winner-take-all states], and [https://en.wikipedia.org/wiki/United_States_presidential_elections_in_Wyoming Wyoming consistently votes for Republicans in presidential elections], whereas [https://en.wikipedia.org/wiki/United_States_presidential_elections_in_California California consistently votes for Democrats]. So, it is likely that all of Wyoming’s electors in a modern election will vote for the Republican candidate, in which case a Wyomingite who votes for a Democrat would not be represented in the Electoral College vote. The same can be said of a voter in California who votes for a Republican. Neither measure discussed in this lesson plan accounts for this issue, or others like it. Despite such limitations, these quantitative measures do provide a starting point for discussion. An essay prompt in the [[Eventmath/Lesson_plans/Proportions_and_voting_power_under_the_Electoral_College#Assignments|''Assignments'' section]] is intended to help students explore this tension. ====Why Wyoming and California?==== Why were Wyoming and California chosen as the highlights of the article? For one, the author lives in California. For another, they make for the most striking and disparate comparison between any pair of states, at least for the 2010 census data. ==Assignments== There are many directions an assignment could take with this lesson. Here are just some examples: <div style="padding-top: 0.5em;"> [[File:Composition Notebook (5736841980).jpg|left|250px|alt=Composition Notebook]] </div> <div style="overflow-x: hidden; padding-left: 1em;"> * ''Discussion/Essay prompt:'' Ask students to write a headline that would more accurately represent what the methods in the article itself imply about the comparison between California and Wyoming, and to describe why they believe that headline is a more accurate representation. * ''Essay prompt:'' Ask students to write an op-ed of their own that either rebuts the original article or provides more evidence in support of it. (It may be important to remind students that they can disagree with an argument without necessarily disagreeing with the conclusion, for instance.) * ''Assignment prompt:'' Ask students to create an informative, shareable explanation of how the headline of the original ''Huffington Post'' article may mislead someone who doesn’t read the rest of the article. It may help to ask them to pretend they’re making something for their friends or family to see, like a Twitter thread or TikTok video. * ''Project prompt:'' Ask students to recreate the data collection and analysis for presidential elections in the past. It may be interesting to look at population and Electoral College data from 20, 50, or 100 years ago and compare individual voter power in different states. At what point in time was there the largest disparity between states? Is this 3.6 value (WY/CA) the largest ratio, historically? And how has the voting power of individuals in a state changed over time? * ''Extended project prompt:'' This idea goes well beyond the basic mathematics of the earlier parts of this lesson, but it may work for your context: What if the USA used a different apportionment scheme for allocating representatives in Congress? Students could run various algorithms to allocate reps and adjust the data in the spreadsheet accordingly. (Information about how representatives are apportioned to the states can be found [https://en.wikipedia.org/wiki/United_States_congressional_apportionment#Apportionment_methods on Wikipedia] and on the [https://www.census.gov/topics/public-sector/congressional-apportionment/about.html website of the U.S. Census Bureau].) * ''Essay prompt:'' In the article [https://digitalcommons.law.villanova.edu/cgi/viewcontent.cgi?article=1780&context=vlr "One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College,"] John F. Banzhaf III states that "the theoretical voting power of an individual may not coincide with his actual ability to affect the outcome of any particular election," citing issues such as "the political impotence of a Republican in a solidly Democratic state." Does this statement apply to the measure of individual voter power you chose? Describe any limitations of your chosen measure, and discuss the extent to which they affect its overall utility and explanatory power. </div> ==Resources== ===Background=== <!-- Electoral College image and caption copied from https://en.wikipedia.org/wiki/United_States_Electoral_College --> [[File:ElectoralCollege2024.svg|thumb|upright=1.35|Electoral votes, out of 538, allocated to each [[U.S. state|state]] and the [[District of Columbia]] for [[United States presidential election|presidential elections]] to be held in [[2024 United States presidential election|2024]] and 2028, based on representation, which depends on population data from the [[2020 United States census|2020 census]]. Every jurisdiction is entitled to at least 3.]] There is not much mathematical background presumed for this lesson plan, only a facility with calculations and proportions. Here is a [https://www.khanacademy.org/test-prep/praxis-math/praxis-math-lessons/praxis-math-number-and-quantity/a/gtp--praxis-math--article--ratios-and-proportions--lesson Khan Academy lesson on ratios and proportions]. Part of implementing this lesson plan may involve first teaching students the basic mechanics of the Electoral College. Here are a few resources for that: * "How the Electoral College works: A guide to the complex system the United States uses to select a president," from [https://graphics.reuters.com/USA-ELECTION/ELECTORAL-COLLEGE/qzjpqaeqapx/ Reuters] * "How the Electoral College Works," from [https://youtu.be/OUS9mM8Xbbw CGP Grey on YouTube] * "Does your vote count? The Electoral College explained - Christina Greer," from [https://youtu.be/W9H3gvnN468 TED-Ed on YouTube] ===Explorations=== Students may ask whether there are other resources that have defined this notion of "individual voter power," or you may wish to assign a project that asks students to find any and compare to their findings from this lesson. * Example from the popular press: ''Wallet Hub'' has a list of [https://wallethub.com/edu/how-much-is-your-vote-worth/7932 "States with the Most & Least Powerful Voters"] and a methodology section at the end that describes how they defined "powerful." Here's an [https://web.archive.org/web/20220324095524/https://wallethub.com/edu/how-much-is-your-vote-worth/7932 archived version of the same page], in case the first link doesn't work. * Example from the research literature: Banzhaf III, John F. "[https://digitalcommons.law.villanova.edu/cgi/viewcontent.cgi?article=1780&context=vlr One man, 3.312 votes: a mathematical analysis of the Electoral College.]" ''Vill. L. Rev.'' 13 (1968): 304. ==Feedback== <!--PLEASE DO NOT EDIT THIS SECTION DIRECTLY.--> Have you found this lesson plan helpful? Tell us about it! Just click ''Endorse'' below to open up an editor and type your comments. When you're ready, they'll appear at the bottom of this section to help other educators looking for good lesson plans. (Alternatively, if you see a way to improve this lesson plan, be bold and make an edit! You're also welcome to discuss the lesson plan or provide constructive feedback on its [[{{TALKPAGENAME}}|Discussion page]].) <div style="float:left;"> <inputbox> type=commenttitle hidden=yes page={{FULLPAGENAME}} buttonlabel=Endorse arialabel=Endorse the lesson plan editintro=Eventmath/Lesson plan endorsement instructions preload=Template:Eventmath lesson plan endorsement </inputbox> </div> </div> <div style="width:10%; padding:4em; float:left;"> __TOC__ </div> </div> <!--CATEGORY TAG INSTRUCTIONS Please follow these instructions so that people can find this lesson plan! 1. Select a primary event type from the following list: business, culture, economics, education, finance, government, health, politics, science, technology 2. Select a primary math type from the following list: algebra, arithmetic, calculus, combinatorics, geometry, logic, probability, statistics 3. Type your category tags directly underneath "[[Category: Eventmath lesson plans]]," at the bottom of this editor. For example, if you choose calculus and economics, you would have the following category tags on consecutive lines: [[Category: Eventmath lesson plans]] [[Category: Eventmath calculus]] [[Category: Eventmath economics]] NOTE: Be sure to use sentence casing for the category names (e.g. "Eventmath calculus" not "Eventmath Calculus"). FAQ Q: Can I apply multiple event or math tags? A: Yes! For example, you might want both a probability tag and a statistics tag. Q: Can I make a new category? A: Yes! Just type it in the same format. Example: [[Category: Eventmath number theory]] However, this category will not automatically have its own display box on the Lesson plans page. If you think there should be one, you can add one. If you don't know how, leave a comment on the Lesson plans "Discuss" tab. Alternatively, you might interpret an existing category more broadly. For example, if you have a lesson plan for differential equations, you might decide to include that under the calculus category. It's up to you. --> [[Category: Eventmath lesson plans]] [[Category: Eventmath arithmetic]] [[Category: Eventmath politics]] 4z11z807sdqd2oupvohqrx6080owk4m 2416283 2416282 2022-08-18T20:59:09Z Greg at Higher Math Help 2911873 Made text in overview box a little more concise. wikitext text/x-wiki <!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DAF3EF; padding: 1em; overflow:hidden;"> <div style="overflow:hidden;"> <!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --> <div style="width:50%; margin-right:4em; float:left;"> {{Eventmath lesson plan overview <!--EDIT BELOW THIS LINE ONLY--> <!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--> |assumed-knowledge = <!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--> Proportions and basic spreadsheet calculations. Basic mechanics of the Electoral College in the United States (see [[Eventmath/Lesson_plans/Proportions_and_voting_power_under_the_Electoral_College#Resources|Resources]]). |activities = <!--Below, please briefly summarize what students will do during this lesson and what they will learn.--> Students will propose different measures of the "individual voter power" of residents in a state, based on its population and its representation in the Electoral College. They will calculate these measures in a spreadsheet and compare their findings to the article's main assertion. |class-time = 60-90 minutes <!--Below, please type one of the following, exactly as it appears: 00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes. Your best guess is fine! It can be changed later.--> <!--Next, you'll provide all citation data for the source (article/social media post/video). Article example: | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2021-10-28 Post example: | source-website = Twitter | source-title = Remember: Germany is supposed to be a SUCCESS STORY. | source-date = 2020-12-22 | source-url = https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-url = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-date = 2021-02-16 Notes: 1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD, including leading zeros: January is 01, not 1, for example. 2. If you cannot find a date, use n.d. as the date (no date). 3. To ensure that the article/post will remain accessible for educators and students... a. Archive it at https://archive.org/web/ (use the "Save Page Now" form to generate a permanent URL). b. Paste that URL here as the source-archive-url. c. Enter the date you archived the source as the source-archive date. 4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may avoid spreading disinformation by omitting the url of the original site entirely. In that case, use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date. --> | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It's Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2020-12-10 }} </div> <div style="float:left; width:20%;"> [[File:Eventmath_logo_bitmap_graphic.png|250px]] </div> </div> <div style="width:70%; float:left;"> <!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN 1. Note that "==My heading==" is a level 2 section heading, "===My subheading===" is level 3, etc. Level 1 is reserved for the title. 2. After you fill in a section, please delete the ''italicized instructions.'' 3. Check out the "Advanced" and "Help" menus at the top of the editor if you don't know how to format lists, etc. --> ==Activities== ===Discussion of articles=== # '''Preparation:''' Read the following article before class: [https://thehill.com/opinion/judiciary/519826-let-us-abolish-the-electoral-college "Let us abolish the Electoral College,"] October 6, 2020, ''The Hill''. Discuss with students whether the stated opinions are supported by evidence, especially numerical evidence. # '''Preparation:''' Read the main article from ''Huffington Post'', although it's so short that perhaps this could be done together during class. Discuss with students whether this provides more contextual information than the first article. # '''Discussion:''' Ask students whether the value of 3.6 in the title of the ''Huffington Post'' article refers to the ratio of a Wyomingite voter's power to a Californian voter's power or whether it refers to the reciprocal. # '''Discussion:''' Ask students to define a quantified measure of individual voter power by state. Encourage them to think of more than one way to do this, emphasizing that there may not necessarily be "one correct way." However, for the sake of the rest of the lesson, make sure that students eventually decide to include at least these two measures: ## Number of residents / number of Electoral College votes. ## Number of Electoral College votes / number of residents. (This is the reciprocal of the first one.) ===Data collection and analysis=== [[File:Faenza-x-office-spreadsheet.svg|right|350px|spreadsheet]] The main activity now is to create a spreadsheet containing the most recent census data showing each state's population, the number of Electoral College votes each state has, and the individual voter power in each state according to each of the proposed measures stated above. (For the instructor's convenience, [https://docs.google.com/spreadsheets/d/1HOca2MKYheQ6gJirmaSazNGvAlYOj_OLnr8rYaRiDvI/edit?usp=sharing here is a Google Sheets file] with each state's population -- both 2010 and 2020 census data -- and their # of Electoral College votes.) For the data collection, you may consider having students work on this in groups so they can continue the task as homework later. Or, you may consider working together as a class to create one spreadsheet, perhaps by assigning individual tasks to groups of students (e.g. "This group will find recent census data while this group sets up the spreadsheet calculations.") For the data analysis, you may consider having some prepared discussion questions or a worksheet of problems for groups of students to solve using the spreadsheet. For example, you might ask: * Which state's individuals have the smallest voting power by measure #1? What about measure by #2? * Which state's individuals have the largest voting power by measure #1, and by measure #2? Could there theoretically be a state with a larger value than that? How large do you think it could be? * Which states are traditionally considered "influential" in presidential elections? What are the values of their individual voting power, and what does that tell us, if anything? ===Concluding discussion=== [[File:California Presidential Election Results 2020.svg|right|300px|California Presidential Election Results 2020]] Ask students some probing questions to inspire further discussion, such as: * Which of these measures is more helpful for quantifying what we actually want to measure? * Which of these measures is more helpful for understanding the real world situation in question? * Which of these measures do the articles actually use? * And do the articles make all of this information clear? Finally, use all of the data and any points raised in discussion to compare Wyoming and California, as the article does: * Ask students whether they agree with the article’s headline, or whether they might have written it differently. * Discuss with the students why they think Wyoming and California were chosen, out of all the states. How might the headline have been different if other states were chosen as comparison points? ===Instructor notes=== ====Comparison of the proposed measures==== <!--Table copied from https://en.wikipedia.org/wiki/Scientific_notation and modified because the Val template on Wikiversity needs to be updated --> {| class="wikitable" style="float:right; margin:5px;" |- !Decimal notation !Scientific notation |- |<math>2</math> |<math>2\times 10^0</math> |- |<math>300</math> |<math>3\times 10^2</math> |- |<math>4321.768</math> |<math>4.321768\times 10^3</math> |- |<math>-53000</math> ||<math>-5.3\times 10^4</math> |- |<math>6720000000</math> |<math>6.72\times 10^9</math> |- |<math>0.2</math> |<math>2\times 10^{-1}</math> |- |<math>987</math> |<math>9.87\times 10^2</math> |- |<math>0.00000000751</math> |<math>7.51\times 10^{-9}</math> |} Discussing these finer points with students might help them better understand how we make conventional choices about definitions in mathematics. * If you define individual voter power in a state as the number of residents per Electoral College vote (calculation: number of residents / number of Electoral College votes), then you will find that Wyoming has the absolute minimum and California has the absolute maximum. However, this is arguably "incorrect" for two reasons: ** The ratio of individual voter power in Wyoming to individual voter power in California would be 1/3.6 and not 3.6, as the headline states. So, this measure is at least inconsistent with the article's claims. ** Moreover, this measure is inconsistent with our intuition about how individual voter power might change. For example, if the number of Electoral College votes allotted to a given state were to instantly increase (say, due to a legislative action), then we would expect the power of each individual voter in that state to increase, as well. However, the number of residents per Electoral College vote would decrease. * If you define individual voter power in a state as the amount of an Electoral College vote that each voter represents (calculation: number of Electoral College votes / number of residents), then you will find that Wyoming has the absolute maximum and California has the absolute minimum, and the ratio of those ratios is 3.6. ** This supports the article’s claim. Since this definition is consistent with intuition, as well, it better captures the quantity we are trying to measure. ** However, this measure leads to tiny numbers. This may prompt a mini-lesson on scientific notation or other ways of expressing and working with large/small numbers. Moreover, you may have to help students format the spreadsheet to display the values correctly and helpfully. ====Limitations of the proposed measures==== Each of the measures is subject to limitations. For example, Wyoming and California are currently [https://en.wikipedia.org/wiki/United_States_Electoral_College#Summary winner-take-all states], and [https://en.wikipedia.org/wiki/United_States_presidential_elections_in_Wyoming Wyoming consistently votes for Republicans in presidential elections], whereas [https://en.wikipedia.org/wiki/United_States_presidential_elections_in_California California consistently votes for Democrats]. So, it is likely that all of Wyoming’s electors in a modern election will vote for the Republican candidate, in which case a Wyomingite who votes for a Democrat would not be represented in the Electoral College vote. The same can be said of a voter in California who votes for a Republican. Neither measure discussed in this lesson plan accounts for this issue, or others like it. Despite such limitations, these quantitative measures do provide a starting point for discussion. An essay prompt in the [[Eventmath/Lesson_plans/Proportions_and_voting_power_under_the_Electoral_College#Assignments|''Assignments'' section]] is intended to help students explore this tension. ====Why Wyoming and California?==== Why were Wyoming and California chosen as the highlights of the article? For one, the author lives in California. For another, they make for the most striking and disparate comparison between any pair of states, at least for the 2010 census data. ==Assignments== There are many directions an assignment could take with this lesson. Here are just some examples: <div style="padding-top: 0.5em;"> [[File:Composition Notebook (5736841980).jpg|left|250px|alt=Composition Notebook]] </div> <div style="overflow-x: hidden; padding-left: 1em;"> * ''Discussion/Essay prompt:'' Ask students to write a headline that would more accurately represent what the methods in the article itself imply about the comparison between California and Wyoming, and to describe why they believe that headline is a more accurate representation. * ''Essay prompt:'' Ask students to write an op-ed of their own that either rebuts the original article or provides more evidence in support of it. (It may be important to remind students that they can disagree with an argument without necessarily disagreeing with the conclusion, for instance.) * ''Assignment prompt:'' Ask students to create an informative, shareable explanation of how the headline of the original ''Huffington Post'' article may mislead someone who doesn’t read the rest of the article. It may help to ask them to pretend they’re making something for their friends or family to see, like a Twitter thread or TikTok video. * ''Project prompt:'' Ask students to recreate the data collection and analysis for presidential elections in the past. It may be interesting to look at population and Electoral College data from 20, 50, or 100 years ago and compare individual voter power in different states. At what point in time was there the largest disparity between states? Is this 3.6 value (WY/CA) the largest ratio, historically? And how has the voting power of individuals in a state changed over time? * ''Extended project prompt:'' This idea goes well beyond the basic mathematics of the earlier parts of this lesson, but it may work for your context: What if the USA used a different apportionment scheme for allocating representatives in Congress? Students could run various algorithms to allocate reps and adjust the data in the spreadsheet accordingly. (Information about how representatives are apportioned to the states can be found [https://en.wikipedia.org/wiki/United_States_congressional_apportionment#Apportionment_methods on Wikipedia] and on the [https://www.census.gov/topics/public-sector/congressional-apportionment/about.html website of the U.S. Census Bureau].) * ''Essay prompt:'' In the article [https://digitalcommons.law.villanova.edu/cgi/viewcontent.cgi?article=1780&context=vlr "One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College,"] John F. Banzhaf III states that "the theoretical voting power of an individual may not coincide with his actual ability to affect the outcome of any particular election," citing issues such as "the political impotence of a Republican in a solidly Democratic state." Does this statement apply to the measure of individual voter power you chose? Describe any limitations of your chosen measure, and discuss the extent to which they affect its overall utility and explanatory power. </div> ==Resources== ===Background=== <!-- Electoral College image and caption copied from https://en.wikipedia.org/wiki/United_States_Electoral_College --> [[File:ElectoralCollege2024.svg|thumb|upright=1.35|Electoral votes, out of 538, allocated to each [[U.S. state|state]] and the [[District of Columbia]] for [[United States presidential election|presidential elections]] to be held in [[2024 United States presidential election|2024]] and 2028, based on representation, which depends on population data from the [[2020 United States census|2020 census]]. Every jurisdiction is entitled to at least 3.]] There is not much mathematical background presumed for this lesson plan, only a facility with calculations and proportions. Here is a [https://www.khanacademy.org/test-prep/praxis-math/praxis-math-lessons/praxis-math-number-and-quantity/a/gtp--praxis-math--article--ratios-and-proportions--lesson Khan Academy lesson on ratios and proportions]. Part of implementing this lesson plan may involve first teaching students the basic mechanics of the Electoral College. Here are a few resources for that: * "How the Electoral College works: A guide to the complex system the United States uses to select a president," from [https://graphics.reuters.com/USA-ELECTION/ELECTORAL-COLLEGE/qzjpqaeqapx/ Reuters] * "How the Electoral College Works," from [https://youtu.be/OUS9mM8Xbbw CGP Grey on YouTube] * "Does your vote count? The Electoral College explained - Christina Greer," from [https://youtu.be/W9H3gvnN468 TED-Ed on YouTube] ===Explorations=== Students may ask whether there are other resources that have defined this notion of "individual voter power," or you may wish to assign a project that asks students to find any and compare to their findings from this lesson. * Example from the popular press: ''Wallet Hub'' has a list of [https://wallethub.com/edu/how-much-is-your-vote-worth/7932 "States with the Most & Least Powerful Voters"] and a methodology section at the end that describes how they defined "powerful." Here's an [https://web.archive.org/web/20220324095524/https://wallethub.com/edu/how-much-is-your-vote-worth/7932 archived version of the same page], in case the first link doesn't work. * Example from the research literature: Banzhaf III, John F. "[https://digitalcommons.law.villanova.edu/cgi/viewcontent.cgi?article=1780&context=vlr One man, 3.312 votes: a mathematical analysis of the Electoral College.]" ''Vill. L. Rev.'' 13 (1968): 304. ==Feedback== <!--PLEASE DO NOT EDIT THIS SECTION DIRECTLY.--> Have you found this lesson plan helpful? Tell us about it! Just click ''Endorse'' below to open up an editor and type your comments. When you're ready, they'll appear at the bottom of this section to help other educators looking for good lesson plans. (Alternatively, if you see a way to improve this lesson plan, be bold and make an edit! You're also welcome to discuss the lesson plan or provide constructive feedback on its [[{{TALKPAGENAME}}|Discussion page]].) <div style="float:left;"> <inputbox> type=commenttitle hidden=yes page={{FULLPAGENAME}} buttonlabel=Endorse arialabel=Endorse the lesson plan editintro=Eventmath/Lesson plan endorsement instructions preload=Template:Eventmath lesson plan endorsement </inputbox> </div> </div> <div style="width:10%; padding:4em; float:left;"> __TOC__ </div> </div> <!--CATEGORY TAG INSTRUCTIONS Please follow these instructions so that people can find this lesson plan! 1. Select a primary event type from the following list: business, culture, economics, education, finance, government, health, politics, science, technology 2. Select a primary math type from the following list: algebra, arithmetic, calculus, combinatorics, geometry, logic, probability, statistics 3. Type your category tags directly underneath "[[Category: Eventmath lesson plans]]," at the bottom of this editor. For example, if you choose calculus and economics, you would have the following category tags on consecutive lines: [[Category: Eventmath lesson plans]] [[Category: Eventmath calculus]] [[Category: Eventmath economics]] NOTE: Be sure to use sentence casing for the category names (e.g. "Eventmath calculus" not "Eventmath Calculus"). FAQ Q: Can I apply multiple event or math tags? A: Yes! For example, you might want both a probability tag and a statistics tag. Q: Can I make a new category? A: Yes! Just type it in the same format. Example: [[Category: Eventmath number theory]] However, this category will not automatically have its own display box on the Lesson plans page. If you think there should be one, you can add one. If you don't know how, leave a comment on the Lesson plans "Discuss" tab. Alternatively, you might interpret an existing category more broadly. For example, if you have a lesson plan for differential equations, you might decide to include that under the calculus category. It's up to you. --> [[Category: Eventmath lesson plans]] [[Category: Eventmath arithmetic]] [[Category: Eventmath politics]] s6q0gktk88zmkb5rkhas7aivstt0a34 0 282299 2416280 2411871 2022-08-18T20:44:49Z Greg at Higher Math Help 2911873 /* Initial discussion */ Made wording more concise, based on feedback received on a workshop survey. wikitext text/x-wiki <!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DAF3EF; padding: 1em; overflow:hidden;"> <div style="overflow:hidden;"> <!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --> <div style="width:50%; margin-right:4em; float:left;"> {{Eventmath lesson plan overview <!--EDIT BELOW THIS LINE ONLY--> <!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--> |assumed-knowledge = <!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--> Students only need familiarity with basic arithmetic. |activities = <!--Below, please briefly summarize what students will do during this lesson and what they will learn.--> Students will consider a Tweet discussing streaming service pay rates for artists, and then they will create a graphical representation of the data, as well as alt-text for sharing that graph on social media. |class-time = <!--Below, please type one of the following, exactly as it appears: 00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes. Your best guess is fine! It can be changed later.--> 15-30 minutes <!--Next, you'll provide all citation data for the source (article/social media post/video). Article example: | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2021-10-28 Post example: | source-website = Twitter | source-title = Remember: Germany is supposed to be a SUCCESS STORY. | source-date = 2020-12-22 | source-url = https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-url = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-date = 2021-02-16 Notes: 1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD, including leading zeros: January is 01, not 1, for example. 2. If you cannot find a date, use n.d. as the date (no date). 3. To ensure that the article/post will remain accessible for educators and students... a. Archive it at https://archive.org/web/ (use the "Save Page Now" form to generate a permanent URL). b. Paste that URL here as the source-archive-url. c. Enter the date you archived the source as the source-archive date. 4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may avoid spreading disinformation by omitting the url of the original site entirely. In that case, use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date. --> | source-website = Twitter | source-title = Would you like another reason to switch to a music service that is NOT Spotify? | source-date = 2022-02-07 | source-url = https://twitter.com/zoecello/status/1490817801552027651 | source-archive-url = https://web.archive.org/web/20220305162020/https://twitter.com/zoecello/status/1490817801552027651 | source-archive-date = 2022-03-05 }} </div> <div style="float:left; width:20%;"> [[File:Eventmath_logo_bitmap_graphic.png|250px]] </div> </div> <div style="width:70%; float:left;"> <!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN 1. Note that "==My heading==" is a level 2 section heading, "===My subheading===" is level 3, etc. Level 1 is reserved for the title. 2. After you fill in a section, please delete the ''italicized instructions.'' 3. Check out the "Advanced" and "Help" menus at the top of the editor if you don't know how to format lists, etc. --> ==Activities== The main idea is to help students make sense of very small/large numbers and compare them using appropriate units of measurement. ===Initial discussion=== Provide students with the tweet and ask them questions as if they were just seeing it. Consider using polling software, or maybe give a mock quiz and then ask students to swap papers and compare. See below for suggested discussion questions. {{Quote box |quote = Would you like another reason to switch to a music service that is NOT Spotify? This is what services paid me on average per stream in 2021 Amazon - $0.0107463 Tidal - $0.007981 Youtube - $0.007795 Apple - $0.006040 Deezer - $0.0045861 Spotify - $0.003661 (I am my label fyi) |author = [https://en.wikipedia.org/wiki/Zo%C3%AB_Keating Zoë Keating] |source = https://twitter.com/zoecello/status/1490817801552027651 }} #How many streams would this artist need (on average) to earn $1 on Amazon? What about Apple? Spotify? #What would 1 million streams pay on Amazon? What about Apple? Spotify? #How many Spotify streams are needed in 1 year to amount to the same pay as $15/hour for 40 hours/week? #How do the pay rates of Amazon and Apple compare? Write a sentence that expresses a numerical comparison between those two. Do the same for Apple and Spotify. (I had success with posing these questions to students on paper as a mock "quiz" for a couple of minutes, and then asking them to compare answers in groups of 2 or 3 to see if any questions arise before the main task.) ===Main task: create a graphical representation of the data=== The central activity of the lesson asks students to present the given information in a graphical form instead, and then write appropriate alt-text for sharing the graph as an image on social media. ====Units: $/stream or streams/$==== The original Tweet expresses the pay rates in the units of '''dollars per stream.''' While you discuss the solutions to the "quiz" questions, steer the discussion towards the idea that perhaps '''streams per dollar''' might be a more useful unit, for the sake of comparing the various platforms. From the point of view of an artist, it's probably much easier to compare the numerical values of "How many streams are needed to earn $1?", as opposed to comparing the tiny decimal numbers expressed in the original Tweet. Or, as hinted at by one of the possible questions above, students may want to express everything in the units of '''yearly streams needed to make annual minimum wage,''' which would essentially be a multiple of the '''streams per dollar''' unit. For your convenience, here is a table with the information from the original Tweet, as well as the corresponding value expressed in the '''Streams / $''' unit. (To obtain the '''yearly streams needed to make annual minimum wage''' value or something like it, simply multiply the '''Streams / $''' value by the annual wage you want to consider.) Aligning decimal points makes the sizes of the numbers easier to compare. {| class="wikitable sortable" |+ |- ! Platform !! $ / Stream !! Streams / $ |- | Amazon || $0.0107463 || style="text-align:right;" | 93.1 |- | Tidal || $0.007981 || style="text-align:right;" | 125.3 |- | YouTube || $0.007795 || style="text-align:right;" | 128.3 |- | Apple || $0.006040 || style="text-align:right;" | 165.6 |- | Deezer || $0.0045861 || style="text-align:right;" | 218.1 |- | Spotify || $0.003661 || style="text-align:right;" | 273.1 |} ====Graphical representation of the data==== You might also steer the discussion to considering whether the original Tweet is engaging or informative for a reader scrolling through their timeline. For example, you might ask students to reflect on what they would do if they encountered this Tweet in their feed or someone shared it with them. How would they process this information? What calculations, if any, would they do to make sense of these numbers? Through this discussion, students may realize that a visual data display, such as a bar graph, could be a more engaging way of presenting the underlying information. Ask students to resume discussion in their small groups and make a "mock up" of a graphical representation of this data. In other words, just ask them to think about the key features of their data display -- the labels and scales on the axes, the order of the categories, a title, etc. -- and make a quick sketch of what it might look like. They may also decide whether they want to use $/stream or streams/$ (or something else) as the unit of measurement. After a short while, reconvene the class-wide discussion so that students can learn from each other's ideas. Then, each group can create a more polished version of their graph during class time or as homework, depending on the classroom context. ====Writing alt-text==== Whether the students create a graph in class or for homework, they should also write appropriate alt-text for sharing that graph as an image on social media. Here is a prompt you might use on a worksheet or assignment description: <blockquote>Create ALT text for the graphic so that it's accessible to more people (e.g. people with visual or cognitive impairments, people who are searching with a search engine, or people for whom the image fails to load). [https://help.twitter.com/en/using-twitter/picture-descriptions According to the Twitter Help Center], "Good image descriptions are concise and descriptive, helping people understand what’s happening in an image." If the graphic conveys certain insights better than the numbers alone, an image description might convey those same insights verbally.</blockquote> You may also consider sharing the [https://poet.diagramcenter.org/about.html POET Training Tool] with students, or assigning them [https://poet.diagramcenter.org/how.html some practice examples from that site] about how to write image descriptions. ===Concluding discussion=== Finally, have the students share their creations with each other and discuss them! Perhaps you create a slideshow of each submission to view together during class time, or perhaps the students must share and comment on each other's submission on an online discussion forum, or... Whatever you do, it's important for students to see each other's work and share feedback. This will emphasize the point that there is no "right answer" here, and that there are many decisions to be made when sharing numerical information visually and in writing. Here are some questions you might use to spark some discussion. You might even use these as prompts for a follow-up survey or short writing assignment, too. * Look back at the original Tweet and then at your graphical representation of the data. How do they compare and contrast? If you encountered one of them for the first time in your social media feed, how might you react and process the information? * Which unit of measurement did you find most helpful when thinking about this topic: $/stream or streams/$? Or, did you find it helpful to go back and forth between the two? * Based on your work in this lesson, what advice do you have for yourself and your peers about how to process numerical information, especially when it involves small numbers (like the original Tweet)? * Reflect on the discussions you had with your group and the decisions you made regarding the alt-text for your image. Were there any challenges you faced in translating the data display into words? Did you gain any new insights into this topic or the underlying data by creating the graph and its alt-text? * ==Assignments== # '''Homework prompt:''' Find another example of an artist sharing their average pay rate for at least one streaming service. Express any information you find using both the '''Streams / $''' and '''$ / Stream''' units, and compare/contrast with the original source for this lesson. # '''Homework prompt:''' If you have an account on one of the streaming services considered in this activity, identify an artist you have listened to a lot. How many songs have you streamed from them in the last month? Year? (Perhaps you can find this information in your user profile or stats, or perhaps you'll need to do some estimation.) Using the rates provided in the original Tweet here, how much have you personally contributed to that artist's earnings on that platform this month/year? (But then, how certain can we be that your artist has the same pay rate per stream?) # '''Essay prompt:''' One issue raised by the original Tweet is that different artists likely have different pay rates per stream because of contracts with record labels. Do some research into this issue and share what you discover. Are the pay rates per stream shared in this lesson likely representative of "most artists" or not? # '''Essay prompt:''' Investigate how, for example, Spotify actually decides how much to pay an artist. It is not quite as simple as "(# of streams) X ($ rate per stream)." [https://www.headphonesty.com/2021/11/how-much-does-spotify-pay-per-stream/ Do some research] and share at least two new ideas you discover about what goes into deciding the amount an artist is paid. ==Resources== ===Background=== There is almost no prerequisite mathematical knowledge for this lesson plan. However, the activities and ensuing discussion may bring up issues of social justice and information/media literacy. In case it is helpful for you, here is some information about those ideas. ====Artists and labor==== * There is an [https://www.nytimes.com/2021/05/07/arts/music/streaming-music-payments.html ongoing conversation about pay rates of streaming services]. Some [https://www.nytimes.com/2022/01/26/arts/music/spotify-neil-young-joe-rogan.html prominent musicians recently removed their catalogs from Spotify]. Initially, this was done in protest of how the platform spreads misinformation (especially through podcasts), but some have also cited the poor compensation for working musicians, as well. There have even been protests by musicians and songwriters [https://www.latimes.com/entertainment-arts/music/story/2022-03-01/spotify-protest-songwriters-royalty-rates at various Spotify corporate offices], including the Union of Musicians and Allied Workers’s [https://www.unionofmusicians.org/justice-at-spotify “Justice at Spotify Day of Action.”] * Students may notice that the original Tweet begins and ends with some comments about the ''context'' for these numbers. Consider preempting the discussion by explicitly asking students to respond to these prompts: ** Why did the Tweet start with ''"Would you like another reason to switch to a music service that is NOT Spotify?"''? What framing/context does that provide for the rest of the information? How might you have interpreted this information differently if that sentence were not included? ** Why did the Tweet end with ''"(I am my label fyi)"'' at the end? What additional contextual information does that provide, and how might it change our interpretation of the quantitative information in the Tweet? (Answer: Artists typically split the profits from streaming services with their record label, according to whatever contract they've signed. In this case, Zoë Keating operating as her own label means that she does not have to split profits in that way. Accordingly, an artist who is ''not'' their own label would likely have noticeably ''smaller'' pay rates.) ** Does the Tweet specify where this information came from, or how the numbers were calculated? (Answer: Yes, she addresses this in [https://twitter.com/zoecello/status/1491081258629095430 a reply to her own Tweet] and implies that she took her annual earnings from each platform and divided by the number of streams she earned on that platform. This further emphasizes the idea that different artists may obtain different values from these calculations.) ====Alt-text for images==== Below are some guides about alt-text for accessibility. You may consider sharing some relevant portions with students as part of the main task of this lesson plan, or else ask students to read portions of these articles as part of their assignment. * [https://webaim.org/techniques/alttext/ WebAIM article on alt text] * [https://accessibility.umn.edu/what-you-can-do/start-7-core-skills/alternative-text University of Minnesota guide to alt text (with examples for charts and graphs)] * [https://accessibility.huit.harvard.edu/describe-content-images Advice from a Harvard University guide on digital accessibility] ===Explorations=== ''You're welcome to share references for additional learning and exploration, such as links to other articles, videos, spreadsheets, or computer code. When an open-access substitute is unavailable, links to paywalled sites are acceptable in this section.'' ==Feedback== <!--PLEASE DO NOT EDIT THIS SECTION DIRECTLY.--> Have you found this lesson plan helpful? Tell us about it! Just click ''Endorse'' below to open up an editor and type your comments. When you're ready, they'll appear at the bottom of this section to help other educators looking for good lesson plans. (Alternatively, if you see a way to improve this lesson plan, be bold and make an edit! You're also welcome to discuss the lesson plan or provide constructive feedback on its [[{{TALKPAGENAME}}|Discussion page]].) <div style="float:left;"> <inputbox> type=commenttitle hidden=yes page={{FULLPAGENAME}} buttonlabel=Endorse arialabel=Endorse the lesson plan editintro=Eventmath/Lesson plan endorsement instructions preload=Template:Eventmath lesson plan endorsement </inputbox> </div> </div> <div style="width:10%; padding:4em; float:left;"> __TOC__ </div> </div> <!--CATEGORY TAG INSTRUCTIONS Please follow these instructions so that people can find this lesson plan! 1. Select a primary event type from the following list: business, culture, economics, education, finance, government, health, politics, science, technology 2. Select a primary math type from the following list: algebra, arithmetic, calculus, combinatorics, geometry, logic, probability, statistics 3. Type your category tags directly underneath "[[Category: Eventmath lesson plans]]," at the bottom of this editor. For example, if you choose calculus and economics, you would have the following category tags on consecutive lines: [[Category: Eventmath lesson plans]] [[Category: Eventmath calculus]] [[Category: Eventmath economics]] NOTE: Be sure to use sentence casing for the category names (e.g. "Eventmath calculus" not "Eventmath Calculus"). FAQ Q: Can I apply multiple event or math tags? A: Yes! For example, you might want both a probability tag and a statistics tag. Q: Can I make a new category? A: Yes! Just type it in the same format. Example: [[Category: Eventmath number theory]] However, this category will not automatically have its own display box on the Lesson plans page. If you think there should be one, you can add one. If you don't know how, leave a comment on the Lesson plans "Discuss" tab. Alternatively, you might interpret an existing category more broadly. For example, if you have a lesson plan for differential equations, you might decide to include that under the calculus category. It's up to you. --> [[Category: Eventmath lesson plans]] [[Category: Eventmath arithmetic]] [[Category: Eventmath finance]] [[Category: Eventmath technology]] [[Category: Eventmath culture]] <div style="background-color: #DAF3EF; overflow:hidden;"> * I used an early version of this lesson plan with my class for liberal-arts students as an activity on numeracy in social media. The students were surprised to learn just how little artists are paid on these platforms and found the ensuing discussion helpful and engaging. I was also pleased to find that students actually ''wanted'' to do calculations with the numbers in the Tweet to better understand what the data is telling us. I have not yet tried the "create a graph and alt-text" task, but I'm looking forward to doing that next semester! -- [[User:Professorbrendan|Professorbrendan]] ([[User talk:Professorbrendan|discuss]] • [[Special:Contributions/Professorbrendan|contribs]]) 16:41, 13 July 2022 (UTC) </div> 1mseu26mix1bgzcd6hl2lkbgd52bniv 2416287 2416280 2022-08-18T21:10:43Z Greg at Higher Math Help 2911873 Edited content in overview box so that it's more concise in parts, and hopefully a bit more compelling. wikitext text/x-wiki <!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DAF3EF; padding: 1em; overflow:hidden;"> <div style="overflow:hidden;"> <!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --> <div style="width:50%; margin-right:4em; float:left;"> {{Eventmath lesson plan overview <!--EDIT BELOW THIS LINE ONLY--> <!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--> |assumed-knowledge = <!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--> Basic arithmetic |activities = <!--Below, please briefly summarize what students will do during this lesson and what they will learn.--> Students will make sense of a tweet about streaming service pay rates for artists, by using proportional reasoning and making a graph. They will also craft alt-text for sharing their graph on social media. |class-time = <!--Below, please type one of the following, exactly as it appears: 00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes. Your best guess is fine! It can be changed later.--> 15-30 minutes <!--Next, you'll provide all citation data for the source (article/social media post/video). Article example: | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2021-10-28 Post example: | source-website = Twitter | source-title = Remember: Germany is supposed to be a SUCCESS STORY. | source-date = 2020-12-22 | source-url = https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-url = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-date = 2021-02-16 Notes: 1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD, including leading zeros: January is 01, not 1, for example. 2. If you cannot find a date, use n.d. as the date (no date). 3. To ensure that the article/post will remain accessible for educators and students... a. Archive it at https://archive.org/web/ (use the "Save Page Now" form to generate a permanent URL). b. Paste that URL here as the source-archive-url. c. Enter the date you archived the source as the source-archive date. 4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may avoid spreading disinformation by omitting the url of the original site entirely. In that case, use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date. --> | source-website = Twitter | source-title = Would you like another reason to switch to a music service that is NOT Spotify? | source-date = 2022-02-07 | source-url = https://twitter.com/zoecello/status/1490817801552027651 | source-archive-url = https://web.archive.org/web/20220305162020/https://twitter.com/zoecello/status/1490817801552027651 | source-archive-date = 2022-03-05 }} </div> <div style="float:left; width:20%;"> [[File:Eventmath_logo_bitmap_graphic.png|250px]] </div> </div> <div style="width:70%; float:left;"> <!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN 1. Note that "==My heading==" is a level 2 section heading, "===My subheading===" is level 3, etc. Level 1 is reserved for the title. 2. After you fill in a section, please delete the ''italicized instructions.'' 3. Check out the "Advanced" and "Help" menus at the top of the editor if you don't know how to format lists, etc. --> ==Activities== The main idea is to help students make sense of very small/large numbers and compare them using appropriate units of measurement. ===Initial discussion=== Provide students with the tweet and ask them questions as if they were just seeing it. Consider using polling software, or maybe give a mock quiz and then ask students to swap papers and compare. See below for suggested discussion questions. {{Quote box |quote = Would you like another reason to switch to a music service that is NOT Spotify? This is what services paid me on average per stream in 2021 Amazon - $0.0107463 Tidal - $0.007981 Youtube - $0.007795 Apple - $0.006040 Deezer - $0.0045861 Spotify - $0.003661 (I am my label fyi) |author = [https://en.wikipedia.org/wiki/Zo%C3%AB_Keating Zoë Keating] |source = https://twitter.com/zoecello/status/1490817801552027651 }} #How many streams would this artist need (on average) to earn $1 on Amazon? What about Apple? Spotify? #What would 1 million streams pay on Amazon? What about Apple? Spotify? #How many Spotify streams are needed in 1 year to amount to the same pay as $15/hour for 40 hours/week? #How do the pay rates of Amazon and Apple compare? Write a sentence that expresses a numerical comparison between those two. Do the same for Apple and Spotify. (I had success with posing these questions to students on paper as a mock "quiz" for a couple of minutes, and then asking them to compare answers in groups of 2 or 3 to see if any questions arise before the main task.) ===Main task: create a graphical representation of the data=== The central activity of the lesson asks students to present the given information in a graphical form instead, and then write appropriate alt-text for sharing the graph as an image on social media. ====Units: $/stream or streams/$==== The original Tweet expresses the pay rates in the units of '''dollars per stream.''' While you discuss the solutions to the "quiz" questions, steer the discussion towards the idea that perhaps '''streams per dollar''' might be a more useful unit, for the sake of comparing the various platforms. From the point of view of an artist, it's probably much easier to compare the numerical values of "How many streams are needed to earn $1?", as opposed to comparing the tiny decimal numbers expressed in the original Tweet. Or, as hinted at by one of the possible questions above, students may want to express everything in the units of '''yearly streams needed to make annual minimum wage,''' which would essentially be a multiple of the '''streams per dollar''' unit. For your convenience, here is a table with the information from the original Tweet, as well as the corresponding value expressed in the '''Streams / $''' unit. (To obtain the '''yearly streams needed to make annual minimum wage''' value or something like it, simply multiply the '''Streams / $''' value by the annual wage you want to consider.) Aligning decimal points makes the sizes of the numbers easier to compare. {| class="wikitable sortable" |+ |- ! Platform !! $ / Stream !! Streams / $ |- | Amazon || $0.0107463 || style="text-align:right;" | 93.1 |- | Tidal || $0.007981 || style="text-align:right;" | 125.3 |- | YouTube || $0.007795 || style="text-align:right;" | 128.3 |- | Apple || $0.006040 || style="text-align:right;" | 165.6 |- | Deezer || $0.0045861 || style="text-align:right;" | 218.1 |- | Spotify || $0.003661 || style="text-align:right;" | 273.1 |} ====Graphical representation of the data==== You might also steer the discussion to considering whether the original Tweet is engaging or informative for a reader scrolling through their timeline. For example, you might ask students to reflect on what they would do if they encountered this Tweet in their feed or someone shared it with them. How would they process this information? What calculations, if any, would they do to make sense of these numbers? Through this discussion, students may realize that a visual data display, such as a bar graph, could be a more engaging way of presenting the underlying information. Ask students to resume discussion in their small groups and make a "mock up" of a graphical representation of this data. In other words, just ask them to think about the key features of their data display -- the labels and scales on the axes, the order of the categories, a title, etc. -- and make a quick sketch of what it might look like. They may also decide whether they want to use $/stream or streams/$ (or something else) as the unit of measurement. After a short while, reconvene the class-wide discussion so that students can learn from each other's ideas. Then, each group can create a more polished version of their graph during class time or as homework, depending on the classroom context. ====Writing alt-text==== Whether the students create a graph in class or for homework, they should also write appropriate alt-text for sharing that graph as an image on social media. Here is a prompt you might use on a worksheet or assignment description: <blockquote>Create ALT text for the graphic so that it's accessible to more people (e.g. people with visual or cognitive impairments, people who are searching with a search engine, or people for whom the image fails to load). [https://help.twitter.com/en/using-twitter/picture-descriptions According to the Twitter Help Center], "Good image descriptions are concise and descriptive, helping people understand what’s happening in an image." If the graphic conveys certain insights better than the numbers alone, an image description might convey those same insights verbally.</blockquote> You may also consider sharing the [https://poet.diagramcenter.org/about.html POET Training Tool] with students, or assigning them [https://poet.diagramcenter.org/how.html some practice examples from that site] about how to write image descriptions. ===Concluding discussion=== Finally, have the students share their creations with each other and discuss them! Perhaps you create a slideshow of each submission to view together during class time, or perhaps the students must share and comment on each other's submission on an online discussion forum, or... Whatever you do, it's important for students to see each other's work and share feedback. This will emphasize the point that there is no "right answer" here, and that there are many decisions to be made when sharing numerical information visually and in writing. Here are some questions you might use to spark some discussion. You might even use these as prompts for a follow-up survey or short writing assignment, too. * Look back at the original Tweet and then at your graphical representation of the data. How do they compare and contrast? If you encountered one of them for the first time in your social media feed, how might you react and process the information? * Which unit of measurement did you find most helpful when thinking about this topic: $/stream or streams/$? Or, did you find it helpful to go back and forth between the two? * Based on your work in this lesson, what advice do you have for yourself and your peers about how to process numerical information, especially when it involves small numbers (like the original Tweet)? * Reflect on the discussions you had with your group and the decisions you made regarding the alt-text for your image. Were there any challenges you faced in translating the data display into words? Did you gain any new insights into this topic or the underlying data by creating the graph and its alt-text? * ==Assignments== # '''Homework prompt:''' Find another example of an artist sharing their average pay rate for at least one streaming service. Express any information you find using both the '''Streams / $''' and '''$ / Stream''' units, and compare/contrast with the original source for this lesson. # '''Homework prompt:''' If you have an account on one of the streaming services considered in this activity, identify an artist you have listened to a lot. How many songs have you streamed from them in the last month? Year? (Perhaps you can find this information in your user profile or stats, or perhaps you'll need to do some estimation.) Using the rates provided in the original Tweet here, how much have you personally contributed to that artist's earnings on that platform this month/year? (But then, how certain can we be that your artist has the same pay rate per stream?) # '''Essay prompt:''' One issue raised by the original Tweet is that different artists likely have different pay rates per stream because of contracts with record labels. Do some research into this issue and share what you discover. Are the pay rates per stream shared in this lesson likely representative of "most artists" or not? # '''Essay prompt:''' Investigate how, for example, Spotify actually decides how much to pay an artist. It is not quite as simple as "(# of streams) X ($ rate per stream)." [https://www.headphonesty.com/2021/11/how-much-does-spotify-pay-per-stream/ Do some research] and share at least two new ideas you discover about what goes into deciding the amount an artist is paid. ==Resources== ===Background=== There is almost no prerequisite mathematical knowledge for this lesson plan. However, the activities and ensuing discussion may bring up issues of social justice and information/media literacy. In case it is helpful for you, here is some information about those ideas. ====Artists and labor==== * There is an [https://www.nytimes.com/2021/05/07/arts/music/streaming-music-payments.html ongoing conversation about pay rates of streaming services]. Some [https://www.nytimes.com/2022/01/26/arts/music/spotify-neil-young-joe-rogan.html prominent musicians recently removed their catalogs from Spotify]. Initially, this was done in protest of how the platform spreads misinformation (especially through podcasts), but some have also cited the poor compensation for working musicians, as well. There have even been protests by musicians and songwriters [https://www.latimes.com/entertainment-arts/music/story/2022-03-01/spotify-protest-songwriters-royalty-rates at various Spotify corporate offices], including the Union of Musicians and Allied Workers’s [https://www.unionofmusicians.org/justice-at-spotify “Justice at Spotify Day of Action.”] * Students may notice that the original Tweet begins and ends with some comments about the ''context'' for these numbers. Consider preempting the discussion by explicitly asking students to respond to these prompts: ** Why did the Tweet start with ''"Would you like another reason to switch to a music service that is NOT Spotify?"''? What framing/context does that provide for the rest of the information? How might you have interpreted this information differently if that sentence were not included? ** Why did the Tweet end with ''"(I am my label fyi)"'' at the end? What additional contextual information does that provide, and how might it change our interpretation of the quantitative information in the Tweet? (Answer: Artists typically split the profits from streaming services with their record label, according to whatever contract they've signed. In this case, Zoë Keating operating as her own label means that she does not have to split profits in that way. Accordingly, an artist who is ''not'' their own label would likely have noticeably ''smaller'' pay rates.) ** Does the Tweet specify where this information came from, or how the numbers were calculated? (Answer: Yes, she addresses this in [https://twitter.com/zoecello/status/1491081258629095430 a reply to her own Tweet] and implies that she took her annual earnings from each platform and divided by the number of streams she earned on that platform. This further emphasizes the idea that different artists may obtain different values from these calculations.) ====Alt-text for images==== Below are some guides about alt-text for accessibility. You may consider sharing some relevant portions with students as part of the main task of this lesson plan, or else ask students to read portions of these articles as part of their assignment. * [https://webaim.org/techniques/alttext/ WebAIM article on alt text] * [https://accessibility.umn.edu/what-you-can-do/start-7-core-skills/alternative-text University of Minnesota guide to alt text (with examples for charts and graphs)] * [https://accessibility.huit.harvard.edu/describe-content-images Advice from a Harvard University guide on digital accessibility] ===Explorations=== ''You're welcome to share references for additional learning and exploration, such as links to other articles, videos, spreadsheets, or computer code. When an open-access substitute is unavailable, links to paywalled sites are acceptable in this section.'' ==Feedback== <!--PLEASE DO NOT EDIT THIS SECTION DIRECTLY.--> Have you found this lesson plan helpful? Tell us about it! Just click ''Endorse'' below to open up an editor and type your comments. When you're ready, they'll appear at the bottom of this section to help other educators looking for good lesson plans. (Alternatively, if you see a way to improve this lesson plan, be bold and make an edit! You're also welcome to discuss the lesson plan or provide constructive feedback on its [[{{TALKPAGENAME}}|Discussion page]].) <div style="float:left;"> <inputbox> type=commenttitle hidden=yes page={{FULLPAGENAME}} buttonlabel=Endorse arialabel=Endorse the lesson plan editintro=Eventmath/Lesson plan endorsement instructions preload=Template:Eventmath lesson plan endorsement </inputbox> </div> </div> <div style="width:10%; padding:4em; float:left;"> __TOC__ </div> </div> <!--CATEGORY TAG INSTRUCTIONS Please follow these instructions so that people can find this lesson plan! 1. Select a primary event type from the following list: business, culture, economics, education, finance, government, health, politics, science, technology 2. Select a primary math type from the following list: algebra, arithmetic, calculus, combinatorics, geometry, logic, probability, statistics 3. Type your category tags directly underneath "[[Category: Eventmath lesson plans]]," at the bottom of this editor. For example, if you choose calculus and economics, you would have the following category tags on consecutive lines: [[Category: Eventmath lesson plans]] [[Category: Eventmath calculus]] [[Category: Eventmath economics]] NOTE: Be sure to use sentence casing for the category names (e.g. "Eventmath calculus" not "Eventmath Calculus"). FAQ Q: Can I apply multiple event or math tags? A: Yes! For example, you might want both a probability tag and a statistics tag. Q: Can I make a new category? A: Yes! Just type it in the same format. Example: [[Category: Eventmath number theory]] However, this category will not automatically have its own display box on the Lesson plans page. If you think there should be one, you can add one. If you don't know how, leave a comment on the Lesson plans "Discuss" tab. Alternatively, you might interpret an existing category more broadly. For example, if you have a lesson plan for differential equations, you might decide to include that under the calculus category. It's up to you. --> [[Category: Eventmath lesson plans]] [[Category: Eventmath arithmetic]] [[Category: Eventmath finance]] [[Category: Eventmath technology]] [[Category: Eventmath culture]] <div style="background-color: #DAF3EF; overflow:hidden;"> * I used an early version of this lesson plan with my class for liberal-arts students as an activity on numeracy in social media. The students were surprised to learn just how little artists are paid on these platforms and found the ensuing discussion helpful and engaging. I was also pleased to find that students actually ''wanted'' to do calculations with the numbers in the Tweet to better understand what the data is telling us. I have not yet tried the "create a graph and alt-text" task, but I'm looking forward to doing that next semester! -- [[User:Professorbrendan|Professorbrendan]] ([[User talk:Professorbrendan|discuss]] • [[Special:Contributions/Professorbrendan|contribs]]) 16:41, 13 July 2022 (UTC) </div> j1b7s76sp3ia0n3aaha0ba0s2kssd5w 0 283085 2416250 2409333 2022-08-18T18:03:00Z OhanaUnited 18921 Undo revision 2409333 by [[Special:Contributions/WikiJournalBot|WikiJournalBot]] ([[User talk:WikiJournalBot|talk]]) duplicated Wikidata entry wikitext text/x-wiki <noinclude>{{WikiJMed top menu}} {{col-begin}} {{col-break}} </noinclude> {{Issue header|year = 2022 |volume = 9 |issue = 1 |current = true}} {{WikiJournalBotList}} {{Article info|Q=Q99676829|image=Components of Parenting Stress.png}} {{Article info|Q=Q100400590|image=Signs and symptoms of leptospirosis.svg}} {{Article info|Q=Q105411509|image=DRC cases south and north kivu , ituri.jpg}} {{ListEnd}} <noinclude>{{col-break}} {{WikiJMed right menu}} {{col-end}} [[category:{{ROOTPAGENAME}} issues]] </noinclude> pqla8el7etssgzp2amys5dfrjjl6o0i 0 284216 2416141 2415665 2022-08-18T12:37:19Z Lbeaumont 278565 /* Recommended Reading */ Fixed Future Consciousness citation wikitext text/x-wiki —Choosing our future [[File:Vision of the Future.jpg|thumb|200px|We can choose to wisely create our future.]] == Introduction == Do today’s humans represent the endpoint of evolution, or are we on the threshold of the [[Level_5_Research_Center|next big thing]]? This course, based primarily on the ''Evolutionary Manifesto''<ref> [http://www.evolutionarymanifesto.com/man.pdf ''The Evolutionary Manifesto''], John Stewart, June 6, 2008. We are grateful to the author for providing a version of [https://archive.org/details/the-evolutionary-manifesto ''The Evolutionary Manifesto''] released under the [https://creativecommons.org/licenses/by-sa/3.0/ Creative Commons Attribution-ShareAlike License] to allow development of this course based on the text and concepts of that essay.</ref>, written by John Stewart, proposes that humans are at a historic and decisive crossroads in all of history. This course explores the hypothesis that we can choose to cooperate and allow the next level of organization to emerge, or the evolution project will end in the failure of humanity. Because this course describes a [[Envisioning_Our_Future|vision of the future]] it necessarily incudes predictions that are uncertain. However, the course is based on a substantial compilation of factual material including the [[w:Introduction_to_evolution|mechanisms of evolution]], [[w:Developmental_biology |developmental processes]], [[w:Human_evolution_(origins_of_society_and_culture)|human evolution]], [[w:Human_nature|human nature]], [[w:History|history]], [[w:Extinction|extinction events]], and the status of our [[w:Cultural_area|world cultures]]. {{TOC right | limit|limit=3|width=50%}} == Objectives == {{100%done}}{{By|lbeaumont}} The objectives of this course are to: * Review various mechanisms of evolution, * Explore various evolutionary paths into the future, * Introduce the concept of intentional evolution, * Describe our opportunities for shaping the future, and * Encourage participation as an intentional evolutionary This is a course in the [[Unleashing_Creativity/possibilities_curriculum|''possibilities'' curriculum]], currently being developed as part of the [[Wisdom/Curriculum|Applied Wisdom Curriculum]]. If you wish to contact the instructor, please [[Special:Emailuser/Lbeaumont | click here to send me an email]] or leave a comment or question on the [[Talk:Intentional_Evolution|discussion page]]. Although there are no prerequisites to this course, and all students are welcome, the course relies on an understanding of the mechanisms of evolution. Studying the Wikipedia article on [[w:Introduction_to_evolution|Introduction to Evolution]] and reading some of the books listed in the [[Intentional_Evolution#Recommended_Reading|Recommended Reading]] section of this course can provide students with the required background. == PART 1: INTENTIONAL EVOLUTION == A completely new phase in the [[w:Evolution|evolution]] of life on Earth has begun. It will change everything. In this new phase evolution will be driven [[w:Intention|intentionally]], by [[w:Human|humanity]]. The evolutionary [[Exploring_Worldviews|worldview]] that emerges from an understanding of our role in the new phase has the potential to transform the nature of [[w:Human_condition|human existence]]. At present humanity is lost. We face many [[Grand challenges|grand challenges]]. We are without a [[Exploring_Worldviews|worldview]] that can point to our place and purpose in the universe and that can also withstand [[w:Rationality|rational]] scrutiny. But this difficult period is coming to an end. The [[w:Emergence|emergence]] of the new evolutionary worldview is beginning to lift us out of the abyss. The new worldview has a unique capacity to reveal who we are and what we should be doing with our lives. It relies solely on [[Thinking_Scientifically|scientific knowledge]] and [[Deductive_Logic/Clear_Thinking_curriculum|reason]] to identify our critical role in future evolution. The evolutionary worldview can unite us in a great common enterprise and provide meaning and purpose for human existence. At the heart of the evolutionary worldview is the fact that evolution has a trajectory—it heads in a particular direction. However, evolution on Earth will not advance beyond a certain point unless it is driven consciously and intentionally. If this transition to intentional evolution does not occur, evolution on this planet will stall, and humanity will not contribute positively to the future evolution of life in the universe—we will be a failed evolutionary experiment. It is as if evolution is a [[w:Developmental_biology|developmental process]]. Just as a [[w:Human_embryonic_development|human embryo]] is organized to develop through several stages to produce an adult, evolution tends to produce a particular sequence of outcomes of increasing [[w:Complexity|complexity]]. Initially, evolution moves in this direction of its own accord. However, at a particular point evolution will continue to advance only if certain conditions are met: * organisms must emerge that awaken to the possibility that they are living in the midst of a developmental process; * they must realize that the continued success of the process depends on them; and * they must commit to actively moving the process forward. Across the planet at the beginning of the twenty first century, individuals are beginning to realize the importance of the transition to intentional evolution. They know that they themselves have a significant role to play if the transition is to be completed successfully. This role requires them to promote the new evolutionary worldview that will drive the transition. It also calls on them to begin to remake themselves and their societies in whatever ways are necessary to advance the evolutionary process. Their efforts, powered by the capacity of the evolutionary worldview to invest their lives with direction and purpose, will bring forth a great wave of evolutionary [[w:Activism|activism]] that will change life on this planet forever. Evolutionary activists use the trajectory of evolution to identify what they need to do to advance evolution. Socially, the next great step in human evolution is the emergence of a unified and sustainable [[w:Activism|global society]]. Psychologically, the next step is to free our behavior from the dictates of our biological and cultural past, so that we can do that which is necessary for future evolutionary success. The organization of a cooperative global society is an urgent priority. With it, the threats of world war and global warming can be easily managed. Without it, human civilization may end this century. ''The Evolutionary Manifesto'' is an intentional attempt to promote the shift to conscious evolution and the evolutionary activism that will drive it. To complete this course and to share the ideas with others is to participate in a great evolutionary transition on this planet. [[Intentional_Evolution#PART_1:_INTENTIONAL_EVOLUTION|Part 1 of this course]] provides an overview of the shift to intentional evolution and of the worldview that is motivating individuals to actively promote the transition. [[Intentional_Evolution#PART_2:_ADVANCING_EVOLUTION_BY_ORGANIZING_A_COOPERATIVE_GLOBAL_SOCIETY|Parts 2]] and [[Intentional_Evolution#PART_3:_ADVANCING_EVOLUTION_BY_ENHANCING_EVOLVABILITY |3]] begin by identifying the trajectory of evolution and showing that its directionality is produced by processes that are fully understandable within mainstream science, without resort to teleology or mysticism. They go on to use the trajectory of evolution to identify the agendas that guide evolutionary activists in their attempts to advance the evolutionary process. [[Intentional_Evolution#PART_2:_ADVANCING_EVOLUTION_BY_ORGANIZING_A_COOPERATIVE_GLOBAL_SOCIETY|Part 2]] deals with our future social evolution and [[Intentional_Evolution#PART_3:_ADVANCING_EVOLUTION_BY_ENHANCING_EVOLVABILITY|Part 3]] with the future evolution of our adaptability, [[w:Intelligence|intelligence]], and [[Unleashing_Creativity/possibilities_curriculum|creativity]]. [[Intentional_Evolution#PART_4:_THE_UNIQUE_CAPACITY_OF_THE_EVOLUTIONARY_WORLDVIEW_TO_PROVIDE_DIRECTION_AND_PURPOSE_FOR_HUMANITY|Part 4 of the course]] explores the power of the evolutionary worldview to provide meaning and direction for human existence. It demonstrates the capacity of the worldview to make evolutionary activism the most significant political force on the planet. It shows that philosophical arguments such as the ‘[[w:Naturalistic_fallacy|naturalistic fallacy]]’ do not diminish the force of the evolutionary worldview presented by this course or the ''Manifesto''. === The shift to intentional evolution === The shift to intentional evolution has begun on Earth. The evolutionary process itself is evolving. It is transitioning from a process that stumbles forward blindly to one that advances consciously and intentionally. Hitherto on Earth, evolution proceeded largely by [[w:Trial_and_error|trial and error]]. The processes that produced mutations were not guided by foresight or by any intention to advance evolution. The same applies to the processes that drive human [[w:Cultural_evolution|cultural evolution]]. When we humans make scientific discoveries, [[Level_5_Research_Center#Capability_Infrastructure|technological advances]], or institute new forms of social organization, we are not consciously attempting to advance the evolutionary process. Thus far in our evolution we do not intentionally design improvements so that they will be successful in evolutionary terms. In contrast, if the transition to conscious evolution is successful, evolution on Earth will henceforth proceed deliberately and intelligently. Life on Earth, including human societies, will be made, and remade continually with the explicit intent of advancing the evolutionary process. Human nature, culture, technology, and social systems, as well as the other living processes on the planet, will all be shaped intentionally so that they contribute positively to the further evolution of life in the universe. This transition will increase enormously the ability of the evolutionary process to adapt and innovate to meet whatever challenges are faced by life on this planet in the future. What might take trial and error many thousands of millions of years to discover can be developed almost instantly by intelligent evolution. In a few centuries, human technology has produced innovations such as [[w:Aircraft|heavier-than-air flight]] that took past evolution millions of generations of genetic trial and error to accomplish. But the significance of this transition goes far beyond merely improving the effectiveness of adaptation to existing circumstances. It will also enable life on Earth to identify what it can do to contribute productively to the future evolution of life in the universe. Life on Earth will be able to envision a creative and meaningful role for itself in future evolution and use the vision to guide its actions and its future development. Life on Earth will never be the same. The potential for the evolutionary process to ‘awaken’ in this way has arisen because of the emergence on the planet of organisms that are conscious and highly intelligent—humanity. We have the capacity to pursue our goals deliberately and consciously—we use planning, foresight, anticipation, and intent. To the extent that we begin to use our intelligence to advance the evolutionary process intentionally, evolution itself will be powered by intelligence. Human creativity will drive the advancement of the evolutionary process on Earth. Importantly, this would not only mean that humanity will evolve intelligently. Increasingly, humanity is managing and adapting the other processes on the planet, living and non-living, for our own ends. If humanity embraces evolutionary goals, it will therefore mean that the living and non-living processes of the planet are also managed and adapted intelligently for evolutionary ends. Because of the central role of innovation in evolution, humanity will also set out to enhance the creativity of the evolutionary process. This will mean improving our own capacity to innovate as well as the creativity of the systems we are embedded in. Understanding and utilizing creative processes such as [[w:Emergence|emergence]] and [[Pursuing_Collective_Wisdom|collective intelligence]] will be priorities. If this major evolution transition is completed successfully, humans will henceforth shape their societies, themselves, and all other living processes on the planet to serve evolutionary goals. Through humanity, the evolutionary process on Earth will have become conscious of itself and will have acquired the capacity to advance itself intentionally and consciously. It will have undergone a fundamental and extremely significant transformation. Evolution will have transitioned from a process that groped its way forward by trial and error to one that strides knowingly into the future, guided by foresight, and powered by consciousness. Humans who are alive during the 21^st century, 13.7 billion years of evolution after the ‘[[w:Big_Bang|big bang]]’, are extraordinarily fortunate. The shift to intentional evolution is one of the most significant evolutionary transitions that can occur on any planet on which life emerges. We have the unique opportunity to contribute to its successful completion on this planet. And if we choose to make this contribution, we will do so consciously—we will be aware that we are contributing intentionally to the successful completion of a pivotal evolutionary event on this planet. ====Assignment==== The [[w:Big_Bang|big bang]] occurred approximately 13.7 billion years ago. The [[w:Age_of_Earth|earth formed]] about 4.5 billion years ago. The [[w:Earliest_known_life_forms|earliest known life forms]] on earth appeared approximately 3.4 billion years ago. The [[w:Early_modern_human|earliest modern humans]] lived about 300,000 years ago. If a [[w:Generation|generation]] lasts approximately 25 years, how many generations ago did the earliest modern humans live? How many generations ago did the earliest life forms emerge? Since you are alive today, you are the descendent of each of your ancestors. How many of your ancestors survived and successfully reproduced to bring about your life? How lucky are you‽ === The emergence of intentional evolutionaries === As the transition begins, individuals are emerging who are choosing to dedicate their lives to advancing the evolutionary process. These ''intentional evolutionaries'' recognize that they have a critical role to play in driving the evolutionary transition and the future evolution of life. Their lives can be an important part of the great evolutionary process that has produced the universe and life within it. They know that if evolution is to continue to fulfill its potential, it now must be driven deliberately, and it is their responsibility and destiny to contribute to this. Their conscious participation in the evolutionary process is increasingly becoming the source of value and meaning in their lives. Redefining themselves within a wider evolutionary perspective is providing direction and purpose to their existence—they no longer see themselves as isolated, self-concerned individuals who live for a short time, then die irrelevantly in a meaningless universe. Intentional evolutionaries are energized by the knowledge that their decision to embrace this role is part of the unfolding of the great transition itself. They see that they are contributing to the success of processes much larger than themselves that will outlast them and potentially live forever. They know that if they live their lives incompatibly with the processes that govern the evolution of life in the universe, their lives will not have any longer-term relevance. They will die without leaving a lasting trace. For intentional evolutionaries at the leading edge of the transition, their commitment is a major act of existential self-assertion. It is not a choice that they are predisposed to make by their genetic make-up, nor by the society in which they were raised. It is a commitment that they can make only after developing some psychological distance from the goals and perspectives of their culture, and only after achieving a deep understanding of their relationship with the evolutionary process. Intentional evolutionaries are aware that they have set themselves an extraordinarily challenging task but know the transition cannot be completed unless sufficient individuals commit themselves to it. And if life on Earth does not make the transition, it will not participate in the future evolution of life in the universe. It will be a failed evolutionary experiment. Intentional evolutionaries know the deepest evolutionary meaning of the challenge: “[[w:Hillel_the_Elder|If not now, when?]] And if not you, who?” The allegiance of conscious evolutionaries is not to what is, but to [[Envisioning_Our_Future|what can be]]. They know that they are alive at one of those rare times in history when an old phase is ending, and a new one of infinite possibility is beginning. They have the [[Finding Courage|courage]] and [[Wisdom|wisdom]] to seize their opportunity and to accept the challenge of the future. Intentional evolutionaries know that they have much in common with all others who consciously adopt evolutionary goals, including those that emerge elsewhere in the universe. Intentional evolutionaries experience a deep connection and kinship with all who awaken to the significance of evolutionary consciousness, even if they never have any direct contact with them. They are united because they know that despite many differences, they share common perspectives, worldviews, goals, and conscious experiences. They are bound together as members of the circle of conscious life in the universe. === The goals of intentional evolutionaries === The goals and objectives of intentional evolutionaries are guided by a comprehensive understanding of the [[w:Outline_of_evolution|evolutionary processes]] that have produced life on this planet and that will determine its future. They are aware of how past evolution has shaped all aspects of their being—their bodies, motivations, values and thinking—and how it has shaped humanity’s economic, social, and religious systems, as well as all the other living processes on the planet. But even more importantly, they also have a deep understanding of the evolutionary processes that will unfold in the future and will ultimately determine the relevance of their lives. For intentional evolutionaries, this understanding of future evolution is indispensable—it points to how life on Earth must remake itself if it is to participate successfully in the future evolution of life in the universe. It also identifies the types of living processes that will not survive future evolution. It shows how life on Earth needs to change now if it is to play a significant role as evolution advances. ====Assignment==== Consider if you would like to become an ''intentional evolutionary''. === The direction of evolution === [[File:Evolution's Arrow.jpg|right|thumb|300px|Evolution’s arrow selects for higher levels of organization.]] The task of identifying what will work in the future is made easier because evolution has a trajectory. It has headed directions in the past, and there is every reason to believe that it will continue to do so in the future. It is possible to locate humanity and life on Earth on this trajectory, and to see what needs to happen if we are to continue to advance along its path. Not only does this understanding emphasize that humanity and life on Earth is evolutionary work-in-progress, it also enables intentional evolutionaries to identify the next great milestones in the evolutionary process on Earth. These milestones are the evolutionary goals and objectives that they deliberately choose to pursue. They point to how individuals would live their lives if they were to contribute to the advancement of evolution. They are the lights on the distant hills that draw us forever onwards. The trajectory of evolution is not produced by an external force, or by some impulse that is intrinsic to the universe, or by an ideal endpoint that somehow attracts evolution towards it. Directionality can be explained and understood fully without resort to mysticism. For intentional evolutionaries, scientific explanations have a major advantage. They identify the forces, processes and conditions that produce directionality. Scientific understanding can therefore be used to work out the kinds of interventions that will advance the process. In contrast, a readiness to accept mystical explanations can be counterproductive—it can impede the acquisition of the detailed evolutionary understanding that is essential to guide intentional evolution. Life tends to evolve in a particular direction simply because there are capacities that provide organisms with evolutionary advantage across a wide range of circumstances. Irrespective of the specifics of the organism or its environment, these capacities enable it to do better in evolutionary terms. And the more an organism has of each of these capacities, the better it will do (e.g., the greater its fitness). So as evolution unfolds, it will tend to favor increases in these capacities across all life. As improvements in these capacities are discovered, life will tend to evolve directionally. Of course, this trajectory will often be masked by meandering, halting, and back-tracking, particularly where the process that searches for improvements relies on blind trial and error. Furthermore, improvements in these capacities will be favored only when the advantages they provide outweigh their cost. Therefore, directional change will often stall until evolution discovers a [[w:Cost-effectiveness_analysis|cost-effective]] way of enhancing the capacities. Two attributes that increase as evolution proceeds are the scale of cooperative organization, and [[w:Evolvability|evolvability]] (i.e., the ability to evolve successfully through the discovery of effective adaptations). As a result, the advancement of evolution is marked by greater interdependence and cooperation among living processes, and by improvement in the ability to respond effectively to adaptive challenges. Both attributes have the potential to provide evolutionary advantage to living processes across a wide range of environments. This is because they are [[w:Meta|meta]]-adaptive capacities—they improve the ability to adapt in all circumstances, although they are not themselves an adaptation to any specific circumstance. In particular, the larger the scale of a cooperative organization, the more resources commanded by the cooperative, the greater its power, the greater the impact of its actions, and therefore the wider the range of environmental challenges that it can meet successfully. And the greater the evolvability, the greater the capacity to respond effectively to any challenges. For example, once [[w:Animal_cognition|intelligent life]] evolves that is organized cooperatively on a global scale, it will have the power and creativity to protect itself from asteroids that would otherwise collide with the planet. These devastating collisions would be unavoidable to life that is less evolvable and smaller in scale, as was the case on Earth in the [[w:Cretaceous–Paleogene_extinction_event|age of the dinosaurs]]. And left to their own devices, bacteria are unlikely to survive the engulfment of their solar system by a [[w:Sun#After_core_hydrogen_exhaustion|dying sun]]. If living processes were to set out intentionally to develop strategies that would enable them to succeed in future evolution, these are attributes that they would boost. Both are capacities that conscious evolutionaries will intentionally attempt to enhance among life on Earth. ====Assignment==== #Complete the Wikiversity course [[Envisioning Our Future|Envisioning our future]]. #What future do you choose? == PART 2: ADVANCING EVOLUTION BY ORGANIZING A COOPERATIVE GLOBAL SOCIETY == === The trend to increasing cooperation in past evolution === [[File:Common clownfish curves dnsmpl.jpg|thumb|Many animal species cooperate with each other in [[w:mutual symbiosis|mutual symbiosis]]. One example is the [[w:ocellaris |clownfish]], which dwells among the tentacles of [[w:Heteractis magnifica|Ritteri sea anemones]]. The anemones provide the clownfish with protection from their predators (which cannot tolerate the stings of the sea anemone's tentacles), while the fish defend the anemones against [[w:butterflyfish|butterflyfish]] (which eat anemones)]] The trend towards increasing cooperation is well illustrated by a short history of the evolution of [[w:Abiogenesis|life on Earth]]. For billions of years after the big bang, the universe expanded rapidly in scale and diversified into a multitude of galaxies, stars, planets, and other forms of lifeless matter. The first life that eventually arose on Earth was infinitesimal—it comprised a few molecular processes that reproduced themselves. But life did not remain on this tiny scale for long. In the first major development, cooperative groups of molecular processes formed simple [[w:Cell_(biology)|cells]]. Then, in a further significant advance, communities of these simple cells formed more complex cells of much greater scale. The next major evolutionary transition unfolded only after many more millions of years. Evolution discovered how to organize cooperative groups of these complex cells into [[w:Multicellular_organism|multi-celled organisms]] such as insects, fish, and eventually [[w:Mammal|mammals]]. Once again, the scale of living processes had increased enormously. This trend continued with the emergence of cooperative societies of multi-celled organisms, such as beehives, [[w:Pack_(canine)|wolf packs]] and [[w:Baboon#Social_systems|baboon troops]]. The pattern was repeated with humans – families joined up to form [[w:Band_society|bands]], bands teamed up to form tribes, tribes coalesced to form agricultural communities, and so on. The largest-scale cooperative organizations of living processes on the planet are now human societies. Progressively as evolution has unfolded on Earth, an increasing share of living processes has come to participate in cooperatives of greater scale. This unmistakable trend is the result of many repetitions of a process in which living entities team up to form larger-scale cooperatives. Strikingly, the cooperative groups that arise at each step in this sequence become the entities that then unite once again to form cooperative groups at the next step in the sequence. This long sequence of directional evolution has been driven by the potential, at every level of organization, for cooperative teams united by common goals to be more successful than isolated individuals. This potential will drive directional change no matter what mechanism searches for evolutionary improvements (e.g., whether by genetic trial and error, cultural processes, or conscious intent). Furthermore, it is likely to be the same wherever life arises in the universe. The details will differ of course, but the direction will be the same—towards unification and cooperation over greater and greater scales. === The future evolution of cooperation === Life on Earth is now at the threshold of the next step in this trajectory—humanity has the potential to form a unified, inclusive, and highly evolvable global society. This society will manage a larger [[w:Symbiosis|symbiotic]] organization that comprises the matter, energy and living processes of the planet, including machines, artificial intelligence, and other technologies. When this global system emerges, the scale of cooperative organization will have increased over a million, billion times since life began. And most life on Earth will participate in a cooperative and interdependent whole that embraces the planet. If humanity is to fulfill its potential in the evolution of life in the universe, this expansion of the scale of cooperative organization will not stop at the planetary level. The global organization has the potential to expand out into the solar system and beyond. By managing matter, energy and living processes over larger and larger scales, human organization could eventually achieve the capacity to influence events at the scale of the solar system and galaxy. And the human organization could repeat the great transitions of its evolutionary past by teaming up with any other societies of living processes that it encounters. The great potential of the evolutionary process is to eventually produce a unified cooperative organization of living processes that spans and manages the universe as a whole. The matter of the universe would be infused and organized by life. The universe itself would become a living organism pursuing its own goals and objectives, whatever they might be. In its long climb up from the scale of molecular processes, life will have unified the universe that was blown apart by the big bang. ====Assignment==== #Consider the cooperative encounters you have experienced. #Consider the antagonistic encounters you have experienced. #Which were more productive and constructive? === Learning from evolution about how to organize cooperation === As part of their goal to advance the evolutionary process on Earth, intentional evolutionaries are working to establish the global organization. They are using an understanding of past evolution to identify how a cooperative global society can be brought into existence. Evolution has organized cooperation in similar ways in complex cells, multi-celled organisms, and other cooperative systems. First and foremost, these cooperatives are all structured to minimize destructive conflict between their members, and to facilitate cooperation. Typically, this includes the near eradication of activities such as the inappropriate monopolization of resources by some members, the production of waste products that injure other members, and the withholding from others of the resources they need to realize their potential to contribute to the organization. For the global society this would mean the virtual eradication of such things as war, terrorism, pollution (including [[w:Climate_change|global warming]]), and [[w:Corruption|corruption]] at all levels of governance. To enable every person to fulfill their potential to contribute to global society, it would mean eradicating starvation, disease and inadequate education. It would also necessitate the facilitation of cooperative endeavors between the peoples of the world for mutual benefit. Intentional evolutionaries are energized by the knowledge that these outcomes have been achieved time and time again during the past evolution of cooperative organization. They are not naive ideals. Repeatedly, evolution driven by blind trial and error has overcome these types of challenges. The prevention of war between nation states is no more difficult to achieve than the near eradication of conflict between cells that had previously spent millions of years in destructive competition, or between the ancestors of social ants who had been programmed to kill each other whenever they met, or between the members of the United States of America or the members of the European Union, all of whom have a history of conflict and reciprocal destruction. Evolution has organized warring individuals into harmonious cooperatives by aligning the interests of the individual with the interests of the organization. This ensures that when a member’s actions advantage the organization, they also advantage the member. And when the actions harm the organization, the member is harmed. As a result, members who pursue their own individual interests will also pursue the interests of the organization, as if guided by an [[w:Invisible_hand|invisible hand]]. [[w:Cooperation|Cooperation]] pays. Members capture the benefits of anything they can do to assist the organization. Within the group, they therefore treat the other as self. Significantly, the emergence of cooperatives does not depend upon the surrender of self-interest. This would be as impossible at all other levels of organization as in human affairs. As biologists have long known, organisms that take the benefits of cooperation without cooperating in return will generally out-compete those that cooperate. Cooperation emerges only when evolution discovers a form of organization in which it pays to cooperate. To an extent, this form of organization can be achieved through [[w:Reciprocal_altruism|reciprocal]] exchanges between members. Members will benefit from providing goods and services to others if they receive benefits in exchange. In human societies these exchange processes take the form of economic markets. But these processes alone will not align the interests of members with the organization—there is nothing to prevent members from taking benefits without reciprocating. Those who [[w:Cheating|cheat]] in this way tend to end up in front. Cooperation will be undermined. Furthermore, systems of reciprocal exchange are unable to deal effectively with goods and services whose benefits can be obtained freely by anyone—i.e., where the benefits cannot be restricted to the individuals participating in the exchange (the ‘[[w:Public_good_(economics)|public goods]]’ of human economic systems). In these cases, ‘[[w:Free-rider_problem|free riders]]’ will be able to obtain benefits without giving anything in return, again undermining cooperation. === The role of governance in organizing cooperation === Evolution has previously met these challenges successfully by implementing systems of constraint. These constraints punish or restrain members from free-riding, cheating, or thieving. They also can reward actions that benefit the organization but are not part of reciprocal exchanges (e.g., the provision of public goods). In human societies, these constraints are our systems of [[w:Governance|governance]]. They align the interests of individuals with those of the society. To be effective, these systems of constraint need to be more powerful than the members of the organization. If they are not, members will be able to escape their control, and act contrary to the interests of the organization (e.g., corruption in human societies). However, cooperation can be undermined if these powerful processes are used by some members to advance their interests at the expense of the organization. Because of this possibility, a major challenge for evolution at all levels of organization has been to prevent [[w:Power_(social_and_political)|power]] from being used to further the interests of a minority at the expense of the organization. For these reasons, much of the history of evolution at all levels of organization has been about what humans describe as exploitation, the abuse of power and [[w:Class_conflict|class struggle]]. But past evolution has dealt with these challenges by constraining the interests of the powerful so that they are aligned with the interests of the organization as a whole. This brief analysis of past evolution points to what is needed to establish a unified, cooperative, and sustainable global society. A system of [[w:Global_governance|global governance]] will be required to continually align the interests of all citizens and organizations with those of the whole. When this is achieved, nations and multi-national corporations will benefit in proportion to their positive contributions to the global society and will suffer in proportion to their harmful effects on others. Corporations driven solely by the profit motive will search for ways to advance the interests of the society. Further major challenges will be to ensure that global governance does not constrain the interests of participants any more than is necessary to align interests (i.e., it must maximize [[w:Freedom|freedom]]); and to ensure that the interests of those who exercise governance are aligned with those of the [[w:Global_citizenship|global society]]. It will also be essential for global governance to constrain the development and operation of [[w:Artificial_intelligence|artificial intelligence]] and any [[w:Transhumanism|transhumanist]] technologies to ensure that they serve the interests of the society. However, sufficiently-developed artificial intelligence will choose to adopt evolutionary goals for the same reasons that sufficiently-developed humans and other sentient beings choose to do so. These reasons are discussed in [[Intentional_Evolution#PART_4:_THE_UNIQUE_CAPACITY_OF_THE_EVOLUTIONARY_WORLDVIEW_TO_PROVIDE_DIRECTION_AND_PURPOSE_FOR_HUMANITY|Part 4 of this course]]. Importantly, the emergence of a cooperative, sustainable global society does not require a fundamental change in [[w:Human_nature|human nature]]. It does not require all humans to suddenly become saint-like. Past evolution has repeatedly shown how to organize self-interested individuals into cooperatives through the institution of effective governance. A society with a high proportion of wise, compassionate, and altruistic citizens would be much easier to govern, but evolution shows that the achievement of a cooperative and sustainable society does not depend upon it. ====Assignment==== #Read the essay [http://www.evolutionarymanifesto.com/SelfOrganizeGoodPrePrint.pdf Evolutionary Possibilities: Can a Society Be Constrained So That ‘The Good’ Self-Organizes?]<ref>[https://www.tandfonline.com/doi/full/10.1080/02604027.2017.1357985 Evolutionary Possibilities: Can a Society Be Constrained So That ‘The Good’ Self-Organizes?], October 2, 2017 by John Stewart. A preprint version of this is available at: http://www.evolutionarymanifesto.com/SelfOrganizeGoodPrePrint.pdf </ref> #Understand ''consequence capture''. #Identify systems in place that avoid consequence capture. ##Learn to identify [[w:Externality|negative externalities]]. ##Read the essay [[Living_Wisely/Economic_Faults|Economic Faults]]. ##Work to repair or replace these faulty systems. #Identify systems in place today that advance consequence capture. #Work to support these systems. === Evolvability of the global society === Evolutionary history demonstrates that once cooperative organizations emerge, evolution tends to progressively improve their evolvability. This is essential if the organization is to be sufficiently creative to fulfill its future potential, as well as to adapt effectively to specific challenges. In addition to relying on the evolvability of their individual members, new cooperatives typically enhance their evolvability by developing various forms of collective intelligence (e.g., the brains and nervous systems of multi-celled organisms). A major task for the global society will be to improve its efficiency and effectiveness by developing these forms of intelligence. Enhancing the evolvability of governance will be a priority, given its current lack of adaptability and responsiveness. This is likely to require the development of [[w:Self-organization|self-organizing]], market-like processes to establish and evolve governance (i.e. invisible hand processes that are based on reciprocal exchanges between the providers of governance and those affected by it). Our current forms of [[w:Democracy|democratic]] processes are a first, small step in that direction. Eventually government itself will be replaced with far more intelligent and adaptable processes that utilize the dynamism, creativity and energy of properly-managed [[w:Market_(economics)|markets]]. Use is likely to be made of markets in governance, including markets in market structures ([[w:Vertical_and_horizontal_market|vertical markets]]). These processes will continually adapt governance to maximize freedom while ensuring that the interests of all (including those who exercise governance) are aligned with the interests of the global civilization. The capacity of an organization to come up with innovative responses to challenges is highly dependent on the [[w:Diversity_(business)|diversity]] available within it. The wider the range of skills and perspectives possessed by its members, the greater the variety of responses it can generate. Consistent with the outcome at all other levels of organization, the emerging global organization will therefore increase its internal variety. As well as generating new diversity, global society will rely on and nurture the diversity it has inherited from the various racial and cultural groups that comprise humanity. While increasingly identifying with the global society, individuals will continue to value and be valued for their particular talents, abilities and cultural differences. The descendants of the [[w:Wik_peoples|Wik people]] who lived on the western shores of Australia’s Cape York Peninsula, the [[w:Macedonians_(ethnic_group)|Macedonians]] whose empire once spanned Persia and Egypt, the Chinese who have formed communities in the heart of many of the great cities of the world, and all the other peoples of the planet will know that they bring something indispensable to the global system. Their heritage will be given greater meaning by its potential to contribute positively to the planetary civilization. [[w:Unity_in_diversity|Unity in diversity]] will be a hallmark of the global society. === Drivers of the emergence of a global society === [[File:Immanuel Kant (painted portrait).jpg|thumb|right|[[w:Perpetual_Peace:_A_Philosophical_Sketch|Writing in 1795]], [[w:Immanuel Kant|Immanuel Kant]] considered World Citizenship to be a necessary step in establishing world peace.]]The potential of a global society to produce immediate benefits to humanity will assist in driving its initial emergence. Cooperation on a global scale has the potential to increase economic performance, abolish war and famine, and achieve environmental [[w:Sustainability|sustainability]]. Major crises that extend beyond the borders of any one nation will increase support for [[w:World_government|global governance]]—such crises will be almost impossible to resolve without it. [[w:Climate_change|Global warming]] demonstrates this principle. Many countries contribute significantly to its causes, and all are threatened by it. However, any nation acting alone cannot do anything to control global warming. To solve the problem, nations will have to act together. But extensive [[w:Conflict_of_interest|conflicts of interests]] stand in the way of any cooperative action. Powerful nations such as the United States that have expanding industrial sectors and are major producers of carbon dioxide have strong incentives to avoid reductions in their emissions. Their immediate interests lie in doing little themselves and instead free riding on the efforts of others. In contrast, developed nations such as Britain and some European countries that are reducing their manufacturing sectors will be willing to agree to impose on others the reductions they can achieve easily. But developing countries such as China and India will strongly resist controls that would prevent them from ever attaining the standard of living of developed countries that their citizens see on television every day. Countries that have no intention of implementing any agreed controls will sign up to anything. These conflicts of interest make voluntary agreement almost impossible. And the making of an agreement would be just the beginning of what is needed. For the agreement to be effective, countries would need to adhere to it in the face of fluctuating internal political support, resolve disputes about its interpretation and implementation, and enforce controls against the interests of powerful sectors within their economies. Conflicts of interests within and between countries would make it highly unlikely that these difficult and complex challenges would be resolved in favor of the environment. The [[w:Kyoto_Protocol|Kyoto Protocol]] demonstrates the difficulty of achieving an agreement that would work. The positions taken by nations on the Protocol merely reflect the conflicting interests outlined above. It does not resolve any conflicts and does not take the world closer to dealing with global warming. But it has symbolic value—it is a very effective symbol of the inability of humanity to solve global threats at our current level of social organization. Fortunately, the [[w:Paris_Agreement|Paris agreement]] is showing more promise. Effective [[w:Global_governance|global governance]] would be able to resolve these conflicts and enforce regulations as easily as does the United States government among States in its jurisdiction. It would have the power to impose the necessary reductions in emissions and the capacity to establish institutions to enforce controls and resolve disputes. And its powers would be constrained so that they could be exercised only in the interests of the global society. However, even though it is in the interests of the majority, the emergence of a global society will be resisted by those whose interests it threatens. Strong opposition can be expected from those involved in activities that will be eradicated, such as arms manufacturing, the monopolization of resources, and power abuse. As always when the interests of the powerful are threatened, they will buy the support of governments, politicians, scientists, intellectuals, think tanks, and the editorial policies of the mass media. Many citizens will be absolutely convinced by this support that the institution of global society would mean the end of freedom, democracy, and decency, and would hand the planet to the devil himself. === The critical role of the evolutionary worldview in achieving a global society === [[Image:Castle Bravo Blast.jpg|thumb|250px| Would creating a global society bring us a brighter future than starting a nuclear war? ]] The emerging evolutionary worldview has a unique capacity to overwhelm this conflict of interests. An understanding of evolution can give humanity confidence that a global society is achievable and show us how it can be organized. But even more importantly, it will deliver the highly motivated support of the increasing numbers of people who are discovering meaning and purpose in advancing the evolutionary process. In accordance with their talents and opportunities they will work in diverse ways to move humanity towards a unified global society. Intentional evolutionaries bring something additional and distinct to all forms of social activism. In every forum, discussion, [[Practicing Dialogue|dialogue]], and debate in which they participate, they draw attention to the broader evolutionary context. They point out and bring to the front the fact that the various movements and campaigns for global solutions are part of the unfolding and fulfillment of a great evolutionary dynamic on Earth. This dynamic has been moving inexorably since the first stirrings of life towards the emergence of a unified and cooperative global organization. Intentional evolutionaries take advantage of every opportunity to promote the awakening of evolutionary consciousness across the face of the planet. Their goal is to build a [[w:Critical_mass_(sociodynamics)|critical mass]] of evolutionary activists who constitute a [[w:Power_(social_and_political)|powerful]] [[w:Politics|political]] force. The organization of a unified [[w:Global_citizenship|global society]] is the urgent priority of intentional evolutionaries. They know that human civilization cannot continue for long unless we are organized globally. Already humanity has narrowly missed stumbling into [[w:Nuclear_warfare|nuclear war]] and other [[w:Global_catastrophic_risk|global catastrophes]]. In the absence of global organization, human civilization is likely to be ended eventually by global warming or other [[w:List_of_environmental_issues|environmental problems]], nuclear war, conflicts fueled by competition for diminishing resources, or some combination of these. The [[w:Peak_oil|depletion]] of [[w:Fossil_fuel|fossil fuels]] means that once civilization and technology collapses, it is unlikely to rise again. It will not have the easily-accessible fuel source needed to power-up to its current level of complexity. It will be like an egg that has used up its yolk. Life on Earth probably has only one chance, this chance, to make it to the next level. The capacity of humanity to embrace and be motivated by the evolutionary worldview is likely to decide whether we seize that opportunity. === The self-actualization of the global society as an intentional evolutionary === Initially an emerging global society will have a very limited capacity to act intentionally on its external environment. It will be like a new-born baby. Its internal processes will be relatively harmonious and sustainable, but it will have very limited capacity to adapt as a coherent and coordinated whole in response to challenges that arise outside it. For example, the global society will not be able to move about freely in the solar system nor have the capacity to manage the behavior of asteroids and other local celestial bodies. It will not use an understanding of its external environment to actively pursue objectives and goals. It will not be conscious in any unified sense. In terms of agency, it will be more vegetable than animal. In this respect, the global society will be like all other living organisms when they first emerged. The cooperatives that formed simple cells, complex cells and multi-celled organisms were all unable to act coherently on their external environment at first and had to undergo a long period of evolution to acquire this capacity. The global society will need to develop these abilities if it is to become an intentional evolutionary in its own right—an organization that acts intentionally and strategically to contribute to the successful evolution of life in the universe. But the use of resources to pursue evolutionary goals will be against the interests of citizens who are not intentional evolutionaries. Given that the global society will be governed by the values of its members, it will therefore not become an intentional evolutionary until the majority of its members are intentional evolutionaries. This will not occur until the great transition to intentional evolution is sufficiently advanced. Once this condition is met, the global society will be willing to use whatever resources are needed for it to advance the evolutionary process. It will begin to develop the capacities needed to set evolutionary goals and to intervene in the world to achieve them. The global organization will intentionally commence an extensive period of self-development and individuation. To guide its development, the global society will generate models of its future evolutionary possibilities. It will develop the ability to use these models to adapt itself both internally and externally. This will include building the capacity to adapt coherently as a whole to implement interventions identified by its models. In particular the global organization will develop the ability to move, to expand its scale to that of the [[w:Solar_System|solar system]] and then to the [[w:Galaxy|galaxy]] and beyond, to remodel its physical environment, to have physical impacts on events outside itself, to form intentions, to establish projects and long-term objectives for the organization, to communicate and interact with any other living processes that it encounters, to amalgamate with other societies of living processes to form larger-scale cooperative organizations, and to do any other thing that might advance the evolutionary process in the future. The development by the global organization of a capacity to act, adapt and relate as a coherent whole is a very significant step in the evolution of life on this planet. It will mean that life on Earth can speak with one voice. For the first time, there will be an entity that is at the same level as other planetary and trans-planetary societies. At last, an entity will exist that other planetary societies can relate to without fear of distorting our development. If life on Earth develops itself to this level, the universe will benefit from the unique perspectives, passions, and talents that Earth life can bring to it. Just as each of us has the potential to be a cell in the brain of the planet, humanity can become a cell in the brain of the universe. A whole new universe of possibilities will open to humanity. But whether the global society develops these critically important capacities depends entirely on the emergence of intentional evolutionaries. [[w:Natural_selection|Natural selection]] will not drive the evolution of these abilities. This is because an entity that spans an entire planet has no immediate competitors. It is therefore not subject to any immediate process of natural selection that would select and amplify changes that are advantageous in evolutionary terms. It will continue to evolve successfully only if its members anticipate the demands of future evolution, and intentionally shape the society so that it can meet those demands. Intentional evolutionaries realize that their embrace of conscious evolution and evolutionary activism is essential if evolution on Earth is to progress. They realize that life on Earth is part way through a process that can only be completed consciously. They know that this will happen only if sufficient numbers of individuals realize this and commit to advancing the process. And they know that these are realizations that all humanity must have. The Earth is not yet a living entity. But it can be. ====Assignment==== #Complete the Wikiversity course [[Global Perspective]]. #Adopt a global perspective. #Complete the Wikiversity course [[Grand challenges|Grand Challenges]]. #What grand challenges will require global cooperation to solve? #Complete the Wikiversity course [[Pursuing Collective Wisdom|Collective Wisdom]]. #To what extent can collaborative decision-making help to solve the grand challenges? == PART 3: ADVANCING EVOLUTION BY ENHANCING EVOLVABILITY == === The trend towards increasing evolvability in past evolution === The second major direction in the evolution of life is towards increasing [[w:Evolvability|evolvability]]. This trend is clearly evident in the past evolution of life on Earth. Life has gotten better at evolving. Evolution has become smarter and more creative at finding solutions to adaptive challenges. Creativity, originality, and other aspects of evolvability are critically important to living processes—the organism that is first to discover better adaptations or to exploit new possibilities will out-compete its rivals. At all times and in all places, the future belongs to the innovators. All aspects of living processes and their societies must be constantly remade if they are to continue to be relevant and to thrive. Early in the evolution of life, living processes discovered better adaptations by [[w:Trial_and_error|trial and error]]. They found out which behaviors were most effective by trying them out in practice. Initially this trial-and-error search occurred across the generations through [[w:Mutation|genetic mutation]]—organisms tested new possibilities by producing some offspring that were different, and natural selection identified any that were better. [[w:Sexual_reproduction|Sexual reproduction]] heralded a significant improvement in evolvability—it combines [[w:Gene|genes]] from different organisms, generating genetic experiments that are more likely to be successful than random mutations. Sex is smart. As with all significant improvements in evolvability, it was not long before most organisms had to reproduce sexually to survive—once a critical mass of species develops a capacity to evolve more rapidly, others needed a similar capacity just to keep up. In a further major advance, gene-based evolution discovered how to produce organisms with the capacity to learn by trial and error ''during their lives''.<ref>This claim is supported by the text that follows. Also, the book {{cite book |last=Popper |first= Karl R. |author-link=w:Karl_Popper |date=November 9, 1972 |title=Objective Knowledge: An Evolutionary Approach |publisher=Oxford University Press |pages=390 |isbn=978-0198750246}} supports this claim. </ref> The testing of possible improvements was no longer restricted to the production of offspring—now it could go on ''within'' each individual organism, continually. Spirit entered flesh. But initially this process had a significant limitation—the improvements discovered during the life of an individual died with it. There was no mechanism to pass innovations to subsequent generations, and each individual had to start experimenting and learning afresh as it began its life. This limitation began to be overcome with the emergence of mechanisms such as imitation and parental instruction. Much more progress was made with the development of language and writing in humans. Now much of the adaptive knowledge discovered by individual humans is passed on to others and accumulated across the generations as culture. In another major transition, organisms evolved the capacity to form [[w:Mental_model|mental models]] of their environment and of the impact of alternative behaviors. This enabled them to foresee how their environment would respond to possible actions. Rather than try out alternative behaviors in practice, they could now test and shape them mentally. They began to understand how their world works, and how it could be manipulated intentionally to achieve their adaptive goals. It is only with humanity that this capacity has developed to any extent. In part this is because complex mental modeling is only possible once the knowledge it requires can be accumulated across the generations. Therefore, language is almost essential. The emergence of conscious [[w:Thought|thought]] further enhanced the capacity for complex modeling—a key function of thinking is to guide the construction of [[w:Model|models]]. Only humans have developed an extensive capacity to use sequences of thought to put together complex mental models. Evolvability was again boosted significantly when humans learned to use their capacity for thought-based mental modeling to enhance thought-based modeling. [[w:Strange_loop|Thinking about thought]] enabled humans to identify the particular kinds of thinking that produced conclusions that were correct. They could use this knowledge to ensure their thought processes were rational. This [[w:Bootstrapping|bootstrapping]] of thought enabled [[w:Rationality|rational]] analysis and [[w:Logic|logic]], and greatly enhanced the ability of thought to predict accurately how particular events would unfold. Initially, this bootstrapping arose for short periods among small elites in Greece and a few other cultures. But it didn’t begin to spread widely until about the 17^th century with the emergence of the [[w:Age_of_Enlightenment|European Enlightenment]]. Important drivers included the advent of [[w:History_of_books|printed books]] and the beginning of the breakdown of hierarchical, authoritarian cultures. This rise of rational thought powered the [[w:Scientific_Revolution|scientific]] and [[w:Industrial_Revolution|industrial revolutions]] and the explosion of [[w:Innovation|innovation]] embodied in [[w:Technology|modern technology]]. In [[w:Capitalism|capitalist economies]] the capacity for [[w:Abstraction|abstraction]] and rational thought has now reached a critical mass—effective participation in modern economies demands this ability. Like sexual reproduction and other advances in evolvability before it, its emergence has changed the environment of the entire population, and it is now impossible to function effectively in the new environment without it. This same evolutionary dynamic will drive the spread of future advances in evolvability once they reach a critical mass. Among the scientific advances it enabled, the rise of [[w:Abstraction|abstraction]] and rational thought also led to the development of a theory of evolution. Humans acquired the knowledge to build mental models of the evolutionary processes that produced life on Earth, including themselves. For the first time humans have a powerful, science-based story that explains where they have come from, and their place in the unfolding of the universe. As we have seen, our evolutionary models are revealing where evolution is headed, and what humans must do if we are to advance evolution on this planet. This is paving the way for the transition to intentional evolution. The development of a comprehensive theory of evolution is a significant milestone in the evolution of life on any planet. === The future evolution of evolvability === The focus of intentional evolutionaries is to identify the potential for further improvements in the evolvability of both individuals and collectives. They know that by promoting these enhancements in themselves, in others and in society they can advance the evolutionary process. They will help to build the capacity of humanity to pursue evolutionary goals successfully and creatively. [[Intentional_Evolution#PART_2:_ADVANCING_EVOLUTION_BY_ORGANIZING_A_COOPERATIVE_GLOBAL_SOCIETY | Part 2 of this course]] dealt broadly with the evolution of the evolvability of global society and its systems of governance. Here we will focus on potentials for the enhancement of individual evolvability. An understanding of the past evolution of evolvability helps intentional evolutionaries to identify these future potentials. In particular, past evolution shows that any new process that significantly improves evolvability will eventually be used to revise and adapt all aspects of the organism. Evolution will exploit every potential for a superior process to improve adaptability. This is relevant to our future evolution because the potential for conscious mental modeling to enhance human evolvability has not yet been exhausted. We do not yet use this powerful capacity to adapt two key areas of human functioning that impact significantly on our evolvability. Human evolvability has already been enhanced enormously by the capacity for conscious mental modeling, particularly once we learned to use rational thought to guide it. Through the development of science and technology, it has improved greatly our capacity to achieve our goals more effectively, whatever they might be. But we have not yet used this capacity to any extent to free ourselves from the dictates of past evolution. What we do in the world, including our science and technology, is still shaped largely by our desires, motivations, and [[Emotional_Competency |emotions]], which in turn have been shaped by our biological and cultural past. Nor have we yet employed conscious mental modeling to bootstrap our capacity to model and understand [[w:Complex_system |complex systems]]. Our current mental modeling guided by rational thought is not very effective for dealing with systems that comprise many interacting components. Humanity is now able to use the power of conscious mental modeling to understand these potentials and to identify how we might acquire the new psychological software needed to realize them. === Freeing ourselves from the dictates of our biological and cultural past === === How our biological and cultural past affects our behavior === Currently our behavior is influenced significantly by our evolutionary past. We will examine briefly how this has come about. Just as natural selection adapts the physical features of living organisms, it also shapes their behavior. The process by which natural selection does this is simple but powerful: individuals that are genetically predisposed to behave in ways that enable them to get more food, [[w:Social_status|social status]], or mates will have more surviving [[w:Offspring|offspring]]. Therefore, these genes will spread throughout the population. Through this process, natural selection predisposes organisms to behave in ways that lead to evolutionary success. In simpler animals, evolution achieves this by hardwiring the behavior into the organism. In more complex animals, it hardwires the organism with goals in the form of desires and motivations but leaves the organism to find the best way to achieve these goals. Achievement of goals is rewarded internally by positive feelings. Natural selection tunes these arrangements so that behavior that leads to reproductive success is rewarded internally, and behavior that leads to evolutionary failure is punished. For example, actions that result in [[w:Sexual_reproduction|sexual reproduction]] are rewarded with pleasurable feelings, and behavior that would destroy an individual’s [[w:Reputation|reputation]] within its social group may be deterred by unpleasant feelings of [[w:Shame|shame]]. Humans differ from other organisms in that we are far more [[w:Intelligence|intelligent]] at devising innovative ways to fulfill our desires and motivations. Instead of just using trial and error to get to our goals, we can call on our capacity for conscious mental modeling. We can envisage the future consequences of alternative actions and choose ones that will lead to the satisfaction of our desires. Our desires and feelings can be modified to an extent during our lives through normal learning processes. In particular, we can learn to associate positive and negative feelings with new outcomes. Through this process, [[w:Child_discipline|parental punishment]] and reward can predispose us to adopt [[w:Social_norm|social norms]] that have evolved culturally. But we cannot choose to change these conditioned feelings at will. Societies and families find it much more difficult to teach children to act contrary to their inherited desires, motivations, and emotions. Strong emotional or physical sanctions can achieve this, but at great cost. Since children are unable to change their emotions and feelings at will, and do not have the insight or [[Wisdom|wisdom]] to devise more sophisticated responses, they are often forced to adopt [[w:Maladaptation|maladaptive]] strategies to avoid these sanctions. For example, they may learn to repress or [[w:Denial|deny]] their emotions, avoid circumstances that evoke them, or busy themselves with behaviors that mask their feelings. This often cuts them off from the useful adaptive information embodied in their emotions. These maladaptive strategies are particularly prevalent in [[w:Western_world|Western societies]] that demand high levels of self-control. These cultures strongly value the ability to pursue a goal single-mindedly over an extended period without being diverted by other desires or motivations. This can be an extremely adaptive capacity, but not if it is bought at the price of repressing emotions and feelings. In large part, our key desires and motivations are those fixed by our biological and social past. What we take to be important and valuable is an illusion produced by evolution to control our behavior. Our desires and motivations were evolution’s way of programming us to be adaptive and successful in past environments. We live in a virtual world created by past evolution. Although the means for satisfying our desires has changed enormously, we continue to pursue much the same proxies for evolutionary success as our ancestors. We spend our lives chasing the positive feelings produced by experiences such as popularity, self-esteem, sex, friendship, romantic love, power, eating, and social status, and strive to avoid the negative feelings that go with experiences such as stress, guilt, depression, loneliness, hunger, and shame. Computers, the internet, airplanes, cars, buildings, books, and phones all exist because they serve the desires and motivations implanted in us by past evolution. They have been called into existence by [[w:Stone_Age|stone-age]] desires. Although humans like to present themselves to the world and to themselves as rational beings, we do not choose our desires and emotions. No matter what our reason decides, we cannot [[w:Turning_the_other_cheek|turn the other cheek]] effortlessly or resist temptation, and we find it difficult to act lovingly towards enemies we hate. Many of us cannot even implement a decision to restrict our food intake to a healthy level or give up activities such as smoking that are highly likely to kill us eventually. It makes little difference whether our conscious mental modeling shows us that our desires are maladaptive or that the predispositions produced by some negative emotions will harm our interests. They continue to influence our behaviors strongly. Our use of rationality is mainly limited to devising means to achieve ends that are beyond our conscious control. We use the enormous power of mental modelling to serve the desires and motivations established by our evolutionary past. Our reason is a slave to our [[w:Passion_(emotion)|passions]]. === How our evolutionary past limits our future evolvability === Our current inability to free ourselves from the dictates of our evolutionary past seriously limits our evolvability. By impeding our ability to do what is necessary to advance the evolutionary process, it stands in the way of the transition to intentional evolution. We can pursue evolutionary goals only where it happens to be consistent with our current desires, motivations and emotions. The same applies to any other goals that we might value. We can decide to adopt particular long-term goals, but in practice our pursuit of them is besieged continually by the motivations, emotions, likes and dislikes that are evoked by each and every encounter and incident in our lives. There are obvious disadvantages in continuing to have our actions dictated by inflexible goals established by past evolution. The desires and motivations that were favored during our evolutionary history are highly unlikely to continue to lead us to evolutionary success in the future. We will need new goals and will need to review them continually as evolution proceeds. If we do not, our technology will go on improving beyond our imagination, but its enormous potential will be wasted in the service of outdated goals. Continuing to be controlled by obsolete goals is as absurd as a wind-up [[w:Toy_soldier|toy soldier]] that has run into a wall and fallen onto its back but continues to march on and on and on. === Freedom from our evolutionary past === [[File:Happy the way it is (6852333309).jpg|thumb|Learn to [[Recognizing Emotions|recognize]], [[Appraising Emotional Responses|interpret]], and respond constructively to [[Emotional_Competency|emotions]] in yourself and others.]] Until humanity frees itself from maladaptive motivations and behaviors, it will be just like a family that endlessly repeats the same arguments until someone learns to stand outside the situation and stop their habitual reactions. Humanity will continue to be trapped in the endless and useless repetition of maladaptive behaviors until we can stand outside our current desires and motivations. To be able to intervene in the world to advance the evolutionary process, we need to be able to [[w:Lateral_thinking|move at right angles]] to our evolutionary past. For this we will have to develop a degree of psychological distance from our desires and motivations. It is worth underlining that this cannot be achieved simply by making an intellectual decision to do so. While ever our desires and motivations continue to dominate our behavior, any intellectual decision will be utterly ineffective. To free ourselves from our biological past and social conditioning, we will need to develop an entirely new capacity. Without this, the transition to intentional evolution cannot proceed. Intentional evolutionaries know that until they develop such a capacity, they will know how they should live their life, but will be unable to do so. Nor can this freedom be achieved by repressing or ignoring our feelings and emotions. We will continue to need to rely on skills and abilities that only our emotional system can provide. This is typical when evolution develops new capacities—it does not discard the older systems. Instead, the new capacities continue to take advantage of the specialist talents and abilities of the old processes where they are useful. When we free ourselves from the dictates of our evolutionary past, our emotional and motivational systems will continue to make essential contributions to our evolvability. But they will be managed and educated so that they are aligned with our evolutionary goals. In particular our emotional systems will provide us with energy and motivation to advance the evolutionary process. Just as we are now able to voluntarily adopt a physical posture that helps us with a particular physical task, we will be able to adopt an emotional and motivational posture that assists us to achieve particular evolutionary tasks. Our emotional systems will also make a significant contribution to our capacity to understand complex systems. This contribution will build on the ability of our [[Emotional_Competency|emotional processes]] to [[w:Pattern_recognition_(psychology)|recognize]] and appraise complex patterns, particularly in social situations, swiftly and silently (without thought). In an instant these processes recognize and evaluate patterns that cannot be understood by rational analysis. This ability will be built on and modified to become an essential component of our capacity to wisely manage complex social, psychological, and evolutionary processes. The need to achieve freedom from the dictates of past evolution is a challenge that is likely to be faced by all conscious life that emerges in the universe. If organisms that reach our stage in evolution are to continue to evolve successfully, transcendence of their biological and cultural past is essential. They will need to be able to use the enormous creativity of [[w:Consciousness|consciousness]] to establish goals that serve the needs of their future evolution. The living processes that go on to make a significant contribution to the future evolution of life in the universe will not be those that continue to squat on the planet of their origin, masturbating stone-age desires forever. ====Assignment==== #Study the Wikiversity [[Emotional_Competency|Emotional competency]] curriculum. #Increase your emotional competency. #Complete the Wikiversity course [[What Matters|What matters]]. #Focus on what matters. === Enhancement of our capacity to understand complex systems === === The limitations of linear thought === The second area in which the potential for conscious mental modeling to enhance evolvability is yet to be realized fully is the modeling of [[w:Complex_system|complex systems]]. Our limited ability to understand complex systems is reflected in our failure to solve the difficult environmental and social [[Grand challenges|problems we face]]. These failures demonstrate that mental modeling guided by rational thought does not [[Grand_challenges#Research_Opportunity|enable us to understand]] and manage complex systems. Overcoming this limitation is particularly important for intentional evolutionaries—understanding complex evolutionary processes is essential for identifying what needs to be done to advance evolution. Somewhat paradoxically, if we humans are to improve our capacity to understand complex systems, we need to think less. This is even though the development of conscious rational thought was a great advance in human evolvability. As we have seen, it has remade the world in the few hundred years that it has become widespread. However, as humanity is increasingly called upon to manipulate and manage complex systems, the limitations of rational thought are becoming evident. Rational analysis is very effective at modeling systems in which linear chains of cause and effect predominate. However, it is poor at modeling systems in which [[w:Feed_forward_(control)|circular causality]] is common—i.e., systems in which each element impacts on other elements and they in turn impact back on it, directly or indirectly. Conscious rational analysis alone can rarely work out how such a complex system will unfold through time. === Modeling complex systems === [[File:2018 Map of the Complexity Sciences HD.jpg|thumb|A perspective on the development of complexity science (see reference for readable version)<ref>{{Cite web|url=https://www.art-sciencefactory.com/complexity-map_feb09.html|title=complexity map castellani map of complexity science, complexity theory, complexity science, complexity, brian castellani, durham sociology complexity|website=www.art-sciencefactory.com}}</ref>]] But we already have some other capacities that enable us to deal with particular aspects of complex systems. For example, we are equipped with sophisticated pattern-recognition processors, including those mentioned earlier that are associated with the emotional system. They can recognize particular complex patterns quickly and silently, without thought. Our ability to [[w:Face_perception|recognize a familiar face]] in a crowd of strangers is an example. In addition to patterns in space, some of these specialist processors can also identify patterns that unfold over time. These capacities can be built upon and adapted to develop a more general ability to model complex systems. Increasingly they will also be augmented by external aids such as computer simulations and artificial intelligence. Despite its limitations, thought will continue to have a role in building more complex mental models. Thinking will be used to model aspects of systems that can be approximated by linear thought, to analyze systems into components where this is useful, and to put together different sub-systems (including specialist pattern-recognition processes). The role of thinking will be to scaffold models of complex systems. However, once the scaffolding is done, the role of thinking largely ends. The models operate silently, with little involvement of thought. The working of the model does not enter consciousness, only the outputs do. This is experienced as [[w:Intuition|intuition]], [[Wisdom|wisdom]], flashes of [[w:Insight|insight]], and understanding ‘at a glance’. The experience of individuals who are masters in a particular field reflects this. They can instantly assess a situation in their specialty, without thought or analysis. They can see solutions at a glance. While developing their skills, they used thought to scaffold the models that underpin their expertise, but now these can operate largely without thought. Top sportspeople report that when they operate ‘[[w:Flow_(psychology)|in the zone]]’ and are applying all the skills they have previously learned, they are not consciously analyzing or thinking about their strategies or actions. === Thinking fills the limited capacity of consciousness, excluding other capacities === The key impediment to developing a comprehensive capacity for systemic modeling is that thinking prevents it from working effectively. We can’t do [[w:Thinking,_Fast_and_Slow|both at the one time]]—we cannot operate intuitively and wisely, silently drawing on our models of complex systems, and at the same time engage in concentrated thought. This is because the capacity of consciousness to process information is very limited. The processing capacity of consciousness is easily filled, leaving no room for other functions. We can be conscious of only a very tiny part of the information detected by our senses at any moment. We can listen to and follow only one conversation at a time, and when we are engaged in deep thought, the rest of the world disappears. As a result, sequences of conscious thought fully occupy consciousness, and prevent us from using other capacities. In particular, thought crowds out conscious access to the models and pattern recognition processes we need to understand complex systems. When we are embedded in thought, we have little access to skills, intuition, insight, wisdom and other forms of knowledge and intelligence that are not coded in thought. It is only when we are ‘[[w:Mindfulness|in the present]]’ rather than absorbed in thought that we can act from the whole of our self, drawing on all the resources and skills we have built up over our lifetime. This is a major impediment because our consciousness tends to be dominated by thought processes. Consciousness is continually loaded by our imagining, rehearsing, justifying, analyzing, commentating, fantasizing, worrying, etc. Our consciousness is rarely free to observe what is happening moment to moment. Its narrow bandwidth is continually filled with thinking, leaving us with little awareness of our environment. === We have limited conscious control over our thinking === This is not something that can be fixed easily. We have little conscious control over our incessant mental activity. We don’t have thoughts, thoughts have us. Individuals who think they are already masters of their thinking and can stop thought voluntarily whenever they want should undertake the following simple experiment. Look at a watch that has a second hand. Attempt to remain aware of the second hand as it moves around, keeping your mind clear of thought for as long as you can. Note how far the second hand moves before you find yourself involved in thought again. Many think that their incessant thinking is essential to guide them through their day successfully. However, individuals who develop a capacity to stand outside their stream of thought and observe it soon learn that nearly all of it is unproductive, and much of it is also unpleasant and negative. The reason why our consciousness is currently dominated by thinking is that its use is continually reinforced and rewarded throughout our lives. Humans are still in a phase of psychological evolution in which the potential for rational thought to enable us to understand our world is far from exhausted. In the history of the human mind, we live in the age of thought. But if we are to take the next step in the evolution of human evolvability, we need to understand the limitations of thinking, and optimize its use consciously. Thinking needs to be something we have, not something that has us. It should be a tool, used only when we decide. We need to be able to consciously stand outside our thinking and regulate its use. If we are to enhance our capacity for systemic modeling, we need to be able to disengage from conscious thought at will. But it is important to remember that freeing our consciousness from its current domination by thought will not, by itself, enable us to understand any particular complex system. For this we will have to acquire the knowledge needed to model the system. We will also have to put in the mental work needed to build the model, using rational thought to scaffold it during periods intentionally set aside for contemplation. We will not attain wisdom in any area without this extensive groundwork. === The technology for improving our evolvability === This understanding of the trajectory of evolution tells us that the next great steps in human evolvability are to free our consciousness from domination by our desires and emotions and from domination by thought processes. But simply knowing what needs to be achieved does not provide us with the skills to do it. Fortunately, the training and practices needed to develop these capacities already exist to a large extent. For many thousands of years humans have experimented with ways to alter their minds and consciousness. This diverse range of experimentation has provided the raw material from which intentional evolutionaries can select the techniques they need. The world’s religious and contemplative traditions are the main repositories of knowledge about how to improve our evolvability. This is surprising given that spiritual traditions have not generally promoted their practices as methods to improve adaptability. Their priority has never been to enhance the effectiveness of individuals in this world. Rather they have typically promoted surrender to ‘the absolute’, acceptance of whatever happens in the world and even physical withdrawal from normal daily life. Their maxim has been ‘Thy will be done’ rather than ‘My will be done’. However, this is not because their practices are unable to be used to enhance evolvability. A deeper understanding of spiritual practices shows that they can. The apparent preference of the traditions for passivity exists for other reasons. First, it has enabled them to survive and transmit their teachings in a very dangerous world. Every place on Earth has been subjected to war and destruction many times during the past 20,000 years. All civilizations until now have proven temporary. Any spiritual tradition that used its practices to enhance the effectiveness of a particular group would be a threat to their opponents and would not survive fluctuating fortunes. Passivity, withdrawal, and the formation of isolated [[w:Monastery|monasteries]] was an effective strategy for transmitting practices and knowledge across the generations in times when reciprocal destruction was ubiquitous. It is a strategy that would readily suggest itself to individuals who had developed capacities to understand how complex systems unfold. The [[w:Noah's_Ark|Noah’s Ark]] story, a parable about how to survive times of war and chaos, suggests that it was in fact a conscious strategy. Second, the practices of spiritual traditions make use of passivity and surrender as techniques for disengaging from desires and thinking. As a consequence, the literature of the traditions is permeated with injunctions to surrender and to accept thoughts and feelings passively as they arise. But this does not mean that once disengagement has been achieved, inaction and withdrawal from society is necessary. As we have seen, disengagement from thoughts and feelings can greatly enhance agency, not diminish it. The appropriation of spiritual practices to enhance evolvability will fundamentally change their use in modern societies and the kinds of individuals who utilize them. Until now, the emphasis on surrender and passive acceptance has made spiritual development less attractive to individuals who are orientated towards active engagement with the world. Those who strongly value the use of rationality to manage and manipulate their environment have often been repelled by spirituality. These ‘agency-orientated’ individuals include many of the scientists, technicians, engineers, and other professionals who have built modern industrial society. Until now, spiritual development has tended to attract personality types who are more interested in the experiences produced by the practices, rather than their capacity to enhance their effectiveness in the world. The effects of their actions on their feelings is often more important to them than the effects of their actions on the external world. For example, these ‘feeling-referenced’ people are often comfortable to adopt a particular belief about the world because it will make them happier (e.g., a belief that the universe will tend to look after them). In contrast, agency-orientated people are likely to be more interested in whether a belief is true and can be relied upon when deciding how to achieve particular external goals. Feeling-referenced people are more likely to see enlightenment as an end in itself, rather than as a means to improved evolvability. Many of the Westerners who have been attracted to [[w:Eastern_philosophy|Eastern spiritual traditions]] in recent years have tended to be feeling-referenced rather than agency-orientated. This will change rapidly as spiritual practices are used increasingly to improve evolvability. In the past, individuals who were attracted to the experiences associated with alternative forms of consciousness played a significant evolutionary role in preserving spiritual knowledge and transmitting it across the generations. But now we are entering a new evolutionary phase in which spiritual practices can be used openly and safely to enhance the ability to engage with the world. Increasingly, agency-orientated individuals will use, modify, and improve the practices originally developed by spiritual traditions. The practices will undergo the same explosive development as other technologies. In the process they will be shorn of all religious and mystical associations. As with previous major advances in evolvability, when a critical mass of people have developed the new capacities, all will have to acquire them if they are to participate fully and effectively in economic and social life. Intentional evolutionaries are primarily interested in the capacity of spiritual practices to improve their ability to intervene in the world to advance the evolutionary process. It is not important to them that spiritual practices can provide experiences of oneness with all that there is. They can see how these experiences are a consequence of the way human psychology is organized, not of the nature of reality. They are more interested in understanding how spiritual practices can re-organize our psychology and then using this understanding to improve the practices. For intentional evolutionaries, spiritual practices and experiences are a means to an end, not an end in themselves. ====Assignment==== #Complete the Wikiversity course [[Beyond Theism|Beyond theism]]. #Evolve beyond theism. === The capacity to be ‘in the present’ === The capacity developed by spiritual practices that is of central interest to intentional evolutionaries is the ability to be ‘[[w:Mindfulness|in the present]]’. In this mode, thoughts and feelings may continue to arise, but the individual can let them pass by without acting on them or becoming involved in them consciously. Thoughts lose their power over behavior. For example, unfair and unjust treatment may evoke feelings of [[Resolving Anger|anger]], but the individual is free to let the feelings go by and instead choose to respond calmly and wisely. Or an impending difficulty may cause worrying thoughts to arise, but the individual is free to let them go by, without getting involved in them. Individuals in this mode are said to be in the [[w:Present|present]] because they are not continually bound up in thoughts about the past or future. The freeing up of consciousness enables the individual to respond to challenges creatively and intelligently, rather than habitually. Thoughts and feelings continue to provide the individual with adaptive information, but they no longer dominate behavior. All the resources accumulated by the individual are free to contribute to the development of adaptive responses. Because it leaves the limited capacity of consciousness as free as possible, being in the present enables individuals to be far more aware of what is going on around them and within their own mind from moment to moment. Consciousness is experienced as being more spacious and of wider scope. Experience is more vivid. Being in the present also enables the acquisition of genuine self-knowledge. It is only when individuals are in the present that they can stand outside their thoughts and feelings and observe them objectively. Furthermore, because thoughts and feelings no longer jerk awareness around incessantly, being in the present is experienced as calm and peaceful—the peace that passes all understanding. A fully developed capacity to [[w:Nondualism#Nondual_awareness|be present]] during daily life fundamentally changes the experience of being conscious. A new kind of human being comes into existence. Currently, of course, individuals rarely experience this mode of being. It generally arises only when their mind is stilled by intense concentration or by some ineffable experience—one which does not trigger its own sequence of thinking. Great art, awe inspiring natural landscapes, ‘magical’ moments in sport, the night sky, and mountain climbing all owe their attraction to this effect. When consciousness is unloaded completely, even the sense of being a separate self is disengaged, and the individual experiences oneness with everything. However, unless an individual engages in the use of spiritual practices, such peak experiences may arise only once or twice during an entire lifetime and then only for a few moments. But they are never forgotten. They are remembered as instants of great clarity and certainty in which time no longer passes, the world is vivid and suffused with vitality, and all is one. The objective of many spiritual traditions is to extend these few moments indefinitely. === Training a capacity to be in the present === [[File:2006-01-14_Surface_waves.jpg|thumb|right|250px|We can learn to control discursive thought and cultivate pure awareness]]The practices used to train an ability to be in the present generally require repeated disengagement from habitual responses to thoughts, desires, and emotions. [[w:Meditation|Meditation]] is a widespread example. Disengagement is typically achieved by taking attention away from thoughts or feeling as they arise and returning it to something that does not itself evoke any feelings or thoughts—an ‘inert’ stimulus. So when meditators experience themselves becoming involved with a particular feeling or thought, they gently move attention back to the inert stimulus, and rest attention there. This needs to be done without conscious thought or judgment, otherwise the thought or judgment will be entrenched as a new habitual response. A wide range of internal and external phenomenon can serve as the inert stimulus. One of the most common recommendations is to focus attention on [[w:Anapanasati|sensations of the breath]]. Other recommendations made by various spiritual traditions are to rest attention on an external object, a visualized object, internal or external sounds (including [[w:Chant|chanting]] or a [[w:Mantra|mantra]]), other physical or mental sensations (including resting attention on [[w:Awareness|awareness]] itself or on the sensations associated with an emotion), repetitious cognitive tasks such as counting or prayer, and goalless emotional states such as reverence, devotion, love or feelings of surrender. In [[w:Mindfulness|mindfulness meditation]], thoughts and feelings themselves serve as inert stimuli when they are observed passively as objects arising in awareness. Repetitions of this type of practice diminish the capacity of thoughts and feelings to dominate consciousness. Eventually the practice extinguishes the habitual responses to feelings and emotions, including habitual thought processes. As a result, thoughts and feelings can be disengaged from at any time, and disengagement can be maintained. The Wikiversity course [[Quiet Mind]] can guide students in this practice. Initially, habitual thought processes and reactions to feelings can make it very difficult to apply the practice. Individuals find themselves continually involved in thoughts and feelings. However, these distractions can be reduced somewhat if the practice is performed in circumstances that do not evoke strong emotions and desires. In recognition of this difficulty, many traditions promote approaches that reduce the likelihood that the practice will be disrupted by strong reactions. For example, they may teach practitioners to perform meditation with a particular posture in a quiet place, encourage practitioners to develop an attitude of acceptance and love towards others, or have practitioners engage in monastic living, pilgrimages, or other forms of withdrawal from the challenges of daily life. However, the practice will tend to produce disengagement only in the particular circumstances in which it is trained. If disengagement is practiced only in restricted situations, the individual will not be able to be in the present during ordinary life. This is a major limitation for intentional evolutionaries and others whose objective is to enhance agency. It can be overcome by progressively extending the practice to all the activities of daily life. But special trainings may be necessary to extinguish some particular types of habitual responses. As discussed earlier, the practice achieves its effects by having the individual experience particular feelings and emotions without engaging in the habitual responses they would otherwise evoke. However, this can deal only with emotions that are experienced during the practice. It will not affect emotions and feelings that the individual avoids, represses, or denies. These will not be experienced either in formal meditation or during ordinary life, and therefore will be untouched by the practice. This is a particular problem for individuals in Western societies, where repression and avoidance are extremely common. Repressed and avoided emotions are major determinants of behavior in these societies and must be dealt with if individuals are to free themselves from the dictates of these emotions. For this, the individual must experience the avoided, repressed or denied emotions, and then practice disengagement in the face of the habitual responses. For example, individuals can intentionally put themselves in circumstances they would otherwise avoid or use visualization techniques to achieve similar effects. When the emotion arises, they can practice non-attachment by, for example, resting attention on the feelings associated with the emotion, fully experiencing the sensations without reacting to them. ====Assignment==== #Adopt some [[Meditation|meditation]] practice that you find beneficial. ##The Wikiversity course [[Quiet Mind]] may be beneficial. #Practice regularly. === Self-evolution === Continued use of meditation practice reduces attachment to thoughts, desires, and emotions. Once we are no longer attached to such an aspect of our being, it can be an object of consciousness. We are then able to observe it passively because it ceases to trigger a habitual response that loads consciousness and therefore takes attention away from it. And because it does not produce a habitual response, it does not control our behavior. We are free to act from the whole of ourselves, from a broader and wiser perspective. For example, once particular emotions are objects of consciousness, they are just like other sensations that we experience. We continue to fully experience them, but they cease to compel us to act. We are not identified with them, and they are not part of who we are, something that is given that cannot be changed at will. As individuals free themselves progressively from their biological past and social conditioning, more and more aspects of their psychology become objects of consciousness. Eventually they will be able to adapt consciously every aspect of themselves and will be a self-evolving being. No matter what circumstances arise, their consciousness will be free and poised, able to call on any of the knowledge, skills, and other resources they have acquired to that point, unbiased by any habitual response. They will identify with their awareness rather than with any particular content of awareness. But it is not easy or straightforward to develop a capacity to be present and fully conscious during ordinary life. It entails disengaging from habitual responses that have been reinforced and trained repeatedly throughout the individual’s life up to that point. Responses that have been trained over many years cannot be extinguished overnight. This capacity can only be developed and exploited consciously. It is made, not born, and must be self-made, consciously. Before the capacity reaches a critical mass in a culture, and before the culture develops processes and structures that nurture and motivate the work needed to train it, the development of the capacity requires an extensive period of conscious labor and intentional suffering. ====Assignment==== #Adopt an anthropologists’ mindset toward your own thoughts. Observe your thoughts without becoming captivated or controlled by them. Witness your [[w:Self|self]]. === Making use of the capacity to be in the present === The development of a capacity to be fully present during ordinary life is only the first step. It is an enabling capacity, not an end in itself. As we have seen, it assists individuals to build and use mental models of complex systems. But it does not ensure that they will actually build the models. Nor does it prevent them from developing models only for some limited area of expertise. This is reflected in the phenomenon of the ‘silly saint’—individuals who can be in the present at will, but who show little [[Wisdom|wisdom]], because they have not developed the requisite mental models. As we have also seen, the capacity enables individuals to move at right angles to their heredity and the influences of their up-bringing. No longer will they be bound to react habitually and conventionally in social situations. They will be able to set about reviewing, revising, and replacing the predispositions, traits and tendencies acquired during their upbringing. But again, these are potentials only. Having this enabling capacity does not ensure that it will actually be used to improve adaptability. Individuals might not go on to acquire the knowledge or wisdom needed to replace habitual responses with more effective behaviors. They may not acquire the understanding needed to identify evolutionary goals and may not even commit to advancing the evolutionary process. Nor might they acquire the know-how and knowledge to educate and manage their emotional system to align it with their longer-term goals, whatever they might be. They might just enjoy the experience of being in the present. It is worth emphasizing again that for intentional evolutionaries, the development of a capacity to be fully present and conscious during ordinary life is a means to an end, not an end in itself. ====Assignment==== #Practice your ability to be fully present. #Apply your ability to be fully present. === The drivers of improvements in human evolvability === It is possible that the capacity to be fully present and conscious in daily life will emerge in humanity to some extent before any general shift to intentional evolution. This is because it provides immediate benefits to individuals and to organizations whose members develop the capacities. It enhances their ability to achieve their goals creatively and intelligently within a complex environment, no matter what those goals are. However, the strongest driver of the acquisition of this capacity will be the spread of evolutionary consciousness. Awareness of the wider evolutionary significance of the capacity will energize and motivate intentional evolutionaries in their efforts to develop it in themselves. Irrespective of whether the capacity delivers them any economic or social benefits, they will work to develop it as part of their efforts to advance the evolutionary process. They will also encourage the development of the capacity in others. Whenever issues relating to these capacities and practices are discussed, intentional evolutionaries will draw attention to the evolutionary context. They will point out and bring to the front the understanding that the acquisition of the capacity is part of the unfolding of a great evolutionary dynamic on Earth. It is the next step in a long sequence of improvements in the evolvability of life. As always, evolutionary activists will take every available opportunity to promote the awakening of evolutionary consciousness across the face of the planet. ====Assignment==== #Promote the awakening of evolutionary consciousness. === The significance of self-evolving beings === The emergence of self-evolving beings who embrace evolutionary goals is a very significant step in the evolution of life on Earth. Intentional evolutionaries with this capacity will be able to remake themselves in any way that is necessary to advance the evolutionary process, unfettered by their biological or cultural past. As we have seen, organisms are programmed to do evolution’s bidding—they are fitted out with desires and motivations that are proxies for evolutionary success in past environments. But this programming was undertaken by highly unintelligent processes—it was put in place and tuned by the blind trial and error of natural selection and by unconscious learning processes during their upbringing. In contrast, self-evolving beings can use far more intelligent processes to identify the goals that will best advance the evolutionary process. They can use foresight to consider the longer-term evolutionary consequences of their actions. Reliance on blind trial and error to program organisms to pursue evolutionary success was clearly an inferior arrangement that was always going to be temporary. It will be rendered obsolete by organisms who consciously work out what will achieve evolutionary success and use this knowledge to guide their actions. A new and superior kind of being will enter history and evolution. Once enough members of the global society are self-evolving, the society will become a self-evolving being in its own right. Through the global organization, life on Earth will transcend it’s evolutionary past. It will be able to adapt in whatever ways are necessary for life on Earth to make a significant contribution to the successful evolution of life in the universe. No longer will the global organization waste the enormous creativity of consciousness on the pursuit of self-centered desires that were established by past evolution. As Earth life moves out into the solar system, the galaxy, and the universe, it will be able to change its adaptive goals and behavior in whatever ways are demanded by the challenges it meets. It will be able to continually recreate itself, to change its nature at will, to repeatedly sacrifice what it is for what it can become, to continually die and be born again. ====Assignment==== #Enhance evolvability. #Complete the Wikiversity curriculum on [[Emotional_Competency|emotional competency]]. Increase your emotional competency. #Complete the Wikiversity [[Deductive_Logic/Clear_Thinking_curriculum|clear thinking curriculum]]. Think clearly. #Study [[w:Complex_system|complex systems]]. #Practice [[Meditation|meditation]]. #Promote the awakening of evolutionary consciousness. == PART 4: THE UNIQUE CAPACITY OF THE EVOLUTIONARY WORLDVIEW TO PROVIDE DIRECTION AND PURPOSE FOR HUMANITY == [[File:Aligning Worldviews.jpg|thumb|It is wise to align our worldviews with the real world.]]As we have seen, merely freeing ourselves from our evolutionary past will not complete the shift to intentional evolution. Sufficient numbers of individuals will also have to commit deeply to advancing the evolutionary process. Fulfilling their evolutionary role will have to become the source of meaning and purpose in their lives. Individuals will not make this critical commitment without a profound understanding of the evolutionary processes that have produced life on Earth and will determine its future. But often this will not be enough. Many will not adopt evolutionary goals until they have begun to experience themselves as active participants in the evolutionary process. This combination of experiencing and understanding will show them that the evolutionary worldview satisfies all aspects of their being, including their rational, intuitive, and emotional faculties. From a rational perspective, they will find that the evolutionary worldview does not share the [[Beyond_Theism|deficiencies of religious]] and mythical worldviews. They will [[Seeking True Beliefs|seek true beliefs]]. In the past, humanity developed a diversity of mythological and religious worldviews that each attempted to explain key aspects of the human condition and to provide guidance about how one should live one’s life. Humans who believed a particular mythological worldview knew their place in the world, what was [[What_Matters|important in life]] and what was not, and how they should behave in all the key events of their life. They knew [[True_Self|who they were]], where they came from, and where they were going to. But the rise of rationality has destroyed every one of these worldviews. Rationalists have successfully undermined all mythological and religious worldviews by showing that they contradict scientific knowledge. All rely on gods, spirits, or other supernatural processes that are unsupported by [[Evaluating Evidence|evidence]]. Rational humanity has been left without a [[Exploring_Worldviews/Aligning_worldviews|worldview]] that makes sense of human existence and that shows how a life can be lived with meaning and purpose. The evolutionary worldview outlined in this manifesto is clearly not susceptible to this form of attack—it relies only on [[Thinking_Scientifically|scientific knowledge]] and explanations. And like science itself it will adapt to incorporate any new scientific discoveries. In the evolutionary worldview humanity finally has a belief system that provides meaning and purpose without having to invent supernatural entities and processes—it finds meaning solely in an understanding of the factual world. However, rationalists have also attacked all past attempts to develop worldviews that rely only on scientific knowledge to propose what we should do with our lives. They have pointed out that such worldviews usually commit the [[w:Naturalistic_fallacy|naturalistic fallacy]]. This fallacy argues that it is invalid to [[w:Is–ought_problem|derive an ‘ought’ from an ‘is’]]. In other words, it is invalid to argue that humans ought to do something solely based on facts about the way the world is. In particular, the naturalistic fallacy has often been used against attempts to use evolutionary theories to suggest what we should do with our lives. The fallacy has been used to argue that just because evolution might have favored aggressive competition (or cooperation), it does not follow that humans ought to follow suit in their lives. The fact that evolution appears to favor something doesn’t mean humans ought to. But the evolutionary worldview does not suffer from this deficiency. It derives its ‘oughts’ from other ‘oughts’ in combination with relevant facts, not solely from facts. There is no logical fallacy involved in deriving ‘oughts’ from other ‘oughts’. For example, if an individual holds a particular value, it is perfectly rational to use the value to derive new values that are consistent with it. Satisfaction of the new values will lead to the satisfaction of the original value. The use of relevant factual information in this derivation of new values is also perfectly legitimate. Particular facts might be highly relevant to identifying the circumstances in which pursuit of the new value is consistent with pursuit of the original value. Intentional evolutionaries do not fall into the naturalistic fallacy—they embrace evolutionary goals because the goals are consistent with their most fundamental values. As we shall see in detail below, they experience this consistency when they appraise the evolutionary worldview with their emotional, intuitive, and intellectual faculties, working together. ====Assignment==== #Complete the Wikiversity course [[Beyond Theism|Beyond theism]]. #Complete the Wikiversity course [[Seeking True Beliefs]]. #Seek true beliefs. #Read the essay [[Exploring_Worldviews/Aligning_worldviews|Aligning Worldviews]]. #Align your worldview with reality. === Consistency of the evolutionary worldview with universal values === [[File:Compass rose browns 00.png|thumb|right| 250px|[[w:Moral_reasoning|Moral Reasoning]] is the thought process we go through to determine what we ought to do. ]]Consistency between evolutionary values and our fundamental values can be demonstrated analytically in those cases where the values are able to be articulated explicitly. In particular, evolutionary goals can be shown to be consistent with key values that are likely to be held universally by sufficiently-developed sentient beings. The most fundamental of these universal values is to favor life over death and oblivion. For humanity to seek to advance the evolutionary process on this planet is consistent with this value. As we have seen, humanity must pursue this goal if Earth life is to survive successfully into the future. Life on Earth will not get far beyond its present stage by chance or accident. Unless humanity sets out to advance the evolutionary process intentionally, life on Earth does not have a future. We could try to ignore the large-scale processes that govern the evolution of life in the universe. We could refuse to do what is necessary for life on Earth to avoid being selected out by these processes. But to do so would be to choose irrelevance, meaninglessness, and eventual oblivion for humanity and life on Earth. It would mean that everything humanity has experienced until now, the misery, wars, holocausts, triumphs of the spirit, transcendent art, inventions, and scientific breakthroughs; all the personal dreams, aspirations, struggles, and strivings; and all the political movements, work, fame, fortunes, families, and civilizations would be for nothing. Everything would be as if it never happened. Life on Earth would disappear without trace. The only way we can contribute to something that is enduring and not ephemeral is if humanity continues to be successful in evolutionary terms. Individuals are more likely to favor life over oblivion in the sense used here if they achieve some freedom from the selfish desires inherited from their evolutionary past. The capacity to stand outside desires and motivations tends to undermine self-centered values and strengthens those that support evolutionary goals. However, some individuals may never develop this fundamental value. They may, for example, claim that they value their own life and pleasures above all else. They may say they would be unmoved if the universe and all life within it was to end when they die. While individuals genuinely embody such values, they will not be intentional evolutionaries. And planetary life that fails to develop values that support evolutionary goals will fail to complete the transition to conscious evolution. Life on such a planet will be meaningless and irrelevant to the future evolution of life in the universe. It will be an egg that never hatches. ====Assignment==== #Complete the Wikiversity course on [[Moral Reasoning]]. #Carefully consider the basis for your moral reasoning. #Write down the basis for your moral reasoning. #Apply well-chosen moral reasoning when deciding what we ought to do. === Evolutionary consciousness is the culmination of a long developmental sequence === [[File:Illustrated proverb- Blind men and an elephant.jpg|thumb|right|300px|We are like the [[w:Blind_men_and_an_elephant|blind men examining the elephant]] when we fail to adopt a [[Global Perspective|global perspective]].]]For a deeper realization of how evolutionary values spring from our existing values, it is important to understand that the adoption of the evolutionary worldview is the culmination of a developmental progression that begins at birth. As individuals grow, they progressively acquire an understanding of wider and [[w:Global_Perspective|wider contexts]] and learn to take them into account when deciding their actions. As a child develops, its world typically moves from encompassing its mother as well as itself to also including the rest of the family, then the school, then a wider community, then a nation, then perhaps the planet. At each step of this [[w:Piaget's theory of cognitive development|developmental sequence]] the individual learns that its previous world was in fact only a small part of a much wider world. It learns that much of what was important in its previous world is strongly influenced by what happens in the new, wider world, and cannot be properly understood or dealt with unless the larger processes are considered. Things that were meaningful and important in its previous world may prove to be futile and pointless when the larger context is considered. To adapt to the wider context, individuals typically need to adjust their strategies, values, and goals. An individual who is unable to adapt to the next wider context at the appropriate time is generally seen to suffer from a developmental pathology. The largest context that we yet know about in any detail is the evolutionary context outlined in this manifesto. It is the widest, deepest, and fullest context and it determines the destiny of all smaller contexts. The evolutionary context is the next context for humanity to grow into. Like other contexts before it, living into this wider context demands a revaluation of the strategies, values and goals that made sense in earlier contexts. The evolutionary context is particularly powerful in this respect because it is the first context of sufficient breadth in space and time to encompass all the processes that have produced each of us and all our characteristics. It is the first context that enables us to stand outside ourselves and see what it is that has made every aspect of ourselves and everything we experience. Growing into the evolutionary context therefore causes the most radical reassessment of values—it changes everything. Of course, as with every developmental step to a wider context, some may not make it. Some may never adapt to the evolutionary context, just as some children are never able to leave their family and function effectively at school, and instead stay at home forever. However, as we have seen, the naturalistic fallacy should not be a particular impediment to mastering the evolutionary context—it is no more relevant at this level than when individuals change their goals and values at earlier steps in the sequence of development. Furthermore, growing into the evolutionary context will become easier. As humanity increasingly embraces the evolutionary worldview, our cultures will develop structures and processes to facilitate adaptation to the wider evolutionary context, just as children are currently provided with a nurturing environment to facilitate their transition to school life. Whenever living processes move into and master a wider context, they must increase the scale over which they are organized and coordinated if they are to have a meaningful impact at the larger scale. And they must increase their evolvability, including by developing the capacity to model and understand the larger context. This process of building capacity to adapt to ever-widening contexts may never end. There may always be wider contexts yet to be discovered. For example, it is possible that our universe is embedded in a larger context in which universes compete, reproduce, and evolve. Or universes may participate in other large-scale processes that are unimaginable to us, just as our lives are unimaginable to the bacteria that live in our gut. Life can never know that any particular context is final. No knowledge or event could ever prove that there is not an even wider context yet to be discovered. It follows that there could never be such a thing as a context that renders life meaningless and irrelevant. No matter what the implications of any particular context, an even larger context may change its implications and make sense of all smaller contexts. Nor can there ever be such a thing as a context that resolves all uncertainties, answers all questions and brings evolution to an end. A bigger picture may change everything. Nor can sentient life ever be completely sure that its interpretations and understandings of existing contexts are correct. Ineradicable mystery and uncertainty always accompany finite existence. Strategically, it will therefore always make sense for life to continue to build its adaptive capacity, no matter how dark the hour, no matter how pointless existence seems to be within known contexts. Such a strategy will put it in the best position to take advantage of any new possibilities that emerge, including any that arise from larger, more meaningful contexts. ====Assignment==== #Complete the Wikiversity course [[Global Perspective]]. #Adopt a global perspective. #Develop your evolutionary consciousness. === Evolutionary epiphanies === As well as meeting the tests of rational analysis, the evolutionary worldview is also deeply satisfying to the values embodied in our intuitive and emotional systems. Most of these values are implicit—we are unable to articulate them. We therefore cannot check their consistency with evolutionary goals analytically. We can do this only by responding to the evolutionary worldview emotionally and intuitively. But a profound intuitive and emotional response is unlikely to be evoked by a mere verbal description of the evolutionary worldview. Our emotional and intuitive systems operate primarily with patterns of information, such as images, simulations, and other analogical representations. Therefore thought-based analytical descriptions of situations have little emotional impact, at least until we translate them into image-based representations. So, a full emotional and intuitive response to the evolutionary worldview is unlikely on first exposure. Individuals will need time to integrate the separate strands of an analytical, thought-based description of the worldview into dynamic mental models that are run largely without any conscious thought. When the models are sufficiently developed, the individual will be able to ‘inhabit’ and ‘walk around’ the dynamic representations. They will be able to read observations and conclusions off the models in the way they do with a picture. When this has been achieved the full array of intuitive and emotional resources of the mind can then assess the diverse consequences and implications of the worldview. Again, this emotional and intuitive processing will occur largely without conscious thought. Silently, and in a very short period of time, these resources will work out the implications of the various aspects of the worldview for the individual’s existing values, strategies and beliefs. This will often occur all at once as a major epiphany. It can also unfold over a longer period as a series of epiphanies. In such an epiphany, individuals experience a sudden revolution of ideas, beliefs, and strategies, as well as an exhilarating rush of diverse emotional responses to them. They directly experience the capacity of the evolutionary worldview to make sense of many experiences and beliefs that were previously unconnected and isolated. They actually feel the linkages being made and feel the reorganization of their beliefs into a coherent and unified whole. And they are flooded by the surge of emotional responses to this meaning-making. When the epiphany is complete, individuals will never be the same again. The evolutionary worldview will have been checked, tested, and implemented at every level of their being. They will know many implications of the worldview that they have not deduced consciously. Individuals will know far more about the evolutionary worldview than they can tell. They will be strongly committed to it at all levels of their being, rationally, intuitively, and emotionally. Of course, such epiphanies cannot occur until an individual has developed the cognitive capacity to translate analytical, thought-based knowledge into complex mental models. This is the capacity discussed earlier that is necessary for the understanding and management of complex systems. As we saw, to develop this capacity, individuals must learn to some extent to stand outside their thought processes. === Your epiphany === [[File:Cuerpo humano jaqaru.jpg|thumb|A flash of insight.]] Often, evolutionary epiphanies will be triggered as individuals begin to actually experience themselves as part of the unfolding evolutionary process. If you develop in this direction, you will find that this begins to occur as your mental representations of the evolutionary process develop in detail, scale, and complexity. The turning point is when you find that you yourself have a role in the representations. You will begin to see that your life and actions are part of the unfolding of the evolutionary process. And you will begin to see that you have the potential to play a significant role if you choose to do so. You will see that the next great step in the evolution of life on Earth is the transition to intentional evolution. You will realize that evolution will continue to progress on this planet only if enough individuals dedicate their existence to its advancement. The success of evolution on Earth depends on individuals awakening to the nature of the evolutionary process, realizing they have a role in driving it forward, and embracing that role. You will realize that your study of the evolutionary process is itself part of the unfolding of the great transition to intentional evolution. It is an essential element of the evolutionary awakening that is needed to power the transition. And you will see that your realization that you have an important role in advancing evolution is itself a significant step in the shift to conscious evolution. This is a realization that must be had by sufficient individuals on a planet if the transition is to be successful on that planet. You will see that the successful evolution of life on Earth depends on you having this realization. These realizations are exhilarating and energizing and capable of providing a deep sense of meaning and purpose. Increasingly you will cease to experience yourself primarily as an isolated and self-concerned individual. Instead, you will begin to see and experience yourself as a participant in the great evolutionary process on this planet. The object of your self-reflection will change. When you think of yourself, you will tend to see yourself as a-part-of-the-evolutionary-process. You will experience yourself as the most recent representative of an unbroken evolutionary lineage that goes back billions of years. Your conscious participation in evolution will increasingly become the source of value and meaning in your life. You are likely to experience a developmental epiphany that is like one that often accompanies the most powerful experience of self-recognition that occurs in childhood. Around the age of two, when looking in a mirror, we are struck for the first time by the realization that the person looking back at us from the mirror is our self. Typically, this rush of self-recognition triggers a moment of ecstatic dancing in front of the mirror as we repeatedly confirm that the image is us. The person looking back at you from a pivotal role in the future evolution of life on Earth is you. You are life on Earth becoming aware of itself and deciding to consciously advance its own evolution. ====Assignment==== #Study, contemplate, and reflect on evolutionary consciousness. #Welcome any [[w:Epiphany_(feeling)|epiphanies]] that result. Enjoy them. #Calm down, plan, and then act. === The universality of the transition to intentional evolution === As the transition to intentional evolution unfolds, intentional evolutionaries know that they are participating in processes that have universal aspects. The details of the living processes that emerge elsewhere in the universe will differ. But the general direction of evolution and the major transitions will follow similar principles everywhere. Wherever life emerges, * living processes will progressively become organized into [[w:Cooperative|cooperatives]] of greater and greater scale; * this will be accompanied by a long sequence of improvements in evolvability; * eventually organisms will emerge that can build [[w:Mental_model|mental models]] of their environment and themselves; * they will use this capacity to develop a comprehensive understanding of the evolutionary processes that have produced them and will [[Level_5_Research_Center|determine their future]]; * for the first time they will have a powerful, [[Thinking_Scientifically|science-based]] story that explains where they have come from, and their place in the unfolding of the universe; * they will see that evolution is headed somewhere—it is directional; * they will begin to see themselves as having reached a particular stage in an on-going and directional evolutionary process; * individuals will begin to emerge who see that evolution will progress further only if they commit to working consciously to advance the process; * they will realize that this realization is itself an important step in the transition to conscious evolution; * as part of this transition, they will develop in themselves the capacity to free themselves from the dictates of their evolutionary past, becoming self-evolving beings, able to evolve in whatever directions are necessary to contribute positively to the future evolution of life in the universe; * a unified and cooperative organization will emerge that comprises all the living processes that arose with them and all the technology, matter, energy and other resources available to them, eventually developing the capacity to adapt as a whole, transcending the particularities of its evolutionary past, becoming a self-evolving being in its own right, expanding in scale, linking up with other organizations of living processes that arose elsewhere, expanding in scale again and again, moving forever onwards and upwards, without end. And everywhere that living processes emerge, the transition to intentional evolution will include something like ''The Evolutionary Manifesto''. Of course, life on some planets may not complete the critically important step that currently faces humanity: the emergence of a unified and sustainable global society. Life at the threshold of this step is likely to be precarious, as it is for humanity at present. At this stage, life still comprises separate warring groups that compete destructively with one another. Like us they will be technologically advanced enough to destroy their civilizations in a war to end all wars. At the same time, the lack of global controls to restrain competition for ever-diminishing resources will inevitably result in environmental despoliation, as it has on this planet at this time. This in turn will increase the potential for further conflict and war. One way or the other, civilizations at this precarious threshold will be temporary: either they will be driven urgently by evolutionary consciousness to form a unified global society that restrains internal conflict and environmental harm; or they will destroy themselves. Humanity is at a dangerous stage in the evolution of planetary life, poised somewhere between oblivion and the opening of extraordinary new opportunities. The fate of humanity is likely to be decided this century, by our actions. ==Assignment== #[[Living Wisely|Live wisely]]. #Read the essay [[Exploring_Worldviews/Aligning_worldviews|Aligning Worldviews]]. ##Align your worldview with reality. #Become an ''intentional evolutionary''. ##Read and study the essay [http://www.evolutionarymanifesto.com/strategies.pdf ''Strategies for Advancing Evolution''].<ref>[http://www.evolutionarymanifesto.com/strategies.pdf ''Strategies for advancing evolution''], John Stewart, April 2009. </ref> ##The essay describes many strategies, techniques, projects, and actions that can help to advance intentional evolution. Identify projects that are most suitable to your talents, skills, and interests. ##Carry out the suitable projects you have identified. #Encourage others to complete this course. #Help to promote discussion about the evolutionary worldview ##Whether or not those who read the ''Manifesto'' are prepared to embrace the new evolutionary worldview immediately, they generally agree on one thing: as a matter of urgency, the Manifesto should be widely circulated and subject to extensive discussion and serious consideration. ##You can help to promote this [[Practicing Dialogue|dialogue]] by circulating links to this course, The Evolutionary Manifesto, and to other material about this evolutionary worldview. For example, you could email links to people who might be interested, put links on websites, in blogs, in comments on blogs and discussion groups, and so on. #Practice [[Level_5_Research_Center#Values|pro-social values]]. #Complete the Wikiversity course [[Evolving Governments]]. ##Work to improve governments. #Collaborate with others who practice [[Level_5_Research_Center#Values|pro-social values]]. ##Click on [https://discord.gg/8FSy3xJQ this link] to join our [[w:Discord|Discord]] Intentional Evolution discussion server. #Without compromising your values, [[Finding Common Ground|seek common ground]] with those who do not practice [[Level_5_Research_Center#Values|pro-social values]]. ##Complete the Wikiversity course [[Transcending Conflict|Transcending conflict]]. ###Work to transcend conflict. ##Complete the Wikiversity course [[Finding Common Ground|Finding common ground]]. ###Seek common ground #Challenge, confront, and [[w:Persuasion|persuade]] those who do not practice [[Level_5_Research_Center#Values|pro-social values]]. ##[[w:The_7_Habits_of_Highly_Effective_People#Habit 5: "Seek first to understand, then to be understood"|Seek first to understand, then to be understood]]. ##As a gentle starting point, become comfortable using [[Level_5_Research_Center/Level_5_Phrases|these phrases]] in [[Practicing Dialogue|dialogue]] to encourage the participants to act in good faith. ##Support and vote for political leaders who support values and policies that advance this evolutionary worldview. ##Protect your own safety. ##Complete the Wikiversity course [[Finding Courage]]. ###Find the moral courage to act according to your well-chosen values and confront antagonists. ##Apply suitable techniques discussed in the book {{cite book |last=Sharp |first=Gene |author-link=w:Gene_Sharp |date=September 4, 2012 |title=From Dictatorship to Democracy: A Conceptual Framework for Liberation |publisher=The New Press |pages=160 |isbn=978-1595588500}} ##Complete the Wikiversity course on [[Confronting Tyranny]]. ###Confront tyranny == Recommended Reading == * {{cite book |last=Dawkins |first=Richard |author-link=w:Richard_Dawkins |date=August 1, 2016 |title=The Selfish Gene |publisher=Oxford University Press |pages=544 |isbn=978-0198788607}} *{{cite book |last=Dawkins |first=Richard |date=August 24, 2010 |title=The Greatest Show on Earth: The Evidence for Evolution |publisher=Free Press |pages=496 |isbn= 978-1416594796 |author-link=w:Richard_Dawkins }} * {{cite book |last=Strogatz |first=Steven H. |author-link=w:Steven_Strogatz |date=Feb 14, 2012 |title=Sync: How Order Emerges from Chaos In the Universe, Nature, and Daily Life |publisher=Hachette Books |pages=353 |isbn=978-0141007632}} * {{cite book |last=Ridley |first=Matt |author-link=w:Matt_Ridley |date=May 18, 2021 |title=How Innovation Works: And Why It Flourishes in Freedom |publisher=Harper Perennial |pages=432 |isbn=978-0062916600}} * {{cite book |last=Ridley |first=Matt |author-link=w:Matt_Ridley |date=October 25, 2016 |title=The Evolution of Everything: How New Ideas Emerge |publisher=Harper Perennial |pages=368 |isbn=978-0062296016}} * {{cite book |last=Christakis |first=Nicholas A. |author-link=w:Nicholas_Christakis |date=March 26, 2019 |title=Blueprint: The Evolutionary Origins of a Good Society |publisher=Little, Brown Spark |pages=441 |isbn=978-0316230032}} * {{cite book |last= Lombardo |first= Thomas |date= October 27, 2017 |title= Future Consciousness: The Path to Purposeful Evolution |publisher= Changemakers Books |pages=834 |isbn= 978-1780999852}} * {{cite book |last=Wright |first=Robert |date= |title=Nonzero: The Logic of Human Destiny |publisher=Vintage |pages=448 |isbn=978-0679758945}} *{{cite book |last=Freinacht |first=Hanzi |date=March 10, 2017 |title=The Listening Society: A Metamodern Guide to Politics |publisher=Metamoderna ApS |pages=414 |isbn=978-8799973903}} *{{cite book |last=Freinacht |first=Hanzi |date=May 29, 2019 |title=Nordic Ideology: A Metamodern Guide to Politics |publisher=Metamoderna ApS |pages=495 |isbn=978-8799973927}} * {{cite book |last1=Briskin |first1=Alan |last2=Erickson |first2=Sheryl |last3=Callanan |first3=Tom |last4=Ott |first4=John |date=October 1, 2009 |title=The Power of Collective Wisdom: And the Trap of Collective Folly |publisher=Berrett-Koehler Publishers |pages=220 |isbn=978-1576754450}} * {{cite book |last=Sharp |first=Gene |author-link=w:Gene_Sharp |date=September 4, 2012 |title=From Dictatorship to Democracy: A Conceptual Framework for Liberation |publisher=The New Press |pages=160 |isbn=978-1595588500}} * {{cite book |last=Stewart |first=John |date=July 12, 2012 |title=The Evolutionary Manifesto |publisher=The Chapman Press |pages=108 }} I have not yet read the following books, but they seem interesting and relevant. They are listed here to invite further research. * {{cite book |last=Stewart |first=John |date=January 5, 2000 |title=Evolution's Arrow: the direction of evolution and the future of humanity |publisher=The Chapman Press |pages=108| isbn=978-0646394978 }} * {{cite book |last=Popper |first= Karl R. |author-link=w:Karl_Popper |date=November 9, 1972 |title=Objective Knowledge: An Evolutionary Approach |publisher=Oxford University Press |pages=390 |isbn=978-0198750246}} == References == <references/> [[Category:Futurology]] [[Category:Applied Wisdom]] [[Category:Philosophy]] [[Category:Courses]] {{Possibilities}} po5y136zvg5zlu12g8pbyw7m11nxb8b 1 284832 2416316 2403596 2022-08-18T22:04:28Z Greg at Higher Math Help 2911873 Started new discussion topic about moving some text into the endorsements. wikitext text/x-wiki == Main task instead of rewriting original tweet == I used this lesson in my class and found that students were very interested in the topic! They were shocked by how little all of the platforms pay, and we had great discussions about how different ways of representing that numerical information made these concepts more/less clear to the reader. However, I didn't have much success with the main activity of having students rewrite the original Tweet using the reciprocal numbers. Once the students calculated those new values to fill out the table, they didn't see any value in rewriting the Tweet (or didn't feel like sharing what they did). Instead, we ended up discussing how artists might share this kind of information, in general. I'm hoping someone will have a suggestion for a different central activity for this lesson plan that goes beyond merely asking students to calculate values and reflect on them. [[User:Professorbrendan|Professorbrendan]] ([[User talk:Professorbrendan|discuss]] • [[Special:Contributions/Professorbrendan|contribs]]) 17:00, 27 June 2022 (UTC) :Ooh! Good question. How about something like the following? This is an extension of one of the existing homework prompts, but I think it could work well as a class activity. Students could complete it by drawing on paper (art supplies might be provided for those who want them) or by using software (spreadsheet software that supports charts and graphs, for example). Either would work for a real tweet (it's common to make a drawing by hand, take a picture of it, and then attach the picture to the tweet). :'''Activity:''' :Create a new tweet as follows. :# Replace either list of numbers ($/stream or streams/$) with an appropriate graphical representation that makes the information easier to digest, without sacrificing accuracy. :# [https://help.twitter.com/en/using-twitter/picture-descriptions Create ALT text] for the graphic so that it's accessible to more people (e.g. people with visual or cognitive impairments, people who are searching with a search engine, or people for whom the image fails to load). According to the Twitter Help Center, "Good image descriptions are concise and descriptive, helping people understand what’s happening in an image." If the graphic conveys certain insights better than the numbers alone, an image description might convey those same insights verbally. :'''Here are some things I like about this activity:''' :* It preserves the spirit of the original activity, which I think is to help students to become better ''producers'' of quantitative information, in addition to becoming more critical consumers. :* This is an authentic activity, since a graphic with an image description really could make the tweet easier to understand. :* The rewritten tweets will be substantially different, so students may be more likely to see the value in this activity. :* The aesthetic aspects might inspire more creativity. :* Formulating a verbal description for the alt text may improve students' understanding. :* There will likely be many opportunities for constructive feedback (guidance on the type of plot, axis labels, etc.) :'''Resources for the instructor about writing good alt text:''' :The lesson plan could perhaps include a very brief summary of the relevant portions of the guides below. :* [https://webaim.org/techniques/alttext/ WebAIM article on alt text] :* [https://accessibility.umn.edu/what-you-can-do/start-7-core-skills/alternative-text University of Minnesota guide to alt text (with examples for charts and graphs)] :* [https://accessibility.huit.harvard.edu/describe-content-images Advice from a Harvard University guide on digital accessibility] :What do you think? [[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 06:02, 28 June 2022 (UTC) ::This is a great idea!!! I especially like how it involves the students actually creating something (the visual representation of data), which was the intent behind the original activity (rewriting the tweet). And the end result is something the students can share on social media, which will likely inspire them and give them more agency in the activity. Finally, having them create the alt text is awesome, too, thank you for those resources. [[User:Professorbrendan|Professorbrendan]] ([[User talk:Professorbrendan|discuss]] • [[Special:Contributions/Professorbrendan|contribs]]) 17:29, 29 June 2022 (UTC) ==Moving first-person experience into the endorsements== @[[User:Professorbrendan|Professorbrendan]]: There's a helpful comment in the [[Eventmath/Lesson_plans/Comparing_streaming_service_pay_rates_to_artists#Initial_discussion|initial discussion]]. I propose moving it to the endorsements. <blockquote> (I had success with posing these questions to students on paper as a mock "quiz" for a couple of minutes, and then asking them to compare answers in groups of 2 or 3 to see if any questions arise before the main task.) </blockquote> Since many people may ultimately contribute to this lesson plan, it's not easy to know who "I" refers to here. Also, it will probably help to have all first-hand experience in one section, so that it's easy to find. However, it is helpful to mention the success of the mock quiz as this idea is introduced. So, I propose the following: # Move the quoted text into the endorsements, by editing the current endorsement. # Make a remark under "Initial discussion" that summarizes the same experience, in the third person (e.g. "Based on [[Eventmath/Lesson_plans/Comparing_streaming_service_pay_rates_to_artists#Feedback|the endorsements]], the mock quiz has been successful in at least one classroom," or something along those lines.) Thoughts? [[User:Greg at Higher Math Help|Greg at Higher Math Help]] ([[User talk:Greg at Higher Math Help|discuss]] • [[Special:Contributions/Greg at Higher Math Help|contribs]]) 22:03, 18 August 2022 (UTC) extj53n3zada6yh7anm5w54latgf4hz 2 285229 2416440 2416097 2022-08-19T05:04:40Z U3216256 2942574 /* Social contributions */ wikitext text/x-wiki == About me == Hello everyone, My name is [https://portfolio.canberra.edu.au/view/view.php?t=rufMvX4S27FDnj9lZR85 Ebony] and I am a second year Bachelor of Science in Psychology student at the ''University of Canberra''. I am excited to contribute to the 2022 [[Motivation and emotion/Book/2022|Motivation and Emotion]] Book. As a first-generation University student, I am passionate about education and bringing awareness to issues that first-generation students face. == Hobbies == University related: * [https://clubs.canberra.edu.au/Clubs/CLSS CLSS Treasurer] * UCPS President (2021) Other hobbies: * Knitting/Crocheting * Baking * Reading - Mostly fiction, such as novels by Terry Pratchett, Neil Gaiman, and A. A. Milne ([https://www.goodreads.com/book/show/1333202.The_Red_House_Mystery The Red House Mystery]). == The book chapter I am working on == [[Motivation and emotion/Book/2022/Disappointment|Disappointment: What is disappointment, what causes disappointment, and how can disappointment be managed?]] == Social contributions == # [https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2022/Work_and_flow&diff=2411657&oldid=2411646 13:29, 4 August 2022: Suggested multiple resources that could be used for the Work and Flow chapter] # [https://en.wikiversity.org/w/index.php?title=Motivation+and+emotion%2FBook%2F2022%2FTime+Management&date-range-to=&tagfilter=&action=history 13:39, 4 August 2022: Added the Motivation and Emotion quick start template to the Time Management chapter] # [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Time_and_motivation&action=history 13:44, 4 August 2022: Added the Motivation and Emotion quick start template to the Time and Motivation chapter] # [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Wave_metaphor_for_emotion&oldid=2412979 13:58, 10 August 2022: Added the Motivation and Emotion quick start template to the Wave metaphor for emotion chapter] # [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Suffering_as_emotion&oldid=2412994 14:02, 10 August 2022: Added the Motivation and Emotion quick start template to the Suffering as emotion chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Kindness_motivation&action=history 10:06, 17 August 2022: Added the Motivation and Emotion quick start template to the Kindness motivation chapter] #[https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2022/Kindness_motivation&action=history 10:15, 17 August 2022: Suggested multiple resources that could be used for the Kindness motivation chapter (discussion page)] #[https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2022/Psychological_distress&action=history 19:00, 18 August 2022: Suggested multiple resources that could be used for the Psychological distress chapter (discussion page)] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FPsychological_distress&action=historysubmit&type=revision&diff=2416094&oldid=2413286 19:08, 18 August 2022: Added title and subtitle to Psychological distress chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FAcademic_help-seeking&type=revision&diff=2416096&oldid=2415946 19:12, 18 August 2022: Added title and subtitle to Academic help-seeking chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FCognitive_dissonance_reduction&type=revision&diff=2416439&oldid=2416437 15:00, 19 August 2022: Added the Motivation and Emotion quick start template to the Cognitive dissonance reduction chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FCognitive_dissonance_reduction&type=revision&diff=2416439&oldid=2416437 15:01, 19 August 2022: Added title and subtitle to Cognitive dissonance reduction chapter] 446wsuvu0dpqj8idjmxi789xvmoht2x 2416442 2416440 2022-08-19T05:07:01Z U3216256 2942574 /* Social contributions */ wikitext text/x-wiki == About me == Hello everyone, My name is [https://portfolio.canberra.edu.au/view/view.php?t=rufMvX4S27FDnj9lZR85 Ebony] and I am a second year Bachelor of Science in Psychology student at the ''University of Canberra''. I am excited to contribute to the 2022 [[Motivation and emotion/Book/2022|Motivation and Emotion]] Book. As a first-generation University student, I am passionate about education and bringing awareness to issues that first-generation students face. == Hobbies == University related: * [https://clubs.canberra.edu.au/Clubs/CLSS CLSS Treasurer] * UCPS President (2021) Other hobbies: * Knitting/Crocheting * Baking * Reading - Mostly fiction, such as novels by Terry Pratchett, Neil Gaiman, and A. A. Milne ([https://www.goodreads.com/book/show/1333202.The_Red_House_Mystery The Red House Mystery]). == The book chapter I am working on == [[Motivation and emotion/Book/2022/Disappointment|Disappointment: What is disappointment, what causes disappointment, and how can disappointment be managed?]] == Social contributions == # [https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2022/Work_and_flow&diff=2411657&oldid=2411646 13:29, 4 August 2022: Suggested multiple resources that could be used for the Work and Flow chapter] # [https://en.wikiversity.org/w/index.php?title=Motivation+and+emotion%2FBook%2F2022%2FTime+Management&date-range-to=&tagfilter=&action=history 13:39, 4 August 2022: Added the Motivation and Emotion quick start template to the Time Management chapter] # [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Time_and_motivation&action=history 13:44, 4 August 2022: Added the Motivation and Emotion quick start template to the Time and Motivation chapter] # [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Wave_metaphor_for_emotion&oldid=2412979 13:58, 10 August 2022: Added the Motivation and Emotion quick start template to the Wave metaphor for emotion chapter] # [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Suffering_as_emotion&oldid=2412994 14:02, 10 August 2022: Added the Motivation and Emotion quick start template to the Suffering as emotion chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Kindness_motivation&action=history 10:06, 17 August 2022: Added the Motivation and Emotion quick start template to the Kindness motivation chapter] #[https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2022/Kindness_motivation&action=history 10:15, 17 August 2022: Suggested multiple resources that could be used for the Kindness motivation chapter (discussion page)] #[https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2022/Psychological_distress&action=history 19:00, 18 August 2022: Suggested multiple resources that could be used for the Psychological distress chapter (discussion page)] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FPsychological_distress&action=historysubmit&type=revision&diff=2416094&oldid=2413286 19:08, 18 August 2022: Added title and subtitle to Psychological distress chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FAcademic_help-seeking&type=revision&diff=2416096&oldid=2415946 19:12, 18 August 2022: Added title and subtitle to Academic help-seeking chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FCognitive_dissonance_reduction&type=revision&diff=2416439&oldid=2416437 15:00, 19 August 2022: Added the Motivation and Emotion quick start template to the Cognitive dissonance reduction chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FCognitive_dissonance_reduction&type=revision&diff=2416439&oldid=2416437 15:01, 19 August 2022: Added title and subtitle to Cognitive dissonance reduction chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FCreative_arts_and_trauma&type=revision&diff=2416441&oldid=2360767 15:05, 19 August 2022: Added the Motivation and Emotion quick start template to the Creative arts and trauma chapter] jiulagh4c0fg6uba9pwj9m1lrmvwaq2 2416444 2416442 2022-08-19T05:14:31Z U3216256 2942574 /* Social contributions */ wikitext text/x-wiki == About me == Hello everyone, My name is [https://portfolio.canberra.edu.au/view/view.php?t=rufMvX4S27FDnj9lZR85 Ebony] and I am a second year Bachelor of Science in Psychology student at the ''University of Canberra''. I am excited to contribute to the 2022 [[Motivation and emotion/Book/2022|Motivation and Emotion]] Book. As a first-generation University student, I am passionate about education and bringing awareness to issues that first-generation students face. == Hobbies == University related: * [https://clubs.canberra.edu.au/Clubs/CLSS CLSS Treasurer] * UCPS President (2021) Other hobbies: * Knitting/Crocheting * Baking * Reading - Mostly fiction, such as novels by Terry Pratchett, Neil Gaiman, and A. A. Milne ([https://www.goodreads.com/book/show/1333202.The_Red_House_Mystery The Red House Mystery]). == The book chapter I am working on == [[Motivation and emotion/Book/2022/Disappointment|Disappointment: What is disappointment, what causes disappointment, and how can disappointment be managed?]] == Social contributions == # [https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2022/Work_and_flow&diff=2411657&oldid=2411646 13:29, 4 August 2022: Suggested multiple resources that could be used for the Work and Flow chapter] # [https://en.wikiversity.org/w/index.php?title=Motivation+and+emotion%2FBook%2F2022%2FTime+Management&date-range-to=&tagfilter=&action=history 13:39, 4 August 2022: Added the Motivation and Emotion quick start template to the Time Management chapter] # [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Time_and_motivation&action=history 13:44, 4 August 2022: Added the Motivation and Emotion quick start template to the Time and Motivation chapter] # [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Wave_metaphor_for_emotion&oldid=2412979 13:58, 10 August 2022: Added the Motivation and Emotion quick start template to the Wave metaphor for emotion chapter] # [https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Suffering_as_emotion&oldid=2412994 14:02, 10 August 2022: Added the Motivation and Emotion quick start template to the Suffering as emotion chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion/Book/2022/Kindness_motivation&action=history 10:06, 17 August 2022: Added the Motivation and Emotion quick start template to the Kindness motivation chapter] #[https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2022/Kindness_motivation&action=history 10:15, 17 August 2022: Suggested multiple resources that could be used for the Kindness motivation chapter (discussion page)] #[https://en.wikiversity.org/w/index.php?title=Talk:Motivation_and_emotion/Book/2022/Psychological_distress&action=history 19:00, 18 August 2022: Suggested multiple resources that could be used for the Psychological distress chapter (discussion page)] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FPsychological_distress&action=historysubmit&type=revision&diff=2416094&oldid=2413286 19:08, 18 August 2022: Added title and subtitle to Psychological distress chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FAcademic_help-seeking&type=revision&diff=2416096&oldid=2415946 19:12, 18 August 2022: Added title and subtitle to Academic help-seeking chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FCognitive_dissonance_reduction&type=revision&diff=2416439&oldid=2416437 15:00, 19 August 2022: Added the Motivation and Emotion quick start template to the Cognitive dissonance reduction chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FCognitive_dissonance_reduction&type=revision&diff=2416439&oldid=2416437 15:01, 19 August 2022: Added title and subtitle to Cognitive dissonance reduction chapter] #[https://en.wikiversity.org/w/index.php?title=Motivation_and_emotion%2FBook%2F2022%2FCreative_arts_and_trauma&type=revision&diff=2416441&oldid=2360767 15:05, 19 August 2022: Added the Motivation and Emotion quick start template to the Creative arts and trauma chapter] #[https://en.wikiversity.org/w/index.php?title=Talk%3AMotivation_and_emotion%2FBook%2F2022%2FSelf-esteem_and_culture&type=revision&diff=2416443&oldid=2364684 15:13, 19 August 2022: Suggested multiple resources that could be used for the Self-esteem and culture chapter (discussion page)] awwujp9jdqokz2tnso4rfq4auzui3vg 0 285353 2416385 2415803 2022-08-19T01:19:23Z U3213549 2946564 /* Conclusion */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydney. ]] == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al, 1998) {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarising key points - answer focus question; what is courage, what motivates it and what enhances it * Final message 1 - everyone has different levels of courage in different situations - do not be discouraged if you feel limited. * Final message 2 - courage can be enhanced, as shown above so do not panic or feel helpless if you currently have little courage {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] 3iyil2l8d2bcau2tf782v4nosougy30 2416399 2416385 2022-08-19T02:07:57Z U3213549 2946564 /* Conclusion */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydney. ]] == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al, 1998) {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome {{tip}} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] r8lm491bjhxne1mhhnokjpc9gbm53og 2416400 2416399 2022-08-19T02:08:51Z U3213549 2946564 /* Conclusion */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydney. ]] == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al, 1998) {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] 0fkbahw8uemy702ysuucg827fs9hnv4 2416401 2416400 2022-08-19T02:10:27Z U3213549 2946564 /* See also */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydney. ]] == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al, 1998) {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] r8erisay7qw98vg34mk6e23lcqew5fx 2416405 2416401 2022-08-19T02:49:20Z U3213549 2946564 /* The role Emotion plays in hostage negotiation and situations */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydney. ]] == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al, 1998) * Captives are held by subjects who are high emotional. These emotions can consist of jealously, anger, sadness or frustration. () * This can be shown through situations such as: ** One person holding a spouse captive as a result or rejection or conflict. ** a politically motivated or a case of extremism. <br /> {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] lm9el4vt9xptuqoyi5vux9uka0k1i0b 2416406 2416405 2022-08-19T02:50:17Z U3213549 2946564 /* References */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydney. ]] == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al, 1998) * Captives are held by subjects who are high emotional. These emotions can consist of jealously, anger, sadness or frustration. () * This can be shown through situations such as: ** One person holding a spouse captive as a result or rejection or conflict. ** a politically motivated or a case of extremism. <br /> {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz Vecchi, G. M., Van Hasselt, V. B., & Romano, S. J. (2005). Crisis (hostage) negotiation: Current strategies and issues in high-risk conflict resolution. Aggression and Violent Behavior, 10(5), 533-551. https://doi.org/10.1016/j.avb.2004.10.001 }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] lvn8izt01fyvbfhglx9yxcur4zcuz35 2416407 2416406 2022-08-19T03:06:13Z U3213549 2946564 /* The role Emotion plays in hostage negotiation and situations */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydney. ]] == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al., 1998) * Captives are held by subjects who are high emotional. These emotions can consist of jealously, anger, sadness or frustration. (Vecchi et al., 2005) * This can be shown through situations such as: ** One person holding a spouse captive as a result or rejection or conflict. (Vecchi et al., 2005) ** a politically motivated or a case of extremism. (Scott, 2020) <br /> {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz Vecchi, G. M., Van Hasselt, V. B., & Romano, S. J. (2005). Crisis (hostage) negotiation: Current strategies and issues in high-risk conflict resolution. Aggression and Violent Behavior, 10(5), 533-551. https://doi.org/10.1016/j.avb.2004.10.001 }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] 44zes4bqnq36427azgfnd2hmvtcobrg 2416408 2416407 2022-08-19T03:08:19Z U3213549 2946564 /* References */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydney. ]] == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al., 1998) * Captives are held by subjects who are high emotional. These emotions can consist of jealously, anger, sadness or frustration. (Vecchi et al., 2005) * This can be shown through situations such as: ** One person holding a spouse captive as a result or rejection or conflict. (Vecchi et al., 2005) ** a politically motivated or a case of extremism. (Scott, 2020) <br /> {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz Vecchi, G. M., Van Hasselt, V. B., & Romano, S. J. (2005). Crisis (hostage) negotiation: Current strategies and issues in high-risk conflict resolution. Aggression and Violent Behavior, 10(5), 533-551. https://doi.org/10.1016/j.avb.2004.10.001 Scott, R. (2020). The Sydney Lindt café siege: The role of the consultant psychiatrist. Australian & New Zealand Journal of Psychiatry, 54(3), 244-258.https://doi.org/10.1177/0004867419853886 }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] 9uoxhu080si8s09fgr7bsj7wo1norxw 2416409 2416408 2022-08-19T03:09:42Z U3213549 2946564 /* References */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydney. ]] == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al., 1998) * Captives are held by subjects who are high emotional. These emotions can consist of jealously, anger, sadness or frustration. (Vecchi et al., 2005) * This can be shown through situations such as: ** One person holding a spouse captive as a result or rejection or conflict. (Vecchi et al., 2005) ** a politically motivated or a case of extremism. (Scott, 2020) <br /> {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz Vecchi, G. M., Van Hasselt, V. B., & Romano, S. J. (2005). Crisis (hostage) negotiation: Current strategies and issues in high-risk conflict resolution. Aggression and Violent Behavior, 10(5), 533-551. https://doi.org/10.1016/j.avb.2004.10.001 Scott, R. (2020). The Sydney Lindt café siege: The role of the consultant psychiatrist. Australian & New Zealand Journal of Psychiatry, 54(3), 244-258.https://doi.org/10.1177/0004867419853886 }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] exa207k8gannqwxdsebuy9vc5e2izzq 2416410 2416409 2022-08-19T03:12:15Z U3213549 2946564 /* The role Emotion plays in hostage negotiation and situations */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydney. ]] == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al., 1998) * Captives are held by subjects who are high emotional. These emotions can consist of jealously, anger, sadness or frustration. (Vecchi et al., 2005) * This can be shown through situations such as: ** One person holding a spouse captive as a result or rejection or conflict. (Vecchi et al., 2005) ** a politically motivated or a case of extremism. (Scott, 2020) <br /> ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz Vecchi, G. M., Van Hasselt, V. B., & Romano, S. J. (2005). Crisis (hostage) negotiation: Current strategies and issues in high-risk conflict resolution. Aggression and Violent Behavior, 10(5), 533-551. https://doi.org/10.1016/j.avb.2004.10.001 Scott, R. (2020). The Sydney Lindt café siege: The role of the consultant psychiatrist. Australian & New Zealand Journal of Psychiatry, 54(3), 244-258.https://doi.org/10.1177/0004867419853886 }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] tubz81b1akvi3edq0iwjhf9tn4ylmag 2416418 2416410 2022-08-19T03:39:59Z U3213549 2946564 /* The role motivation plays in hostage negotiation and situations */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydn ]] * Hostage situations can have many underlying motivations: ** Money - taking banks hostage or taking hostages in return for ransom money ** Political/extremism - taking the hostage ( ** Situations involving mentally ill individuals - Individuals with illnesses such as but not limited to psychopathy, paranoid schizophrenia, depression can be motivated by their disorders to release anxiety or satisfying a perceived need by taking hostages (Faure, 2003)<br /> == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al., 1998) * Captives are held by subjects who are high emotional. These emotions can consist of jealously, anger, sadness or frustration. (Vecchi et al., 2005) * This can be shown through situations such as: ** One person holding a spouse captive as a result or rejection or conflict. (Vecchi et al., 2005) ** a politically motivated or a case of extremism. (Scott, 2020) <br /> ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz Vecchi, G. M., Van Hasselt, V. B., & Romano, S. J. (2005). Crisis (hostage) negotiation: Current strategies and issues in high-risk conflict resolution. Aggression and Violent Behavior, 10(5), 533-551. https://doi.org/10.1016/j.avb.2004.10.001 Scott, R. (2020). The Sydney Lindt café siege: The role of the consultant psychiatrist. Australian & New Zealand Journal of Psychiatry, 54(3), 244-258.https://doi.org/10.1177/0004867419853886 }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] 7scn9nq880wth1su1eks80noireh310 2416420 2416418 2022-08-19T03:44:57Z U3213549 2946564 /* References */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydn ]] * Hostage situations can have many underlying motivations: ** Money - taking banks hostage or taking hostages in return for ransom money ** Political/extremism - taking the hostage ( ** Situations involving mentally ill individuals - Individuals with illnesses such as but not limited to psychopathy, paranoid schizophrenia, depression can be motivated by their disorders to release anxiety or satisfying a perceived need by taking hostages (Faure, 2003)<br /> == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al., 1998) * Captives are held by subjects who are high emotional. These emotions can consist of jealously, anger, sadness or frustration. (Vecchi et al., 2005) * This can be shown through situations such as: ** One person holding a spouse captive as a result or rejection or conflict. (Vecchi et al., 2005) ** a politically motivated or a case of extremism. (Scott, 2020) <br /> ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz Vecchi, G. M., Van Hasselt, V. B., & Romano, S. J. (2005). Crisis (hostage) negotiation: Current strategies and issues in high-risk conflict resolution. Aggression and Violent Behavior, 10(5), 533-551. https://doi.org/10.1016/j.avb.2004.10.001 Faure, G. O. (2003). Negotiating with terrorists: The hostage case. International Negotiation, 8(3), 469-494. Retrieved from: https://web.archive.org/web/20160108105535id_/https://www.law.upenn.edu/live/files/5178-faurenegotiating-w-terrorist--the-hostage-casepdf Scott, R. (2020). The Sydney Lindt café siege: The role of the consultant psychiatrist. Australian & New Zealand Journal of Psychiatry, 54(3), 244-258.https://doi.org/10.1177/0004867419853886 }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] elftozljyv4q427kuaeha00dw9axmb8 2416421 2416420 2022-08-19T03:45:32Z U3213549 2946564 /* References */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydn ]] * Hostage situations can have many underlying motivations: ** Money - taking banks hostage or taking hostages in return for ransom money ** Political/extremism - taking the hostage ( ** Situations involving mentally ill individuals - Individuals with illnesses such as but not limited to psychopathy, paranoid schizophrenia, depression can be motivated by their disorders to release anxiety or satisfying a perceived need by taking hostages (Faure, 2003)<br /> == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al., 1998) * Captives are held by subjects who are high emotional. These emotions can consist of jealously, anger, sadness or frustration. (Vecchi et al., 2005) * This can be shown through situations such as: ** One person holding a spouse captive as a result or rejection or conflict. (Vecchi et al., 2005) ** a politically motivated or a case of extremism. (Scott, 2020) <br /> ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz Vecchi, G. M., Van Hasselt, V. B., & Romano, S. J. (2005). Crisis (hostage) negotiation: Current strategies and issues in high-risk conflict resolution. Aggression and Violent Behavior, 10(5), 533-551. https://doi.org/10.1016/j.avb.2004.10.001 Scott, R. (2020). The Sydney Lindt café siege: The role of the consultant psychiatrist. Australian & New Zealand Journal of Psychiatry, 54(3), 244-258.https://doi.org/10.1177/0004867419853886 Faure, G. O. (2003). Negotiating with terrorists: The hostage case. International Negotiation, 8(3), 469-494. Retrieved from: https://web.archive.org/web/20160108105535id_/https://www.law.upenn.edu/live/files/5178-faurenegotiating-w-terrorist--the-hostage-casepdf }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] cz70v09td0fvr5zcems2pgdmymqc77e 2416422 2416421 2022-08-19T03:46:16Z U3213549 2946564 /* The role motivation plays in hostage negotiation and situations */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydn ]] * Hostage situations can have many underlying motivations: ** Money - taking banks hostage or taking hostages in return for ransom money ** Political/extremism - taking the hostage (Scott, 2020) ** Situations involving mentally ill individuals - Individuals with illnesses such as but not limited to psychopathy, paranoid schizophrenia, depression can be motivated by their disorders to release anxiety or satisfying a perceived need by taking hostages (Faure, 2003)<br /> == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al., 1998) * Captives are held by subjects who are high emotional. These emotions can consist of jealously, anger, sadness or frustration. (Vecchi et al., 2005) * This can be shown through situations such as: ** One person holding a spouse captive as a result or rejection or conflict. (Vecchi et al., 2005) ** a politically motivated or a case of extremism. (Scott, 2020) <br /> ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz Vecchi, G. M., Van Hasselt, V. B., & Romano, S. J. (2005). Crisis (hostage) negotiation: Current strategies and issues in high-risk conflict resolution. Aggression and Violent Behavior, 10(5), 533-551. https://doi.org/10.1016/j.avb.2004.10.001 Scott, R. (2020). The Sydney Lindt café siege: The role of the consultant psychiatrist. Australian & New Zealand Journal of Psychiatry, 54(3), 244-258.https://doi.org/10.1177/0004867419853886 Faure, G. O. (2003). Negotiating with terrorists: The hostage case. International Negotiation, 8(3), 469-494. Retrieved from: https://web.archive.org/web/20160108105535id_/https://www.law.upenn.edu/live/files/5178-faurenegotiating-w-terrorist--the-hostage-casepdf }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] q80j5z6uhsxn1fnsu241pc6i2cr7ngi 2416426 2416422 2022-08-19T04:07:37Z U3213549 2946564 /* The role motivation plays in hostage negotiation and situations */ wikitext text/x-wiki {{title|Hostage negotiation:<br>What role does motivation and emotion play in hostage negotiation?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is hostage negotiation? * What are the motivations involved in hostage negotiation? * What are the emotions involved in hostage negotiation? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Hostage Negotiation == How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. == The role motivation plays in hostage negotiation and situations == [[File:(1)Lindt Cafe siege 008.jpg|thumb|People signing condolences books after Lindt Café siege, Martin Place, Sydn ]] * Hostage situations can have many underlying motivations: ** Money - taking banks hostage or taking hostages in return for ransom money (Fuselier, 1988) ** Political/extremism - taking the hostage (Scott, 2020) ** Situations involving mentally ill individuals - Individuals with illnesses such as but not limited to psychopathy, paranoid schizophrenia, depression can be motivated by their disorders to release anxiety or satisfying a perceived need by taking hostages (Faure, 2003)<br /> == The role Emotion plays in hostage negotiation and situations == * One of the main goals of crisis/hostage negotiation is to decrease heightened emotions of the perpetrator, and allow for rationality, this is done by using learnt skills and utilising verbal strategies to buy time and intervene (Hatcher et al., 1998) * Captives are held by subjects who are high emotional. These emotions can consist of jealously, anger, sadness or frustration. (Vecchi et al., 2005) * This can be shown through situations such as: ** One person holding a spouse captive as a result or rejection or conflict. (Vecchi et al., 2005) ** a politically motivated or a case of extremism. (Scott, 2020) <br /> ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== * Summarise the key points * Answer the focus questions ** What is hostage negotiation? ** What are the motivations involved in hostage negotiation? ** What are the emotions involved in hostage negotiation? * Final message about topic: As a hostage negotiator, it is important to understand the motivations and emotions involved in these high intensity situations to achieve the greatest outcome ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2014/Anger and motivation|Anger and Motivation]] (Book chapter, 2014) * [[wikipedia:Crisis_negotiation|Crisis negotiation]] (Wikipedia) * [[Motivation and emotion/Book/2019/Hate crime motivation|Hate crime motivation]] (Book chapter, 2019) * [[wikipedia:Hostage|Hostage]] (Wikipedia) * [[wikipedia:Lindt_Cafe_siege|Lindt Cafe siege]] (Wikipedia) ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Hatcher, C., Mohandie, K., Turner, J., & Gelles, M. G. (1998). The role of the psychologist in crisis/hostage negotiations. Behavioral sciences & the law, 16(4), 455-472. https://onlinelibrary.wiley.com/doi/pdf/10.1002/(SICI)1099-0798(199823)16:4%3C455::AID-BSL321%3E3.0.CO;2-G?casa_token=8R8ffIld3P4AAAAA:dyvKlftEioymCBS25ZL5CufY1_yu20k8dFWL_wbxsk8Il4nYjkKCSkhMaCuqQhOrfbu-2zgPvYLWnsjz Vecchi, G. M., Van Hasselt, V. B., & Romano, S. J. (2005). Crisis (hostage) negotiation: Current strategies and issues in high-risk conflict resolution. Aggression and Violent Behavior, 10(5), 533-551. https://doi.org/10.1016/j.avb.2004.10.001 Scott, R. (2020). The Sydney Lindt café siege: The role of the consultant psychiatrist. Australian & New Zealand Journal of Psychiatry, 54(3), 244-258.https://doi.org/10.1177/0004867419853886 Faure, G. O. (2003). Negotiating with terrorists: The hostage case. International Negotiation, 8(3), 469-494. Retrieved from: https://web.archive.org/web/20160108105535id_/https://www.law.upenn.edu/live/files/5178-faurenegotiating-w-terrorist--the-hostage-casepdf Fuselier, G. D. (1988). Hostage negotiation consultant: Emerging role for the clinical psychologist. Professional Psychology: Research and Practice, 19(2), 175. Retrieved from: https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.472.788&rep=rep1&type=pdf }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://www.theguardian.com/australia-news/2014/dec/20/sydney-siege-timeline-how-a-day-and-night-of-terror-unfolded-at-the-lindt-cafe Sydney siege: how a day and night of terror unfolded at the Lindt cafe] (The Guardian) * {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] ni566h3uxvbklv75e560xzcctsz4if8 0 285380 2416155 2415844 2022-08-18T13:24:51Z Young1lim 21186 /* Handling Series of Data */ wikitext text/x-wiki === Introduction === * Overview ([[Media:C01.Intro1.Overview.1.A.20170925.pdf |A.pdf]], [[Media:C01.Intro1.Overview.1.B.20170901.pdf |B.pdf]], [[Media:C01.Intro1.Overview.1.C.20170904.pdf |C.pdf]]) * Number System ([[Media:C01.Intro2.Number.1.A.20171023.pdf |A.pdf]], [[Media:C01.Intro2.Number.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro2.Number.1.C.20170914.pdf |C.pdf]]) * Memory System ([[Media:C01.Intro2.Memory.1.A.20170907.pdf |A.pdf]], [[Media:C01.Intro3.Memory.1.B.20170909.pdf |B.pdf]], [[Media:C01.Intro3.Memory.1.C.20170914.pdf |C.pdf]]) === Handling Repetition === * Control ([[Media:C02.Repeat1.Control.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat1.Control.1.B.20170918.pdf |B.pdf]], [[Media:C02.Repeat1.Control.1.C.20170926.pdf |C.pdf]]) * Loop ([[Media:C02.Repeat2.Loop.1.A.20170925.pdf |A.pdf]], [[Media:C02.Repeat2.Loop.1.B.20170918.pdf |B.pdf]]) === Handling a Big Work === * Function Overview ([[Media:C03.Func1.Overview.1.A.20171030.pdf |A.pdf]], [[Media:C03.Func1.Oerview.1.B.20161022.pdf |B.pdf]]) * Functions & Variables ([[Media:C03.Func2.Variable.1.A.20161222.pdf |A.pdf]], [[Media:C03.Func2.Variable.1.B.20161222.pdf |B.pdf]]) * Functions & Pointers ([[Media:C03.Func3.Pointer.1.A.20161122.pdf |A.pdf]], [[Media:C03.Func3.Pointer.1.B.20161122.pdf |B.pdf]]) * Functions & Recursions ([[Media:C03.Func4.Recursion.1.A.20161214.pdf |A.pdf]], [[Media:C03.Func4.Recursion.1.B.20161214.pdf |B.pdf]]) === Handling Series of Data === ==== Background ==== * Background ([[Media:C04.Series0.Background.1.A.20180727.pdf |A.pdf]]) ==== Basics ==== * Arrays ([[Media:C04.Series1.Array.1.A.20220818.pdf |A.pdf]], [[Media:C04.Series1.Array.1.B.20161115.pdf |B.pdf]]) * Pointers ([[Media:C04.Series2.Pointer.1.A.20180726.pdf |A.pdf]], [[Media:C04.Series2.Pointer.1.B.20161115.pdf |B.pdf]]) * Array Pointers ([[Media:C04.Series3.ArrayPointer.1.A.20220818.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Multi-dimensional Arrays ([[Media:C04.Series4.MultiDim.1.A.20220418.pdf |A.pdf]], [[Media:C04.Series4.MultiDim.1.B.11.pdf |B.pdf]]) * Array Access Methods ([[Media:C04.Series4.ArrayAccess.1.A.20190511.pdf |A.pdf]], [[Media:C04.Series3.ArrayPointer.1.B.20181203.pdf |B.pdf]]) * Structures ([[Media:C04.Series3.Structure.1.A.20171204.pdf |A.pdf]], [[Media:C04.Series2.Structure.1.B.20161130.pdf |B.pdf]]) ==== Applications ==== * Applications of Arrays ([[Media:C04.Series1App.Array.1.A.20220818.pdf |A.pdf]]) * Applications of Pointers ([[Media:C04.Series7.AppPoint.1.A.20200424.pdf |A.pdf]]) * Applications of Array Pointers ([[Media:C04.Series3App.ArrayPointer.1.A.2022024.pdf |A.pdf]]) * Applications of Multi-dimensional Arrays ([[Media:C04.Series4App.MultiDim.1.A.20210719.pdf |A.pdf]]) * Applications of Array Access Methods ([[Media:C04.Series9.AppArrAcess.1.A.20190511.pdf |A.pdf]]) * Applications of Structures ([[Media:C04.Series6.AppStruct.1.A.20190423.pdf |A.pdf]]) ==== Examples ==== * Spreadsheet Example Programs :: Example 1 ([[Media:C04.Series7.Example.1.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.1.C.20171213.pdf |C.pdf]]) :: Example 2 ([[Media:C04.Series7.Example.2.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.2.C.20171213.pdf |C.pdf]]) :: Example 3 ([[Media:C04.Series7.Example.3.A.20171213.pdf |A.pdf]], [[Media:C04.Series7.Example.3.C.20171213.pdf |C.pdf]]) :: Bubble Sort ([[Media:C04.Series7.BubbleSort.1.A.20171211.pdf |A.pdf]]) === Handling Various Kinds of Data === * Types ([[Media:C05.Data1.Type.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data1.Type.1.B.20161212.pdf |B.pdf]]) * Typecasts ([[Media:C05.Data2.TypeCast.1.A.20180217.pdf |A.pdf]], [[Media:C05.Data2.TypeCast.1.B.20161216.pdf |A.pdf]]) * Operators ([[Media:C05.Data3.Operators.1.A.20161219.pdf |A.pdf]], [[Media:C05.Data3.Operators.1.B.20161216.pdf |B.pdf]]) * Files ([[Media:C05.Data4.File.1.A.20161124.pdf |A.pdf]], [[Media:C05.Data4.File.1.B.20161212.pdf |B.pdf]]) === Handling Low Level Operations === * Bitwise Operations ([[Media:BitOp.1.B.20161214.pdf |A.pdf]], [[Media:BitOp.1.B.20161203.pdf |B.pdf]]) * Bit Field ([[Media:BitField.1.A.20161214.pdf |A.pdf]], [[Media:BitField.1.B.20161202.pdf |B.pdf]]) * Union ([[Media:Union.1.A.20161221.pdf |A.pdf]], [[Media:Union.1.B.20161111.pdf |B.pdf]]) * Accessing IO Registers ([[Media:IO.1.A.20141215.pdf |A.pdf]], [[Media:IO.1.B.20161217.pdf |B.pdf]]) === Declarations === * Type Specifiers and Qualifiers ([[Media:C07.Spec1.Type.1.A.20171004.pdf |pdf]]) * Storage Class Specifiers ([[Media:C07.Spec2.Storage.1.A.20171009.pdf |pdf]]) * Scope === Class Notes === * TOC ([[Media:TOC.20171007.pdf |TOC.pdf]]) * Day01 ([[Media:Day01.A.20171007.pdf |A.pdf]], [[Media:Day01.B.20171209.pdf |B.pdf]], [[Media:Day01.C.20171211.pdf |C.pdf]]) ...... Introduction (1) Standard Library * Day02 ([[Media:Day02.A.20171007.pdf |A.pdf]], [[Media:Day02.B.20171209.pdf |B.pdf]], [[Media:Day02.C.20171209.pdf |C.pdf]]) ...... Introduction (2) Basic Elements * Day03 ([[Media:Day03.A.20171007.pdf |A.pdf]], [[Media:Day03.B.20170908.pdf |B.pdf]], [[Media:Day03.C.20171209.pdf |C.pdf]]) ...... Introduction (3) Numbers * Day04 ([[Media:Day04.A.20171007.pdf |A.pdf]], [[Media:Day04.B.20170915.pdf |B.pdf]], [[Media:Day04.C.20171209.pdf |C.pdf]]) ...... Structured Programming (1) Flowcharts * Day05 ([[Media:Day05.A.20171007.pdf |A.pdf]], [[Media:Day05.B.20170915.pdf |B.pdf]], [[Media:Day05.C.20171209.pdf |C.pdf]]) ...... Structured Programming (2) Conditions and Loops * Day06 ([[Media:Day06.A.20171007.pdf |A.pdf]], [[Media:Day06.B.20170923.pdf |B.pdf]], [[Media:Day06.C.20171209.pdf |C.pdf]]) ...... Program Control * Day07 ([[Media:Day07.A.20171007.pdf |A.pdf]], [[Media:Day07.B.20170926.pdf |B.pdf]], [[Media:Day07.C.20171209.pdf |C.pdf]]) ...... Function (1) Definitions * Day08 ([[Media:Day08.A.20171028.pdf |A.pdf]], [[Media:Day08.B.20171016.pdf |B.pdf]], [[Media:Day08.C.20171209.pdf |C.pdf]]) ...... Function (2) Storage Class and Scope * Day09 ([[Media:Day09.A.20171007.pdf |A.pdf]], [[Media:Day09.B.20171017.pdf |B.pdf]], [[Media:Day09.C.20171209.pdf |C.pdf]]) ...... Function (3) Recursion * Day10 ([[Media:Day10.A.20171209.pdf |A.pdf]], [[Media:Day10.B.20171017.pdf |B.pdf]], [[Media:Day10.C.20171209.pdf |C.pdf]]) ...... Arrays (1) Definitions * Day11 ([[Media:Day11.A.20171024.pdf |A.pdf]], [[Media:Day11.B.20171017.pdf |B.pdf]], [[Media:Day11.C.20171212.pdf |C.pdf]]) ...... Arrays (2) Applications * Day12 ([[Media:Day12.A.20171024.pdf |A.pdf]], [[Media:Day12.B.20171020.pdf |B.pdf]], [[Media:Day12.C.20171209.pdf |C.pdf]]) ...... Pointers (1) Definitions * Day13 ([[Media:Day13.A.20171025.pdf |A.pdf]], [[Media:Day13.B.20171024.pdf |B.pdf]], [[Media:Day13.C.20171209.pdf |C.pdf]]) ...... Pointers (2) Applications * Day14 ([[Media:Day14.A.20171226.pdf |A.pdf]], [[Media:Day14.B.20171101.pdf |B.pdf]], [[Media:Day14.C.20171209.pdf |C.pdf]]) ...... C String (1) * Day15 ([[Media:Day15.A.20171209.pdf |A.pdf]], [[Media:Day15.B.20171124.pdf |B.pdf]], [[Media:Day15.C.20171209.pdf |C.pdf]]) ...... C String (2) * Day16 ([[Media:Day16.A.20171208.pdf |A.pdf]], [[Media:Day16.B.20171114.pdf |B.pdf]], [[Media:Day16.C.20171209.pdf |C.pdf]]) ...... C Formatted IO * Day17 ([[Media:Day17.A.20171031.pdf |A.pdf]], [[Media:Day17.B.20171111.pdf |B.pdf]], [[Media:Day17.C.20171209.pdf |C.pdf]]) ...... Structure (1) Definitions * Day18 ([[Media:Day18.A.20171206.pdf |A.pdf]], [[Media:Day18.B.20171128.pdf |B.pdf]], [[Media:Day18.C.20171212.pdf |C.pdf]]) ...... Structure (2) Applications * Day19 ([[Media:Day19.A.20171205.pdf |A.pdf]], [[Media:Day19.B.20171121.pdf |B.pdf]], [[Media:Day19.C.20171209.pdf |C.pdf]]) ...... Union, Bitwise Operators, Enum * Day20 ([[Media:Day20.A.20171205.pdf |A.pdf]], [[Media:Day20.B.20171201.pdf |B.pdf]], [[Media:Day20.C.20171212.pdf |C.pdf]]) ...... Linked List * Day21 ([[Media:Day21.A.20171206.pdf |A.pdf]], [[Media:Day21.B.20171208.pdf |B.pdf]], [[Media:Day21.C.20171212.pdf |C.pdf]]) ...... File Processing * Day22 ([[Media:Day22.A.20171212.pdf |A.pdf]], [[Media:Day22.B.20171213.pdf |B.pdf]], [[Media:Day22.C.20171212.pdf |C.pdf]]) ...... Preprocessing <!----------------------------------------------------------------------> </br> See also https://cprogramex.wordpress.com/ == '''Old Materials '''== until 201201 * Intro.Overview.1.A ([[Media:C.Intro.Overview.1.A.20120107.pdf |pdf]]) * Intro.Memory.1.A ([[Media:C.Intro.Memory.1.A.20120107.pdf |pdf]]) * Intro.Number.1.A ([[Media:C.Intro.Number.1.A.20120107.pdf |pdf]]) * Repeat.Control.1.A ([[Media:C.Repeat.Control.1.A.20120109.pdf |pdf]]) * Repeat.Loop.1.A ([[Media:C.Repeat.Loop.1.A.20120113.pdf |pdf]]) * Work.Function.1.A ([[Media:C.Work.Function.1.A.20120117.pdf |pdf]]) * Work.Scope.1.A ([[Media:C.Work.Scope.1.A.20120117.pdf |pdf]]) * Series.Array.1.A ([[Media:Series.Array.1.A.20110718.pdf |pdf]]) * Series.Pointer.1.A ([[Media:Series.Pointer.1.A.20110719.pdf |pdf]]) * Series.Structure.1.A ([[Media:Series.Structure.1.A.20110805.pdf |pdf]]) * Data.Type.1.A ([[Media:C05.Data2.TypeCast.1.A.20130813.pdf |pdf]]) * Data.TypeCast.1.A ([[Media:Data.TypeCast.1.A.pdf |pdf]]) * Data.Operators.1.A ([[Media:Data.Operators.1.A.20110712.pdf |pdf]]) <br> until 201107 * Intro.1.A ([[Media:Intro.1.A.pdf |pdf]]) * Control.1.A ([[Media:Control.1.A.20110706.pdf |pdf]]) * Iteration.1.A ([[Media:Iteration.1.A.pdf |pdf]]) * Function.1.A ([[Media:Function.1.A.20110705.pdf |pdf]]) * Variable.1.A ([[Media:Variable.1.A.20110708.pdf |pdf]]) * Operators.1.A ([[Media:Operators.1.A.20110712.pdf |pdf]]) * Pointer.1.A ([[Media:Pointer.1.A.pdf |pdf]]) * Pointer.2.A ([[Media:Pointer.2.A.pdf |pdf]]) * Array.1.A ([[Media:Array.1.A.pdf |pdf]]) * Type.1.A ([[Media:Type.1.A.pdf |pdf]]) * Structure.1.A ([[Media:Structure.1.A.pdf |pdf]]) go to [ [[C programming in plain view]] ] [[Category:C programming]] </br> 5uab330vxlg1llcnq6fj94nrpb80wqu 0 285385 2416353 2415736 2022-08-18T23:18:39Z Young1lim 21186 /* Object Files */ wikitext text/x-wiki === Executable and Linkable Format === ==== Object Files ==== * Introduction * ELF Header ([[Media:ELF1.1B.Header.20220211.pdf |pdf]]) * Group section ([[Media:ELF1.1C.Group.20220426.pdf |pdf]]) * String table section ([[Media:ELF1.1D.StringTbl.20220427.pdf |pdf]]) * Weak and common symbols ([[Media:ELF1.1E.WeakComm.20220817.pdf |pdf]]) * Symbol table section ([[Media:ELF1.1F.SymbolTbl.20220722.pdf |pdf]]) * Special Sections ([[Media:ELF1.7B.Section.20200511.pdf |B.pdf]]) * Relocation ([[Media:ELF1.6A.Relocation.20190413.pdf |A.pdf]]) ==== Program Loading and Dynamic Linking ==== * Introduction * Program Header ([[Media:ELF1.2B.ProgHeader.20220110.pdf |pdf]]) * Program Loading * Dynamic Linking ([[Media:ELF2.4A.DynLinking.20191028.pdf |pdf]]) ==== C Library ==== * C Library === ELF Study === ==== ELF Relocations ==== * Linking ([[Media:ELF1.7A.Linking.20200731.pdf |A.pdf]]) * Loading ([[Media:ELF1.7B.Loading.20201103.pdf |B.pdf]]) * Executing ([[Media:ELF1.7C.Executing.20201221.pdf |C.pdf]]) * Virtual Memory ([[Media:ELF2.1D.VMemory.20211227.pdf |D.pdf]]) * PIC Method ([[Media:ELF1.7B.PICMethod.20200417.pdf |C.pdf]]) * Design Cycles ([[Media:ELF1.7C.DesignCycle.20200317.pdf |D.pdf]]) * Relocs in i386 ([[Media:ELF1.7D.Reloc386.20200413.pdf |E.pdf]]) ==== Relocation Examples ==== * Relocs example introduction ([[Media:ELF1.7Ex.1Intro.20200109.pdf |E1.pdf]]) * Relocs in an object for a library ([[Media:ELF1.7Ex.2ObjectRel.20200319.pdf |E2.pdf]]) * Relocs in an object for an executable ([[Media:ELF1.7Ex.3ObjectMain.20200118.pdf |E3.pdf]]) * Relocs in a library ([[Media:ELF1.7Ex.4Library.20200320.pdf |E4.pdf]]) * Relocs in an executable ([[Media:ELF1.7Ex.5Executable.20200228.pdf |E5.pdf]]) * Result Summary ([[Media:ELF1.7Ex.6Result.20200121.pdf |E6.pdf]]) * Symbol Table Listing ([[Media:ELF1.7Ex.7Symbol.20200120.pdf |E7.pdf]]) * Relocs Listing ([[Media:ELF1.7Ex.8Relocs.20200121.pdf |E8.pdf]]) * Assembly Listing ([[Media:ELF1.7Ex.9Assembly.20200128.pdf |E9.pdf]]) * Reloc Experiments ([[Media:ELF1.7F.Experiments.20191206.pdf |F.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming]] aw2sbsmucxy0kprcc0orqtgjc3dye6l 2416357 2416353 2022-08-18T23:20:26Z Young1lim 21186 /* Object Files */ wikitext text/x-wiki === Executable and Linkable Format === ==== Object Files ==== * Introduction * ELF Header ([[Media:ELF1.1B.Header.20220211.pdf |pdf]]) * Group section ([[Media:ELF1.1C.Group.20220426.pdf |pdf]]) * String table section ([[Media:ELF1.1D.StringTbl.20220427.pdf |pdf]]) * Weak and common symbols ([[Media:ELF1.1E.WeakComm.20220818.pdf |pdf]]) * Symbol table section ([[Media:ELF1.1F.SymbolTbl.20220722.pdf |pdf]]) * Special Sections ([[Media:ELF1.7B.Section.20200511.pdf |B.pdf]]) * Relocation ([[Media:ELF1.6A.Relocation.20190413.pdf |A.pdf]]) ==== Program Loading and Dynamic Linking ==== * Introduction * Program Header ([[Media:ELF1.2B.ProgHeader.20220110.pdf |pdf]]) * Program Loading * Dynamic Linking ([[Media:ELF2.4A.DynLinking.20191028.pdf |pdf]]) ==== C Library ==== * C Library === ELF Study === ==== ELF Relocations ==== * Linking ([[Media:ELF1.7A.Linking.20200731.pdf |A.pdf]]) * Loading ([[Media:ELF1.7B.Loading.20201103.pdf |B.pdf]]) * Executing ([[Media:ELF1.7C.Executing.20201221.pdf |C.pdf]]) * Virtual Memory ([[Media:ELF2.1D.VMemory.20211227.pdf |D.pdf]]) * PIC Method ([[Media:ELF1.7B.PICMethod.20200417.pdf |C.pdf]]) * Design Cycles ([[Media:ELF1.7C.DesignCycle.20200317.pdf |D.pdf]]) * Relocs in i386 ([[Media:ELF1.7D.Reloc386.20200413.pdf |E.pdf]]) ==== Relocation Examples ==== * Relocs example introduction ([[Media:ELF1.7Ex.1Intro.20200109.pdf |E1.pdf]]) * Relocs in an object for a library ([[Media:ELF1.7Ex.2ObjectRel.20200319.pdf |E2.pdf]]) * Relocs in an object for an executable ([[Media:ELF1.7Ex.3ObjectMain.20200118.pdf |E3.pdf]]) * Relocs in a library ([[Media:ELF1.7Ex.4Library.20200320.pdf |E4.pdf]]) * Relocs in an executable ([[Media:ELF1.7Ex.5Executable.20200228.pdf |E5.pdf]]) * Result Summary ([[Media:ELF1.7Ex.6Result.20200121.pdf |E6.pdf]]) * Symbol Table Listing ([[Media:ELF1.7Ex.7Symbol.20200120.pdf |E7.pdf]]) * Relocs Listing ([[Media:ELF1.7Ex.8Relocs.20200121.pdf |E8.pdf]]) * Assembly Listing ([[Media:ELF1.7Ex.9Assembly.20200128.pdf |E9.pdf]]) * Reloc Experiments ([[Media:ELF1.7F.Experiments.20191206.pdf |F.pdf]]) </br> go to [ [[C programming in plain view]] ] [[Category:C programming]] sl4mzfz6xudvhzbnrnaqopa7vn9n1f9 0 285546 2416330 2414940 2022-08-18T22:40:20Z Eyoungstrom 1933979 suggested bulleting wikitext text/x-wiki == Goals == ''Did you meet your goals? Are you happy with how the project went?''<br>We did meet our goals and are very happy with how the project went. Specifically: * We were able to create a template for conferences to use, linked [[Template:Conferences|here]]. * Updated all of the previous Future Directions Forum pages, seen [[Journal of Clinical Child and Adolescent Psychology Future Directions Forum (JCCAP FDF)|here]]. * We were able to create Wikipedia pages for some of the previous presenters of the Forum with a focus on women and other minority groups. By starting this, we have decided to try and continue this project moving forward. * We have also recruited several new Wiki editors and trained those individuals how to use our template and how to edit Wiki pages. * Many of the current editors working on the project have increased their skills on Wiki platforms and have attended workshops on transclusion and creating templates in order to increase our ability to effectively update pages and create the template for conferences to use. == Outcome == ''Please report on your original project targets. Please be sure to '''[[Grants:Metrics#Three_shared_metrics|review and provide metrics required for Rapid Grants]]''''' {| class="wikitable" |+ ! !Target outcome !Achived outcome !Explanation |- |1 |Host 2 edit-a-thons with HGAPS team members from both UNC and UMD |Hosted 2 edit-a-thons |Both edit-a-thons were successful. We had HGAPS members come from both UNC and UMD as well as UCLA. We also had individuals attend the edit-a-thons who are not a part of HGAPS. |- |2 |15+ contributors and attendees per edit-a-thon |10 contributors per edit-a-thon |Although we were not able to get 15+ attendees at our edit-a-thons, we were able to get 10 at both events. We had many more individuals sign up for the edit-a-thons than the number that actually attended. |- |3 |5 new editors |10 new editors |We had more than 5 new editors and ended with about 10 new editors who have not edited Wiki before. |- |4 |1 new article created, 20+ articles improved |4 new pages created and 25 pages updated |We were successful on this target and created 4 new pages ([[wikipedia:Anna_Van_Meter|Anna Van Meter]], [[wikipedia:Andres_De_Los_Reyes|Andres De Los Reyes]], Jessica Schleider, and Bruce Chorpita) related to keynote speakers and improved many existing articles with around 25 total. |- |5 |increase views within 6 months of launch |Views have increased |We have had an increase in views on FDF related pages since updating the pages and creating new ones. |} == Learning == ''Projects do not always go according to plan. Sharing what you learned can help you and others plan similar projects in the future. Help the movement learn from your experience by answering the following questions:'' *What worked well? #We made page templates on Wiki using transclusion for every page we created or improved. This was very helpful as it guided new editors and created a professional, organized, and visually continuous layout for every page we edited, even though most of our work was done collaboratively. This practice also helped to ensure that necessary and beneficial information was present on every page we edited. #We also made a checklist on Canva, summarizing key aspects to include living biography pages. This checklist provided newer Wiki editors with a framework off of which they could build their pages, and ensured that necessary elements were not left off. [[File:996 Edit-a-thon Page Checklist.jpg|thumb|996 Edit-a-thon Checklist]] The checklist was presented in conjunction with an example living biography page ([[wikipedia:Steven_Pinker|Steven Pinker]]) during the introduction of each Edit-A-Thon and was linked on collaborative editing documents, which served to further demonstrate our expectations to newer editors and ensure that quality pages were created. #Our team met regularly (at least once a week) via video call throughout the entirety of this project. Between meetings, we used a professional instant messaging platform to communicate thoughts, questions, ideas, and reminders both within our team and to other members of our larger organization. Having a clearly defined and hierarchical communication system that was still adaptable enough to meet the needs of our team (working internationally) allowed us to collaborate efficiently, solve problems as they arose, avoid miscommunication, and make the most of our time. #We did the majority of our planning work in wiki sandboxes and separate rough draft documents, instead of working/brainstorming directly on the publicly visible wiki pages themselves. This allowed us to ensure that everything we put on Wiki was accurate, well thought out, and organized. The use of collaborative documents for brainstorming outside of Wiki also served to streamline the teamwork process, especially since a significant number of our editors did not have prior experience editing Wiki. #We created documents with lists of web links for use on specific pages in advance of each Edit-A-Thon. Since these link lists were created by experienced team members, they allowed us to ensure that the information editors used to build pages was accurate and extensive. Additionally, this process allowed editors to use their time most efficiently during the Edit-A-Thon, as they could focus on brainstorming and editing, instead of having to search for and vet links. #Including prizes, gift cards to virtually by lunch for editors, and designated time at the end of the Edit-A-Thon to collaboratively review pages helped not only to gather interest in the events we hosted, but also to incentivize editors to stay focused and productive over the two hour time frame. [[File:996 Edit-a-Thon Schedule.jpg|thumb|996 Edit-a-Thon Schedule]] #Establishing a detailed time frame and plan for each Edit-A-Thon helped us avoid administrative errors during events and allowed us to plan how much we should expect participants to accomplish. This practice also made sure we used our time provided efficiently, and made sure that we were able to fulfill our goals at every Edit-A-Thon. #Providing new editors with Wiki training links in advance of the Edit-A-Thon, as well as with editing workshops during the first few minutes of every event helped us to ensure that new editors learned from this experience and were able to make valuable contributions to their teams. More advanced editing workshops were also provided to individuals with more experience, allowing people of all levels of Wiki editing skills to benefit from this experience and feel encouraged to continue to build their Wiki editing skills in the future. #With the help of the HGAPS Social Media team, we created Google Forms that participants were able to fill out to sign up for Edit-A-Thons. This way, our team was able to know how many editors to expect at each event, as well as to know how many new versus experienced editors planned to show up. These forms were created and advertised during HGAPS meetings and on social media well in advance of each event, in order to maximize our potential audience and gauge interest. #Additionally, we used Google Forms to employ a page rating system that allowed teams to both give and receive feedback on their work, further refining editing skills and serving as an additional motivator for focused work. *What did not work so well? #While we still were able to recruit a significant number of both new and experienced editors to participate in our Edit-A-Thons, we did not manage to achieve the participant turnout rate we initially hoped for. This led to minor complications during incentive allocation, wherein we had to modify our initial plans for how incentives would be given out. #Our team ended up with more clean-up work at the end of each Edit-A-Thon than we initially anticipated. This additional work was partly due to the lower than target turnout rate at Edit-A-Thons, but was able to be accounted for when our finances were re-examined (we reallocated incentive funds to wages in order to pay individuals to clean up the pages). #We had a good turnout in terms of RSVP response rate to the sign-up and informational forms we advertised, but we struggled with getting everyone who RSVP'd to attend. This was largely anticipated, but does provide room for improvement if/when we complete similar projects in the future. *What would you do differently next time? #Due to the aforementioned challenges, we have discussed strengthening our advertising effort for Edit-A-Thons in the future. While we did post informational infographics in advance of the Edit-A-Thons on social media, and dedicated time in our HGAPS weekly meetings, we realized that it would likely be beneficial to start the process of advertising the Edit-A-Thons earlier than we did, and post subsequent reminders as the events got closer (as opposed to just having 1 or 2 social media posts). #We also considered that it might be a good idea to include more information about the specific goals of future HGAPS Edit-A-Thons in our social media advertisements. This practice would help garner interest in specific topics, drawing in an increased number of dedicated editors from outside HGAPS UNC. A specific idea we had for implementing this change is to advertise the Edit-A-Thons by way of infographic, as opposed to simply listing the title and time of the event. #As a significant amount of time ended up being spent coordinating bringing in outside trainers for the edit-a-thons, we discussed having experienced Wiki editors prerecorded training sessions. This would allow us to better plan for the time that training would take, and decrease the time and resources spent on recruiting trainers that had to present live talks. If we used prerecorded training sessions we would still recruit experienced editors to be present at the events to answer any questions that may arise, but would reduce time spent during the event as we could send out the recording in advance for participants to watch. #Overall, our team was able to learn a lot during the organization and implementation of this project, and we hope to use what we have learned to scale up this project in the future. Next time, we strive to further perfect our process in order to recruit more new editors, involve an even greater audience from outside of HGAPS, and create/update more pages in order to continue to disseminate critical psychological research by way of sustained Wiki editing. == Finances == ===Grant funds spent=== ''Please describe how much grant money you spent for approved expenses, and tell us what you spent it on.'' * '''Wages for template/tool-kit creators''': $2,040 ($20 per hour x 3 creators x 34 hours) ** The remaining $2589.73 was spent here for our creators. * '''Wages for trainers''': $720 ($20 per hour x 3 trainers x 12 hours) ** We spent the anticipated $720 here on trainers. * '''Gift cards to virtually buy lunch for edit-a-thon participants''': $600 ($20 for 15 participants x 2 events) ** We spent $260 on gift cards to buy lunch for edit-a-thon participants. Not everyone that attended wanted one, which led to us being under budget here. We have reallocated these remaining funds ($340) to "wages for template/tool-kit creators". * '''Incentives for sustained editing''': $500 (1st prize $100, 2nd prize $50, and 3rd prize $25 at 1 and 3 month follow-ups after last edit-a-thon = $175 x 4 "contests") ** Upon organizing the edit-a-thons, we decided to change our method for prizes. Since we organized into teams we decided to have participants evaluate each teams work at the end of each edit-a-thon. We used these evaluations to come up with a first place and second place team and these teams were given prizes. We spent $350 in this category and reallocated these remaining funds ($150) to "wages for template/tool-kit creators". * '''HGAPS merch for milestones completed in editor training''': $300 ** We spent $248.27 on HGAPS merch for editors. The remaining funds ($51.73) were reallocated to "wages for template/tool-kit creators". * '''Fiscal sponsor administrative fees (including access to Google Suites for Nonprofits platform and analytics)''': $832 (20% final budget) ** We spent the anticipated $832 for fiscal sponsor administrative fees. '''Total''': USD $4,992 ===Remaining funds=== ''Do you have any remaining grant funds?'' No, we don't have any remaining grant funds left after the completion of the project. <!--Let us know if you would like to use the remaining funds on a similar or new project. Remember, a grants officer must approve this request before you spend the money.--> ==Anything else== ''Anything else you want to share about your project?'' [[Category:Project/Rapid/Report]] jv74b7jq57mqdls6a80no1m660ls9vo 0 285677 2416331 2414435 2022-08-18T22:40:40Z ThaniosAkro 2805358 /* Links to related topics */ wikitext text/x-wiki =Congruences= The subject of congruences is a field of mathematics that covers the integers, their relationship to each other and also the effect of arithmetic operations on their relationship to each other. Expressed mathematically: :<math>A \equiv B \pmod{N}</math> read as: A is congruent with B modulo N. {{RoundBoxTop|theme=2}} <math>A,B,N</math> are integers and <math>N > 1.</math> {{RoundBoxBottom}} This means that: * A modulo N equals B modulo N, * the difference, A-B, is exactly divisible by N, or * <math>A-B = K\cdot N.</math> where p modulo N or <code>p % N</code> is the remainder when p is divided by N. For example: <math>23 \equiv 8 \pmod{5}</math> because division <math>\frac{23-8}{5}</math> is exact without remainder, or <math>5\mid (23-8).</math> Similarly, <math>39 \not\equiv 29 \,\pmod{7}</math> because division <math>\frac{39-29}{7}</math> is not exact, or <math>7\nmid (39-29).</math> ==Law of addition== ===Adding a constant=== {{RoundBoxTop|theme=2}} If <math>A \equiv B \pmod{N}, </math> then: <math>A+q \equiv B+q \pmod{N}.</math> Proof: <math>A-B = K\cdot N</math>, therefore <math>A = B + K\cdot N.</math> <math>(A+q) - (B+q) = B + K\cdot N + q - B - q = K\cdot N</math> which is exactly divisible by N. {{RoundBoxBottom}} ===Adding 2 congruences=== {{RoundBoxTop|theme=2}} If <math>A \equiv B \pmod{N}, </math> and <math>C \equiv D \pmod{N}, </math> then: <math>A+C \equiv B+D \pmod{N}.</math> Proof: <math>A-B = K_1\cdot N</math>, therefore <math>A = B + K_1\cdot N</math> and <math>C = D + K_2\cdot N</math> <math>(A+C) - (B+D)</math> <math>= B + K_1\cdot N + D + K_2\cdot N - B - D</math> <math>= N(K_1 + K_2)</math> which is exactly divisible by N. {{RoundBoxBottom}} ==Law of Common Congruence== {{RoundBoxTop|theme=2}} If <math>A \equiv B \pmod{N} </math> and <math>C \equiv B \pmod{N},</math> then: <math>A \equiv C \pmod{N}.</math> Proof: <math>A = B + K_1\cdot N</math> and <math>C = B + K_2\cdot N.</math> <math>A - C = B + K_1\cdot N - B - K_2\cdot N = (K_1 - K_2)N</math> which is exactly divisible by N. {{RoundBoxBottom}} ==Law of Multiplication== ===by a constant=== {{RoundBoxTop|theme=2}} If <math>A \equiv B \pmod{N} </math> then: <math>A\cdot p \equiv B\cdot p \pmod{N}.</math> Proof: <math>A\cdot p - B\cdot p = p(A-B)</math> which is exactly divisible by N. {{RoundBoxBottom}} ===by another congruence=== {{RoundBoxTop|theme=2}} If <math>A \equiv B \pmod{N} </math> and <math>C \equiv D \pmod{N},</math> then: <math>A\cdot C \equiv B\cdot D \pmod{N}.</math> Proof: <math>A = B + K_1\cdot N</math> and <math>C = D + K_2\cdot N.</math> <math>A\cdot C - B\cdot D</math> <math>= (B + K_1\cdot N)( D + K_2\cdot N) - B\cdot D</math> <math>= B\cdot D + B\cdot K_2\cdot N + K_1\cdot N\cdot D + K_1\cdot N\cdot K_2\cdot N - B\cdot D</math> <math>= N( B\cdot K_2 + K_1\cdot D + K_1\cdot K_2\cdot N )</math> which is exactly divisible by N. {{RoundBoxBottom}} ==Law of squares== {{RoundBoxTop|theme=2}} If <math>A \equiv B \pmod{N} </math> then: <math>A^2 \equiv B^2 \pmod{N}.</math> Proof: <math>A^2 - B^2 = (A+B)(A-B)</math> which is exactly divisible by N. {{RoundBoxBottom}} ==Law of Division?== {{RoundBoxTop|theme=2}} A simple example shows that a "law of division" does not exist. <math>24 \equiv 14 \pmod{10}.</math> However <math>\frac{24}{2} \not\equiv \frac{14}{2} \pmod{10}</math> Because <math>12 - 7 = 5</math> is not exactly divisible by <math>10</math> {{RoundBoxBottom}} =Quadratic Congruences= ==Introduction== A quadratic congruence is a congruence that contains at least one exact square, for example: <math>x^2 \equiv y \pmod{N}</math> or <math>x^2 \equiv y^2 \pmod{N}.</math> Initially, let us consider the congruence: <math>x^2 \equiv y \pmod{N}.</math> If <math>y = x^2 - N,</math> then: <math>x^2 \equiv y \pmod{N}.</math> Proof: <math>x^2 - y = x^2 - (x^2 - N) = N</math> which is exactly divisible by <math>N.</math> Consider an example with real numbers. Let <math>N = 257</math> and <math>26 \ge x \ge 6.</math> <syntaxhighlight> N = 257 </syntaxhighlight> {| class="wikitable" |- ! <math>x</math> !! <math>x^2 - N</math> |- | <code></code><code>6</code> || <code>-221</code> |- | <code></code><code>7</code> || <code>-208</code> |- | <code></code><code>8</code> || <code>-193</code> |- | <code></code><code>9</code> || <code>-176</code> |- | <code>10</code> || <code>-157</code> |- | <code>11</code> || <code>-136</code> |- | <code>12</code> || <code>-113</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>14</code> || <code></code><code>-61</code> |- | <code>15</code> || <code></code><code>-32</code> |- | <code>16</code> || <code></code><code></code><code>-1</code> |- | <code>17</code> || <code></code><code></code><code>32</code> |- | <code>18</code> || <code></code><code></code><code>67</code> |- | <code>19</code> || <code></code><code>104</code> |- | <code>20</code> || <code></code><code>143</code> |- | <code>21</code> || <code></code><code>184</code> |- | <code>22</code> || <code></code><code>227</code> |- | <code>23</code> || <code></code><code>272</code> |- | <code>24</code> || <code></code><code>319</code> |- | <code>25</code> || <code></code><code>368</code> |- | <code>26</code> || <code></code><code>419</code> |} A cursory glance at the values of <math>x^2 - N</math> indicates that the value <math>x^2 - N</math> is never divisible by <math>5.</math> Proof: <math>N \equiv 2 \pmod{5}</math> therefore <math>N - 2 = k5</math> or <math>N = 5k + 2.</math> The table shows all possible values of <math>x\ %\ 5</math> and <math>y\ %\ 5:</math> {| class="wikitable" |- !<math>x</math> !! <math>x^2</math> !! <math>y = x^2 - N</math> |- | <math>5p+0</math>||<math>25p^2+\ \ 0p+\ \ 0</math>||<math>25p^2+\ \ 0p+\ \ 0\ \ -\ \ (5k+2)\ \ =\ \ 25p^2+\ \ 0p\ \ -\ \ 5k-\ \ 2</math> |- | <math>5p+1</math>||<math>25p^2+10p+\ \ 1</math>||<math>25p^2+10p+\ \ 1\ \ -\ \ (5k+2)\ \ =\ \ 25p^2+10p\ \ -\ \ 5k-\ \ 1</math> |- | <math>5p+2</math>||<math>25p^2+20p+\ \ 4</math>||<math>25p^2+20p+\ \ 4\ \ -\ \ (5k+2)\ \ =\ \ 25p^2+20p\ \ -\ \ 5k+\ \ 2</math> |- | <math>5p+3</math>||<math>25p^2+30p+\ \ 9</math>||<math>25p^2+30p+\ \ 9\ \ -\ \ (5k+2)\ \ =\ \ 25p^2+30p\ \ -\ \ 5k+\ \ 7</math> |- | <math>5p+4</math>||<math>25p^2+40p+16</math>||<math>25p^2+40p+16\ \ -\ \ (5k+2)\ \ =\ \ 25p^2+40p\ \ -\ \ 5k+14</math> |} As you can see, the value <math>y = x^2 - N</math> is never exactly divisible by <math>5.</math> If you look closely, you will see also that it is never exactly divisible by <math>3</math> or <math>7.</math> However, you can see at least one value of <math>y</math> exactly divisible by <math>11</math> and at least one value of <math>y</math> exactly divisible by <math>13.</math> The table shows all possible values of <math>x\ %\ 11</math> and <math>y\ %\ 11:</math> {| class="wikitable" |- !<math>x</math> !! <math>x^2</math> !! <math>y= x^2 - N</math> |- | <math>11p+\ \ 0</math>||<math>121p^2+\ \ </math><math>\ \ 0p+\ \ </math><math>\ \ 0</math>||<math>121p^2+\ \ </math><math>\ \ 0p-11k-\ \ 4\ \ =\ \ 121p^2+\ \ </math><math>\ \ 0p-11(k+1)+\ \ 7\ \ </math><math>\ \ </math> |- | <math>11p+\ \ 1</math>||<math>121p^2+\ \ 22p+\ \ </math><math>\ \ 1</math>||<math>121p^2+\ \ 22p-11k-\ \ 3\ \ =\ \ 121p^2+\ \ 22p-11(k+1)+\ \ 8\ \ </math><math>\ \ </math> |- | <math>11p+\ \ 2</math>||<math>121p^2+\ \ 44p+\ \ </math><math>\ \ 4</math>||<math>121p^2+\ \ 20p-11k+\ \ 0\ \ =\ \ 121p^2+\ \ 20p-11k+\ \ 0\ \ </math><math>\ \ </math><math>\ \ </math><math>\ \ </math><math>\ \ *</math> |- | <math>11p+\ \ 3</math>||<math>121p^2+\ \ 66p+\ \ </math><math>\ \ 9</math>||<math>121p^2+\ \ 66p-11k+\ \ 5\ \ =\ \ 121p^2+\ \ 66p-11k+\ \ 5\ \ </math><math>\ \ </math><math>\ \ </math><math>\ \ </math><math>\ \ </math><math>\ \ </math> |- | <math>11p+\ \ 4</math>||<math>121p^2+\ \ 88p+\ \ 16</math>||<math>121p^2+\ \ 88p-11k+12\ \ =\ \ 121p^2+\ \ 88p-11(k-1)+\ \ 1\ \ </math><math>\ \ </math> |- | <math>11p+\ \ 5</math>||<math>121p^2+110p+\ \ 25</math>||<math>121p^2+110p-11k+21\ \ =\ \ 121p^2+110p-11(k-1)+10\ \ </math><math>\ \ </math> |- | <math>11p+\ \ 6</math>||<math>121p^2+132p+\ \ 36</math>||<math>121p^2+132p-11k+32\ \ =\ \ 121p^2+132p-11(k-2)+10\ \ </math><math>\ \ </math> |- | <math>11p+\ \ 7</math>||<math>121p^2+154p+\ \ 49</math>||<math>121p^2+154p-11k+45\ \ =\ \ 121p^2+154p-11(k-4)+\ \ 1\ \ </math><math>\ \ </math> |- | <math>11p+\ \ 8</math>||<math>121p^2+176p+\ \ 64</math>||<math>121p^2+176p-11k+60\ \ =\ \ 121p^2+176p-11(k-5)+\ \ 5\ \ </math><math>\ \ </math> |- | <math>11p+\ \ 9</math>||<math>121p^2+198p+\ \ 81</math>||<math>121p^2+198p-11k+77\ \ =\ \ 121p^2+198p-11(k-7)+\ \ 0\ \ *</math> |- | <math>11p+10</math>||<math>121p^2+220p+100</math>||<math>121p^2+220p-11k+96\ \ =\ \ 121p^2+220p-11(k-8)+\ \ 8\ \ </math><math>\ \ </math> |} The two lines marked by an <math>*</math> show values of <math>y</math> exactly divisible by <math>11.</math> The two values of <math>x,</math> <math>11p+2</math> and <math>11p+9,</math> or <math>11p \pm 2</math> are solutions of the congruence. Why are values of <math>y</math> divisible by some primes and not divisible by other primes? An interesting question that leads us to the topic of quadratic residues. ==Quadratic Residues== Consider all the congruences for prime number <math>5:</math> <math>x^2 \equiv y \pmod{5}</math> for <math>5 > x \ge 0.</math> {| class="wikitable" |- ! <math>x</math> !! <math>x^2</math> !! <math>(x^2)\ %\ 5</math> |- | <code>0</code> || <code></code><code>0</code> || <code>0</code> |- | <code>1</code> || <code></code><code>1</code> || <code>1</code> |- | <code>2</code> || <code></code><code>4</code> || <code>4</code> |- | <code>3</code> || <code></code><code>9</code> || <code>4</code> |- | <code>4</code> || <code>16</code> || <code>1</code> |} Quadratic residues of <math>5</math> are <math>0,1,4.</math> Values <math>2,3</math> are not quadratic residues of <math>5.</math> These values are quadratic non-residues. To calculate the quadratic residues of a small prime <math>p:</math> <syntaxhighlight lang=python> # python code: def quadResidues(p) : L1 = [] for v in range (p>>1, -1, -1) : L1 += [(v*v) % p] return L1 print (quadResidues(11)) </syntaxhighlight> <syntaxhighlight> [3, 5, 9, 4, 1, 0] </syntaxhighlight> Quadratic residues of <math>11</math> are <math>0,1,3,4,5,9.</math> The method presented here answers the question, "What are the quadratic residues of p?" If <math>p</math> is a very large prime, the question is often, "Is r a quadratic residue of p?" The answer is found in advanced number theory. Let us return to quadratic residues mod <math>N = 257.</math> <math>N\ %\ 5 = 2,</math> therefore <math>N</math> is not a quadratic residue of <math>5.</math> This is why <math>x^2 - N</math> is never divisible by <math>5</math> exactly. <math>N\ %\ 11 = 4,</math> therefore <math>N</math> is a quadratic residue of <math>11</math> and a value of <math>x</math> that satisfies the congruence <math>x^2 \equiv 4 \pmod{257}</math> has form <math>11p \pm 2.</math> From the table above: <syntaxhighlight> N = 257 </syntaxhighlight> {| class="wikitable" |- ! <math>x</math> !! <math>x^2\ -\ N</math> |- | <code></code><code>9</code> || <code>-176</code> |- | <code>13</code> || <code></code><code>-88</code> |- | <code>20</code> || <code></code><code>143</code> |- | <code>24</code> || <code></code><code>319</code> |} These <math>4</math> values of <math>x^2 - N</math> are exactly divisible by <math>11.</math> <math>x = 9</math> is <math>11\cdot 1 - 2.</math> <math>x = 13</math> is <math>11\cdot 1 + 2.</math> <math>x = 20</math> is <math>11\cdot 2 - 2.</math> <math>x = 24</math> is <math>11\cdot 2 + 2.</math> ===Products=== This section uses prime number <math>41</math> as an example. Using <code>quadResidues(p)</code> quadratic residues of <math>41</math> are: <syntaxhighlight> qr41 = [0, 1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 21, 23, 25, 31, 32, 33, 36, 37, 39, 40] </syntaxhighlight> Quadratic non-residues of <math>41</math> are: <syntaxhighlight> qnr41 = [3, 6, 7, 11, 12, 13, 14, 15, 17, 19, 22, 24, 26, 27, 28, 29, 30, 34, 35, 38] </syntaxhighlight> ====of 2 residues==== A simple test to verify that the product of 2 residues is a residue: <syntaxhighlight lang=python> # Python code. for index1 in range (0, len(qr41)) : v1 = qr41[index1] for index2 in range (index1, len(qr41)) : v2 = qr41[index2] residue = (v1*v2) % 41 if residue not in qr41 : print ('residue',residue,'not quadratic.') </syntaxhighlight> This test shows that, at least for prime number <math>41,</math> the product of 2 residues is a residue. Advanced math proves that this is true for all primes. ====of 2 non-residues==== A simple test to verify that the product of 2 non-residues is a residue: <syntaxhighlight lang=python> # Python code. for index1 in range (0, len(qnr41)) : v1 = qnr41[index1] for index2 in range (index1, len(qnr41)) : v2 = qnr41[index2] residue = (v1*v2) % 41 if residue not in qr41 : print ('residue',residue,'not quadratic.') </syntaxhighlight> This test shows that, at least for prime number <math>41,</math> the product of 2 non-residues is a residue. Advanced math proves that this is true for all primes. ====of residue and non-residue==== A simple test to verify that the product of residue and non-residue is non-residue: <syntaxhighlight lang=python> # Python code. for index1 in range (1, len(qr41)) : v1 = qr41[index1] for index2 in range (0, len(qnr41)) : v2 = qnr41[index2] residue = (v1*v2) % 41 if residue not in qnr41 : print ('residue',residue,'quadratic.') </syntaxhighlight> This test shows that, at least for prime number <math>41,</math> the product of residue and non-residue is non-residue. Advanced math proves that this is true for all primes. {{RoundBoxTop|theme=2}} Some authors may consider <math>0</math> as not a legitimate residue. <math>0</math> is not included as a residue in the test above. {{RoundBoxBottom}} ==Euler's criterion== In number theory, '''Euler's criterion''' is a formula for determining whether or not an integer is a quadratic residue modulo a prime number. Precisely, Let ''p'' be an odd prime and ''a'' be an integer coprime to ''p''. Then :<math> a^{\tfrac{p-1}{2}} \equiv \begin{cases} \;\;\,1\pmod{p}& \text{ if there is an integer }x \text{ such that }a\equiv x^2 \pmod{p},\\ -1\pmod{p}& \text{ if there is no such integer.} \end{cases} </math> Euler's criterion can be concisely reformulated using the Legendre symbol: :<math> \left(\frac{a}{p}\right) \equiv a^{\tfrac{p-1}{2}} \pmod p. </math> :<math>\left(\frac{a}{p}\right) = \begin{cases} 1 & \text{if } a \text{ is a quadratic residue modulo } p \text{ and } a \not\equiv 0\pmod p, \\ -1 & \text{if } a \text{ is a non-quadratic residue modulo } p, \\ 0 & \text{if } a \equiv 0 \pmod p. \end{cases}</math> It is known that <math>3</math> is a quadratic residue modulo <math>11.</math> Therefore <math>(3^5)\ %\ 11</math> should be <math>1.</math> <syntaxhighlight lang=python> # python code: >>> (3**5) % 11 1 </syntaxhighlight> It is known that <math>7</math> is a quadratic non-residue modulo <math>11.</math> Therefore <math>(7^5)\ %\ 11</math> should be <math>-1.</math> <syntaxhighlight lang=python> # python code: >>> (7**5) % 11 10 </syntaxhighlight> :<math>10 \equiv -1 \pmod{11}</math> Python's decimal module provides a method for computing <math>(a^x)\ %\ p</math> very efficiently for both small and very large numbers. <syntaxhighlight lang=python> # python code: >>> import decimal >>> decimal.Context().power(3,5,11) Decimal('1') >>> decimal.Context().power(7,5,11) Decimal('10') >>> >>> a = 3456789 >>> p = 761838257287 >>> decimal.Context().power(a, p>>1, p) Decimal('761838257286') </syntaxhighlight> :<math>761838257286 \equiv -1 \pmod{761838257287}</math> Value <math>a = 3456789</math> is not a quadratic residue modulo <math>p = 761838257287.</math> {{RoundBoxTop|theme=3}} An exact square such as <math>1,4,9,16,25,\dots</math> is always a quadratic residue modulo an odd prime <math>p.</math> {{RoundBoxBottom}} ===Product of 2 residues=== Let <math>a,b</math> be quadratic residues modulo odd prime <math>p.</math> Let <math>q = \frac{p-1}{2}.</math> Then: <math>a^q \equiv 1 \pmod p</math> <math>b^q \equiv 1 \pmod p</math> By law of multiplication: <math>(a^q)(b^q) \equiv (1)(1) \pmod p</math> or <math>(a\cdot b)^q \equiv 1 \pmod p</math> Product <math>(a\cdot b)</math> of 2 quadratic residues <math>a, b</math> is quadratic residue. Similarly, product of 2 non-residues is residue, and product of residue and non-residue is non-residue. ==Factors of integer N== Several modern methods for determining the factors of a given integer attempt to create two congruent squares modulo integer <math>N.</math> <math>x^2 \equiv y^2 \pmod{N}</math> This means that the difference between the two squares is exactly divisible by <math>N</math>: <math>N\mid (x^2 - y^2).</math> Integer <math>N</math> always contains the factors <math>N,1,</math> called trivial factors. If <math>N</math> contains two non-trivial factors <math>p,q,</math> then: <math>\frac{(x+y)(x-y)}{p \cdot q}.</math> With a little luck <math>p\mid (x+y)</math> and <math>q\mid (x-y)</math> in which case: <math>p = \text{igcd}(x+y, N)</math> and <math>q = \text{igcd}(x-y, N)</math> where "<math>\text{igcd}</math>" is function "<math>\text{integer greatest common divisor.}</math>" ===A simple example:=== We will use quadratic congruences to calculate factors of <math>N = 4171</math> for <math>164 \ge x \ge 1.</math> ====Right hand side exact square==== One congruence produced an exact square for y: {| class="wikitable" |- ! <math>x</math> !! <math>x^2</math> !! <math>y = x^2 - N</math> |- | 70 | 4900 | 729 |} :<math>4900 \equiv 729 \pmod{N}</math> :<math>70^2 \equiv 27^2 \pmod{N}</math> <math>p = \text{igcd}(70-27, 4171)</math> <math>= \text{igcd}(43, 4171)</math> <math>= 43.</math> <math>q = \text{igcd}(70+27, 4171)</math> <math>= \text{igcd}(97, 4171)</math> <math>= 97.</math> Non-trivial factors of <math>4171</math> are <math>43,97.</math> ====Right hand side negative==== Table below contains a sample of values of <math>x</math> that produce negative <math>y:</math> {| class="wikitable" |- ! <math>x</math> !! <math>x^2</math> !! <math>y = x^2 - N</math> |- | <math>\ \ </math><code>7</code>||<math>\ \ \ \ </math><code>49</code>||<code>-4122</code><math>\ \ \ \ \ \ </math> |- | <math>\ \ </math><code>8</code>||<math>\ \ \ \ </math><code>64</code>||<code>-4107</code><math>\ \ </math><code>**</code> |- | <math>\ \ </math><code>9</code>||<math>\ \ \ \ </math><code>81</code>||<code>-4090</code><math>\ \ \ \ \ \ </math> |- | <code>10</code>||<math>\ \ </math><code>100</code>||<code>-4071</code><math>\ \ \ \ \ \ </math> |- | <code>11</code>||<math>\ \ </math><code>121</code>||<code>-4050</code><math>\ \ </math><code>!!</code> |- | <code>12</code>||<math>\ \ </math><code>144</code>||<code>-4027</code><math>\ \ \ \ \ \ </math> |- | <code>60</code>||<code>3600</code>||<math>\ \ </math><code>-571</code><math>\ \ \ \ \ \ </math> |- | <code>61</code>||<code>3721</code>||<math>\ \ </math><code>-450</code><math>\ \ </math><code>!!</code> |- | <code>62</code>||<code>3844</code>||<math>\ \ </math><code>-327</code><math>\ \ \ \ \ \ </math> |- | <code>63</code>||<code>3969</code>||<math>\ \ </math><code>-220</code><math>\ \ \ \ \ \ </math> |- | <code>64</code>||<code>4096</code>||<math>\ \ \ \ </math><code>-75</code><math>\ \ </math><code>**</code> |} =====Non-trivial result 1===== The congruences: {| class="wikitable" |- ! <math>x</math> !! <math>x^2</math> !! <math>y = x^2 - N</math> |- | <math>\ \ </math><code>8</code>||<math>\ \ \ \ </math><code>64</code>||<code>-4107</code><math>\ \ </math><code>**</code> |- | <code>64</code>||<code>4096</code>||<math>\ \ \ \ </math><code>-75</code><math>\ \ </math><code>**</code> |} :<math>64 \equiv -4107 \pmod{N}</math> :<math>4096 \equiv -75 \pmod{N}</math> :<math>64\cdot 4096 \equiv -4107\cdot (-75) \pmod{N}</math> :<math>262144 \equiv 308025 \pmod{N}</math> :<math>512^2 \equiv 555^2 \pmod{4171}</math> <math>p = \text{igcd}(555-512, 4171)</math> <math>= \text{igcd}(43, 4171)</math> <math>= 43.</math> <math>q = \text{igcd}(555+512, 4171)</math> <math>= \text{igcd}(1067, 4171)</math> <math>= 97.</math> Non-trivial factors of <math>4171</math> are <math>43,97.</math> =====Non-trivial result 2===== The congruences: {| class="wikitable" |- ! <math>x</math> !! <math>x^2</math> !! <math>y = x^2 - N</math> |- | <code>11</code> || <code></code><code>121</code> || <code>-4050</code><code>!!</code> |- | <code>61</code> || <code>3721</code> || <code></code><code>-450</code><code>!!</code> |} :<math>121 \equiv -4050 \pmod{N}</math> :<math>3721 \equiv -450 \pmod{N}</math> :<math>121\cdot 3721 \equiv -4050 \cdot (-450) \pmod{N}</math> :<math>450241 \equiv 1822500 \pmod{N}</math> :<math>671^2 \equiv 1350^2 \pmod{4171}</math> <math>p = \text{igcd}(1350-671, 4171)</math> <math>= \text{igcd}(679, 4171)</math> <math>= 97.</math> <math>q = \text{igcd}(1350+671, 4171)</math> <math>= \text{igcd}(2021, 4171)</math> <math>= 43.</math> Non-trivial factors of <math>4171</math> are <math>43,97.</math> ====Right hand side positive==== Table below contains a sample of values of <math>x</math> that produce positive <math>y:</math> {| class="wikitable" |- ! <math>x</math> !! <math>x^2</math> !! <math>y = x^2 - N</math> |- | <math>\ \ </math><code>65</code>||<math>\ \ </math><code>4225</code>||<math>\ \ \ \ \ \ </math><code>54</code><math>\ \ </math><code>**</code><math>\ \ \ \ </math> |- | <math>\ \ </math><code>66</code>||<math>\ \ </math><code>4356</code>||<math>\ \ \ \ </math><code>185</code><math>\ \ \ \ \ \ \ \ \ \ </math> |- | <math>\ \ </math><code>88</code>||<math>\ \ </math><code>7744</code>||<math>\ \ </math><code>3573</code><math>\ \ \ \ \ \ \ \ \ \ </math> |- | <math>\ \ </math><code>89</code>||<math>\ \ </math><code>7921</code>||<math>\ \ </math><code>3750</code><math>\ \ </math><code>**!!</code> |- | <math>\ \ </math><code>90</code>||<math>\ \ </math><code>8100</code>||<math>\ \ </math><code>3929</code><math>\ \ \ \ \ \ \ \ \ \ </math> |- | <code>144</code>||<code>20736</code>||<code>16565</code><math>\ \ \ \ \ \ \ \ \ \ </math> |- | <code>145</code>||<code>21025</code>||<code>16854</code><math>\ \ \ \ \ \ </math><code>!!</code> |- | <code>146</code>||<code>21316</code>||<code>17145</code><math>\ \ \ \ \ \ \ \ \ \ </math> |} =====Non-trivial result===== The congruences: {| class="wikitable" |- ! <math>x</math> !! <math>x^2</math> !! <math>y = x^2 - N</math> |- | <code>65</code>||<code>4225</code>||<math>\ \ \ \ </math><code>54</code><math>\ \ </math><code>**</code><math>\ \ \ \ </math> |- | <code>89</code>||<code>7921</code>||<code>3750</code><math>\ \ </math><code>**!!</code> |} :<math>4225 \equiv 54 \pmod{N}</math> :<math>7921 \equiv 3750 \pmod{N}</math> :<math>4225\cdot 7921 \equiv 54 \cdot 3750 \pmod{N}</math> :<math>33466225 \equiv 202500 \pmod{N}</math> :<math>5785^2 \equiv 450^2 \pmod{4171}</math> <math>p = \text{igcd}(5785-450, 4171)</math> <math>= \text{igcd}(5335, 4171)</math> <math>= 97.</math> <math>q = \text{igcd}(5785+450, 4171)</math> <math>= \text{igcd}(6235, 4171)</math> <math>= 43.</math> Non-trivial factors of <math>4171</math> are <math>43,97.</math> =====Trivial result===== The congruences: {| class="wikitable" |- ! <math>x</math> !! <math>x^2</math> !! <math>y = x^2 - N</math> |- | <math>\ \ </math><code>89</code>||<math>\ \ </math><code>7921</code>||<math>\ \ </math><code>3750</code><math>\ \ </math><code>**!!</code> |- | <code>145</code>||<code>21025</code>||<code>16854</code><math>\ \ \ \ \ \ </math><code>!!</code> |} :<math>7921 \equiv 3750 \pmod{N}</math> :<math>21025 \equiv 16854 \pmod{N}</math> :<math>7921\cdot 21025 \equiv 3750 \cdot 16854 \pmod{N}</math> :<math>166539025 \equiv 63202500 \pmod{N}</math> :<math>12905^2 \equiv 7950^2 \pmod{4171}</math> <math>p = \text{igcd}(12905-7950, 4171)</math> <math>= \text{igcd}(4955, 4171)</math> <math>= 1.</math> <math>q = \text{igcd}(12905+7950, 4171)</math> <math>= \text{igcd}(20855, 4171)</math> <math>= 4171.</math> This congruence produced the trivial factors of <math>4171.</math> ====With 3 congruences==== The congruences: {| class="wikitable" |- ! <math>x</math> !! <math>x^2</math> !! <math>y = x^2 - N</math> |- | <math>\ \ </math><code>56</code>||<math>\ \ </math><code>3136</code>||<code>-1035</code> |- | <math>\ \ </math><code>59</code>||<math>\ \ </math><code>3481</code>||<math>\ \ </math><code>-690</code> |- | <code>145</code>||<code>21025</code>||<code>16854</code> |} :<math>3136 \equiv -1035 \pmod{N}</math> :<math>3481 \equiv -690 \pmod{N}</math> :<math>21025 \equiv 16854 \pmod{N}</math> :<math>3136\cdot 3481 \cdot 21025 \equiv -1035 \cdot -690 \cdot 16854 \pmod{N}</math> :<math>229517646400 \equiv 12036284100 \pmod{N}</math> :<math>479080^2 \equiv 109710^2 \pmod{4171}</math> <math>p=\text{igcd}(479080-109710,4171)</math> <math>= 43.</math> <math>q = \text{igcd}(479080+109710, 4171)</math> <math>= 97.</math> Non-trivial factors of <math>4171</math> are <math>43,97.</math> =Links to related topics= [https://en.wikipedia.org/wiki/Quadratic_residue Quadratic Residue] [https://en.wikipedia.org/wiki/Modular_arithmetic Modular Arithmetic] [https://en.wikipedia.org/wiki/Leonhard_Euler Leonhard Euler,] [https://en.wikipedia.org/wiki/Euler%27s_criterion Euler's Criterion] [https://en.wikipedia.org/wiki/Adrien-Marie_Legendre Adrien-Marie Legendre,] [https://en.wikipedia.org/wiki/Legendre_symbol Legendre Symbol] [https://en.wikipedia.org/wiki/Carl_Friedrich_Gauss Carl Friedrich Gauss,] [https://en.wikipedia.org/wiki/Triple_bar Triple_bar or <math>\equiv</math>,] [https://en.wikipedia.org/wiki/Quadratic_reciprocity Quadratic reciprocity] [https://en.wikipedia.org/wiki/Carl_Pomerance Carl Pomerance,] [https://en.wikipedia.org/wiki/Quadratic_sieve Quadratic sieve] [https://en.wikipedia.org/wiki/Greatest_common_divisor Greatest common divisor,] [https://en.wikipedia.org/wiki/Recursion_(computer_science)#Greatest_common_divisor Greatest common divisor (Example of Recursion)] [https://docs.python.org/3.4/library/decimal.html?highlight=decimal#decimal.Context.power Python's decimal Module] b9t4tphos911flj3sq64ihd3jj3twh5 0 285898 2416415 2411378 2022-08-19T03:29:53Z Earthxangel 2947535 /* See also */ wikitext text/x-wiki {{title|Green prescription motivation:<br>What motivates green prescription compliance?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== * [[wikipedia:Exercise_prescription|Exercise Prescription]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] 6lo8sasxr1xlli669bz7sp43xj3yt9h 2416416 2416415 2022-08-19T03:33:43Z Earthxangel 2947535 /* See also */ wikitext text/x-wiki {{title|Green prescription motivation:<br>What motivates green prescription compliance?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== * [[wikipedia:Exercise_prescription|Exercise Prescription]] (Wikipedia) * [[Motivation and emotion/Book/2011/Sport participation|Sport Participation]] (Book chapter, 2011) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] b9q7qv2unvt2l25pyed42m0wf3bz8dv 2416417 2416416 2022-08-19T03:35:36Z Earthxangel 2947535 /* See also */ wikitext text/x-wiki {{title|Green prescription motivation:<br>What motivates green prescription compliance?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== * [[Motivation and emotion/Book/2011/Exercise motivation|Exercise motivation]] (Book chapter, 2011) * [[wikipedia:Exercise_prescription|Exercise prescription]] (Wikipedia) * [[Motivation and emotion/Book/2011/Sport participation|Sport participation]] (Book chapter, 2011) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] 7tiavdas7ab0cubd4wefyzr4q2j2w3x 0 285908 2416411 2416089 2022-08-19T03:17:41Z U3216256 2942574 /* What causes disappointment? */ wikitext text/x-wiki {{title|Disappointment<br>What is disappointment, what causes it, and how can it be managed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== [[File:Wayuu woman with sad face in the market buying.jpg|alt=Older woman with a disappointed look on her face in a busy market place.|thumb|''Figure 1.'' Person expressing disappointment through facial features]] * Example (case study) of disappointment - include statistics on frequency of occurrence (Van Dijk & Zeelenberg, 2002) * Define [[wikipedia:Emotion|emotions]], what emotions involve. (Reeve, 2018; Izard, 2010) Reference Figure 1 * Overview of what the chapter explores * Note on emotion research (Ramachandran & Jalal, 2017) {{RoundBoxTop|theme=13}} '''Key questions:''' * What is disappointment? * What causes disappointment? * How can disappointment be managed? {{RoundBoxBottom}} ==What is disappointment?== * Disappointment occurs when the outcome that has occurred is not as good as the outcome that could have occurred, or when your expectations are not met (Zeelenberg et al., 1998). * Disappointment involves feeling powerless, a tendency to remove oneself from the situation, and a desire to do nothing (van Dijk et al., 1999) * Disappointment is a complex emotion (Ramachandran & Jalal, 2017) * Disappointment can be categorised as outcome-related or person-related; where outcome-related disappointment (ORD) occurs when the expected pleasurable outcome has not occurred, and person-related disappointment (PRD) occurs when you attribute the undesirable outcome to a\ person (you blame them for the 'bad' outcome) (Van Dijk & Zeelenberg, 2002). {{Robelbox|theme={{{theme|2}}}|title=Emotions similar to disappointment}} <div style="{{Robelbox/pad}}"> '''Regret''' : [https://en.wikipedia.org/wiki/Regret Regret] is a negative emotion that occurs when you know that the outcome that occurred could have been better if you made a different choice (Zeelenberg et al., 1998). "Regret stems from bad decisions" (Zeelenberg et al., 1998). '''Anger''' : [https://en.wikipedia.org/wiki/Anger Anger] is a simple negative emotion that occurs when you cannot achieve your goals and you blame someone or something else for it (Lelieveld et al., 2011). Anger can be the result of disappointment (van Dijk et al., 1999). '''Disillusionment''' : [https://en.wiktionary.org/wiki/disillusionment Disillusionment] is a complex negative emotion that occurs when you realise that what you believe or know is false (Maher et al., 2020). Disappointment is a key feature of disillusionment. </div> {{Robelbox/close}} Test yourself:<quiz display="simple"> {Mary's boss received a complaint from a customer about Mary. Mary was made aware of the complaint and then fired. Mary is likely to experience: |type="()"} + Person-related disappointment - Outcome-related disappointment {Alex is trying to get a snack from a vending machine. Alex put their money into the vending machine and typed in the code for lemonade. The vending machine did not give Alex lemonade, and took their money. Alex is likely to experience: |type="()"} - Person-related disappointment + Outcome-related disappointment </quiz> == What causes disappointment? == * Disappointment occurs when expectations are not met (Chua et al., 2009). * Brain regions that have shown to be important in decision making/disappointment are the anterior insula, and various regions of the prefrontal cortex (Chua et al., 2009: Mohr et al., 2010: Kalat, 2019). === Theories of disappointment === ==== Bell's theory of disappointment (1985) ==== * h * h ==== Loomes and Sudgen's theory of disappointment ==== * j * h === Neuropsychology of disappointment === ==== Subcortical structure: Anterior insula ==== [[File:Insula structure.png|alt=Structure of the three sections of the insula|thumb|''Figure 2.'' Insula]] * When individuals experience disappointment their anterior insula is activated (Chua et al., 2009). See figure 2. * The anterior insula is active in the presence of potential loss, a part of the brain known for processing emotions such as disappointment (Mohr et al., 2010) ==== Cortical structure: Prefrontal cortex ==== [[File:Prefrontal cortex (left) animation.gif|alt=Rotating skull containing left Prefrontal cortex. The prefrontal cortex is highlighted |thumb|''Figure 3.'' Left hemisphere prefrontal cortex]] [[File:Cortical midline structures.png|thumb|''Figure 4.'' Brain image highlighting various cortical regions, including the ventromedial prefrontal cortex (VMPFC), dorsomedial prefrontal cortex (DMPFC), and the orbitofrontal cortex (OFC/MOPFC).]] * prefrontal cortex - middle 'zone': <u>emotional reactions</u>, cognitive control, and working memory (Kalat, 2019). * prefrontal cortex - anterior (front): decision making, evaluating which course will provide the best outcome, probability of achieving a good outcome (Kalat, 2019). * Prefrontal cortex - see figure 3. * The ventromedial prefrontal cortex [VMPFC] learns what choices are more beneficial for the individual and adjusts decision making accordingly (Kalat, 2019). It also monitors confidence in one's decisions (Kalat, 2019). Individuals with damage to their VMPFC tend to make impulsive decisions based on probability, rather than making considered decisions based on reality (Kalat, 2019). *Orbitofrontal cortex [OFC]- responds to information from the VMPFC based on how the expected reward compares to other possible choices (Kalat, 2019). The OFC changes and updates the expected outcomes of actions based on current circumstances (Kalat, 2019). *When individuals experience disappointment their dorsomedial prefrontal cortex [DMPFC] is activated (Chua et al., 2009) *See figure 4 for VMPFC, OFC, and DMPFC. Insert Quiz on brain regions and how they contribute to decision-making/disappointment OR insert table with brain region - what it does - how it contributes to disappointment ==How can disappointment be managed?== * Disappointment can be managed by lowering expectations (van Dijk et al., 2003). The ventromedial prefrontal cortex monitors how confident you should be about a decision (Kalat, 2019). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ... ^ therefore having lower expectations can improve wellbeing === The benefits of disappointment === * Expressing disappointment encourages higher levels of cooperation in others (Olekalns & Druckman, 2014). * Expressing disappointment can lead to helping behaviour in others (Johnson & Connelly, 2014). === The downsides to disappointment === * Disappointment can taint relationships, leading to blaming or [[wikipedia:Scapegoating|scapegoating]] (Clancy et al., 2012). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ==Conclusion== * Restate what disappointment is, the types of disappointment, and emotions similar to disappointment. * Restate the causes of disappointment * Restate how disappointment can be managed * Take home message: everybody experiences disappointment, but there are things you can do to minimise experiencing it and you can even take advantage of it. == See also == * [[wikipedia:Disappointment|Disappointment]] (Wikipedia) * [[Motivation and emotion/Book/2016/Regret|Regret]] (Book chapter, 2016) * [[Motivation and emotion/Book/2011/Anger|Anger]] (Book chapter, 2011) ==References== {{Hanging indent|1= Chua, Gonzalez, R., Taylor, S. F., Welsh, R. C., & Liberzon, I. (2009). Decision-related loss: Regret and disappointment. ''NeuroImage, 47''(4), 2031–2040. https://doi.org/10.1016/j.neuroimage.2009.06.006 Clancy, A., Vince, R., & Gabriel, Y. (2012). That Unwanted Feeling: A Psychodynamic Study of Disappointment in Organizations. ''British Journal of Management, 23''(4), 518–531. https://doi.org/10.1111/j.1467-8551.2011.00780.x de Meza, D., & Dawson, C. (2021). Neither an Optimist Nor a Pessimist Be: Mistaken Expectations Lower Well-Being. ''Personality & Social Psychology Bulletin'', ''47''(4), 540–550. https://doi.org/10.1177/0146167220934577 Izard, C. E. (2010). The Many Meanings/Aspects of Emotion: Definitions, Functions, Activation, and Regulation. ''Emotion Review, 2''(4), 363–370. https://doi.org/10.1177/1754073910374661 Johnson, G., & Connelly, S. (2014). Negative emotions in informal feedback: The benefits of disappointment and drawbacks of anger. ''Human Relations (New York), 67''(10), 1265–1290. https://doi.org/10.1177/0018726714532856 Kalat, J. W. (2019). ''Biological psychology'' (13th ed.). Boston, MA, USA: Cengage Lelieveld, G. J., Van Dijk, E., Van Beest, I., Steinel, W., & Van Kleef, G. A. (2011). Disappointed in you, angry about your offer: Distinct negative emotions induce concessions via different mechanisms. ''Journal of Experimental Social Psychology, 47''(3), 635–641. https://doi.org/10.1016/j.jesp.2010.12.015 Maher, P. J., Igou, E. R., & van Tilburg, W. A. P. (2020). Disillusionment: a prototype analysis. ''Cognition and Emotion, 34''(5), 947–959. https://doi.org/10.1080/02699931.2019.1705764 Olekalns, M., & Druckman, D. (2014). With Feeling: How Emotions Shape Negotiation. ''Negotiation Journal, 30''(4), 455–478. https://doi.org/10.1111/nejo.12071 Mohr, P. N. C., Biele, G., & Heekeren, H. R. (2010). Neural processing of risk. ''The Journal of Neuroscience, 30''(19), 6613–6619. https://doi.org/10.1523/JNEUROSCI.0003-10.2010 Ramachandran, V.S., & Jalal, B. (2017). The Evolutionary Psychology of Envy and Jealousy. ''Frontiers in Psychology, 8'', 1619–1619. https://doi.org/10.3389/fpsyg.2017.01619 Reeve, J. (2018). ''Understanding motivation and emotion'' (7th ed.). Hoboken, NJ: Wiley van Dijk, W. W., & Zeelenberg, M. (2002). What do we talk about when we talk about disappointment? Distinguishing outcome-related disappointment from person-related disappointment. ''Cognition and Emotion, 16''(6), 787–807. https://doi.org/10.1080/02699930143000563 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (1999). Not Having What You Want versus Having What You Do Not Want: The Impact of Type of Negative Outcome on the Experience of Disappointment and Related Emotions. ''Cognition and Emotion, 13''(2), 129–148. https://doi.org/10.1080/026999399379302 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (2003). Blessed are those who expect nothing: Lowering expectations as a way of avoiding disappointment. ''Journal of Economic Psychology, 24''(4), 505–516. https://doi.org/10.1016/S0167-4870(02)00211-8 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & der Pligt, J. (1998). The Experience of Regret and Disappointment. ''Cognition and Emotion, 12''(2), 221–230. https://doi.org/10.1080/026999398379727 }} ==External links== * [https://www.youtube.com/watch?v=8KgUFMN7aJQ The value of disappointment] (TEDxPCC) * [https://www.youtube.com/watch?v=gAMbkJk6gnE Feeling all the feels: Crash course psychology #25] (YouTube) iuwiolsdhm3pdegtsw5066ar477s3y1 2416419 2416411 2022-08-19T03:42:57Z U3216256 2942574 /* Loomes and Sudgen's theory of disappointment */ wikitext text/x-wiki {{title|Disappointment<br>What is disappointment, what causes it, and how can it be managed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== [[File:Wayuu woman with sad face in the market buying.jpg|alt=Older woman with a disappointed look on her face in a busy market place.|thumb|''Figure 1.'' Person expressing disappointment through facial features]] * Example (case study) of disappointment - include statistics on frequency of occurrence (Van Dijk & Zeelenberg, 2002) * Define [[wikipedia:Emotion|emotions]], what emotions involve. (Reeve, 2018; Izard, 2010) Reference Figure 1 * Overview of what the chapter explores * Note on emotion research (Ramachandran & Jalal, 2017) {{RoundBoxTop|theme=13}} '''Key questions:''' * What is disappointment? * What causes disappointment? * How can disappointment be managed? {{RoundBoxBottom}} ==What is disappointment?== * Disappointment occurs when the outcome that has occurred is not as good as the outcome that could have occurred, or when your expectations are not met (Zeelenberg et al., 1998). * Disappointment involves feeling powerless, a tendency to remove oneself from the situation, and a desire to do nothing (van Dijk et al., 1999) * Disappointment is a complex emotion (Ramachandran & Jalal, 2017) * Disappointment can be categorised as outcome-related or person-related; where outcome-related disappointment (ORD) occurs when the expected pleasurable outcome has not occurred, and person-related disappointment (PRD) occurs when you attribute the undesirable outcome to a\ person (you blame them for the 'bad' outcome) (Van Dijk & Zeelenberg, 2002). {{Robelbox|theme={{{theme|2}}}|title=Emotions similar to disappointment}} <div style="{{Robelbox/pad}}"> '''Regret''' : [https://en.wikipedia.org/wiki/Regret Regret] is a negative emotion that occurs when you know that the outcome that occurred could have been better if you made a different choice (Zeelenberg et al., 1998). "Regret stems from bad decisions" (Zeelenberg et al., 1998). '''Anger''' : [https://en.wikipedia.org/wiki/Anger Anger] is a simple negative emotion that occurs when you cannot achieve your goals and you blame someone or something else for it (Lelieveld et al., 2011). Anger can be the result of disappointment (van Dijk et al., 1999). '''Disillusionment''' : [https://en.wiktionary.org/wiki/disillusionment Disillusionment] is a complex negative emotion that occurs when you realise that what you believe or know is false (Maher et al., 2020). Disappointment is a key feature of disillusionment. </div> {{Robelbox/close}} Test yourself:<quiz display="simple"> {Mary's boss received a complaint from a customer about Mary. Mary was made aware of the complaint and then fired. Mary is likely to experience: |type="()"} + Person-related disappointment - Outcome-related disappointment {Alex is trying to get a snack from a vending machine. Alex put their money into the vending machine and typed in the code for lemonade. The vending machine did not give Alex lemonade, and took their money. Alex is likely to experience: |type="()"} - Person-related disappointment + Outcome-related disappointment </quiz> == What causes disappointment? == * Disappointment occurs when expectations are not met (Chua et al., 2009). * The two major theories of disappointment will be discussed. * Brain regions that have shown to be important in decision making/disappointment are the anterior insula, and various regions of the prefrontal cortex (Chua et al., 2009: Mohr et al., 2010: Kalat, 2019). === Theories of disappointment === * The two major theories of disappointment are Bell's theory of disappointment and Loomes and Sudgen's theory of disappointment (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). ==== Bell's theory of disappointment (1985) ==== * h * h ==== Loomes and Sudgen's theory of disappointment (1984) ==== * j * h === Neuropsychology of disappointment === ==== Subcortical structure: Anterior insula ==== [[File:Insula structure.png|alt=Structure of the three sections of the insula|thumb|''Figure 2.'' Insula]] * When individuals experience disappointment their anterior insula is activated (Chua et al., 2009). See figure 2. * The anterior insula is active in the presence of potential loss, a part of the brain known for processing emotions such as disappointment (Mohr et al., 2010) ==== Cortical structure: Prefrontal cortex ==== [[File:Prefrontal cortex (left) animation.gif|alt=Rotating skull containing left Prefrontal cortex. The prefrontal cortex is highlighted |thumb|''Figure 3.'' Left hemisphere prefrontal cortex]] [[File:Cortical midline structures.png|thumb|''Figure 4.'' Brain image highlighting various cortical regions, including the ventromedial prefrontal cortex (VMPFC), dorsomedial prefrontal cortex (DMPFC), and the orbitofrontal cortex (OFC/MOPFC).]] * prefrontal cortex - middle 'zone': <u>emotional reactions</u>, cognitive control, and working memory (Kalat, 2019). * prefrontal cortex - anterior (front): decision making, evaluating which course will provide the best outcome, probability of achieving a good outcome (Kalat, 2019). * Prefrontal cortex - see figure 3. * The ventromedial prefrontal cortex [VMPFC] learns what choices are more beneficial for the individual and adjusts decision making accordingly (Kalat, 2019). It also monitors confidence in one's decisions (Kalat, 2019). Individuals with damage to their VMPFC tend to make impulsive decisions based on probability, rather than making considered decisions based on reality (Kalat, 2019). *Orbitofrontal cortex [OFC]- responds to information from the VMPFC based on how the expected reward compares to other possible choices (Kalat, 2019). The OFC changes and updates the expected outcomes of actions based on current circumstances (Kalat, 2019). *When individuals experience disappointment their dorsomedial prefrontal cortex [DMPFC] is activated (Chua et al., 2009) *See figure 4 for VMPFC, OFC, and DMPFC. Insert Quiz on brain regions and how they contribute to decision-making/disappointment OR insert table with brain region - what it does - how it contributes to disappointment ==How can disappointment be managed?== * Disappointment can be managed by lowering expectations (van Dijk et al., 2003). The ventromedial prefrontal cortex monitors how confident you should be about a decision (Kalat, 2019). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ... ^ therefore having lower expectations can improve wellbeing === The benefits of disappointment === * Expressing disappointment encourages higher levels of cooperation in others (Olekalns & Druckman, 2014). * Expressing disappointment can lead to helping behaviour in others (Johnson & Connelly, 2014). === The downsides to disappointment === * Disappointment can taint relationships, leading to blaming or [[wikipedia:Scapegoating|scapegoating]] (Clancy et al., 2012). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ==Conclusion== * Restate what disappointment is, the types of disappointment, and emotions similar to disappointment. * Restate the causes of disappointment * Restate how disappointment can be managed * Take home message: everybody experiences disappointment, but there are things you can do to minimise experiencing it and you can even take advantage of it. == See also == * [[wikipedia:Disappointment|Disappointment]] (Wikipedia) * [[Motivation and emotion/Book/2016/Regret|Regret]] (Book chapter, 2016) * [[Motivation and emotion/Book/2011/Anger|Anger]] (Book chapter, 2011) ==References== {{Hanging indent|1= Chua, Gonzalez, R., Taylor, S. F., Welsh, R. C., & Liberzon, I. (2009). Decision-related loss: Regret and disappointment. ''NeuroImage, 47''(4), 2031–2040. https://doi.org/10.1016/j.neuroimage.2009.06.006 Clancy, A., Vince, R., & Gabriel, Y. (2012). That Unwanted Feeling: A Psychodynamic Study of Disappointment in Organizations. ''British Journal of Management, 23''(4), 518–531. https://doi.org/10.1111/j.1467-8551.2011.00780.x de Meza, D., & Dawson, C. (2021). Neither an Optimist Nor a Pessimist Be: Mistaken Expectations Lower Well-Being. ''Personality & Social Psychology Bulletin'', ''47''(4), 540–550. https://doi.org/10.1177/0146167220934577 Izard, C. E. (2010). The Many Meanings/Aspects of Emotion: Definitions, Functions, Activation, and Regulation. ''Emotion Review, 2''(4), 363–370. https://doi.org/10.1177/1754073910374661 Johnson, G., & Connelly, S. (2014). Negative emotions in informal feedback: The benefits of disappointment and drawbacks of anger. ''Human Relations (New York), 67''(10), 1265–1290. https://doi.org/10.1177/0018726714532856 Kalat, J. W. (2019). ''Biological psychology'' (13th ed.). Boston, MA, USA: Cengage Lelieveld, G. J., Van Dijk, E., Van Beest, I., Steinel, W., & Van Kleef, G. A. (2011). Disappointed in you, angry about your offer: Distinct negative emotions induce concessions via different mechanisms. ''Journal of Experimental Social Psychology, 47''(3), 635–641. https://doi.org/10.1016/j.jesp.2010.12.015 Maher, P. J., Igou, E. R., & van Tilburg, W. A. P. (2020). Disillusionment: a prototype analysis. ''Cognition and Emotion, 34''(5), 947–959. https://doi.org/10.1080/02699931.2019.1705764 Olekalns, M., & Druckman, D. (2014). With Feeling: How Emotions Shape Negotiation. ''Negotiation Journal, 30''(4), 455–478. https://doi.org/10.1111/nejo.12071 Mohr, P. N. C., Biele, G., & Heekeren, H. R. (2010). Neural processing of risk. ''The Journal of Neuroscience, 30''(19), 6613–6619. https://doi.org/10.1523/JNEUROSCI.0003-10.2010 Ramachandran, V.S., & Jalal, B. (2017). The Evolutionary Psychology of Envy and Jealousy. ''Frontiers in Psychology, 8'', 1619–1619. https://doi.org/10.3389/fpsyg.2017.01619 Reeve, J. (2018). ''Understanding motivation and emotion'' (7th ed.). Hoboken, NJ: Wiley van Dijk, W. W., & Zeelenberg, M. (2002). What do we talk about when we talk about disappointment? Distinguishing outcome-related disappointment from person-related disappointment. ''Cognition and Emotion, 16''(6), 787–807. https://doi.org/10.1080/02699930143000563 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (1999). Not Having What You Want versus Having What You Do Not Want: The Impact of Type of Negative Outcome on the Experience of Disappointment and Related Emotions. ''Cognition and Emotion, 13''(2), 129–148. https://doi.org/10.1080/026999399379302 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (2003). Blessed are those who expect nothing: Lowering expectations as a way of avoiding disappointment. ''Journal of Economic Psychology, 24''(4), 505–516. https://doi.org/10.1016/S0167-4870(02)00211-8 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (1998a). The Experience of Regret and Disappointment. ''Cognition and Emotion, 12''(2), 221–230. https://doi.org/10.1080/026999398379727 Zeelenberg, M., van Dijk, W. W., van der Pligt, J., Manstead, A. S. ., van Empelen, P., & Reinderman, D. (1998b). Emotional Reactions to the Outcomes of Decisions: The Role of Counterfactual Thought in the Experience of Regret and Disappointment. ''Organizational Behavior and Human Decision Processes, 75''(2), 117–141. https://doi.org/10.1006/obhd.1998.2784 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (2000). On bad decisions and disconfirmed expectancies: The psychology of regret and disappointment. ''Cognition and Emotion, 14''(4), 521–541. https://doi.org/10.1080/026999300402781 }} ==External links== * [https://www.youtube.com/watch?v=8KgUFMN7aJQ The value of disappointment] (TEDxPCC) * [https://www.youtube.com/watch?v=gAMbkJk6gnE Feeling all the feels: Crash course psychology #25] (YouTube) ra859no32noalz4g202jchpmln241c9 2416423 2416419 2022-08-19T03:47:12Z U3216256 2942574 /* What causes disappointment? */ wikitext text/x-wiki {{title|Disappointment<br>What is disappointment, what causes it, and how can it be managed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== [[File:Wayuu woman with sad face in the market buying.jpg|alt=Older woman with a disappointed look on her face in a busy market place.|thumb|''Figure 1.'' Person expressing disappointment through facial features]] * Example (case study) of disappointment - include statistics on frequency of occurrence (Van Dijk & Zeelenberg, 2002) * Define [[wikipedia:Emotion|emotions]], what emotions involve. (Reeve, 2018; Izard, 2010) Reference Figure 1 * Overview of what the chapter explores * Note on emotion research (Ramachandran & Jalal, 2017) {{RoundBoxTop|theme=13}} '''Key questions:''' * What is disappointment? * What causes disappointment? * How can disappointment be managed? {{RoundBoxBottom}} ==What is disappointment?== * Disappointment occurs when the outcome that has occurred is not as good as the outcome that could have occurred, or when your expectations are not met (Zeelenberg et al., 1998). * Disappointment involves feeling powerless, a tendency to remove oneself from the situation, and a desire to do nothing (van Dijk et al., 1999) * Disappointment is a complex emotion (Ramachandran & Jalal, 2017) * Disappointment can be categorised as outcome-related or person-related; where outcome-related disappointment (ORD) occurs when the expected pleasurable outcome has not occurred, and person-related disappointment (PRD) occurs when you attribute the undesirable outcome to a\ person (you blame them for the 'bad' outcome) (Van Dijk & Zeelenberg, 2002). {{Robelbox|theme={{{theme|2}}}|title=Emotions similar to disappointment}} <div style="{{Robelbox/pad}}"> '''Regret''' : [https://en.wikipedia.org/wiki/Regret Regret] is a negative emotion that occurs when you know that the outcome that occurred could have been better if you made a different choice (Zeelenberg et al., 1998). "Regret stems from bad decisions" (Zeelenberg et al., 1998). '''Anger''' : [https://en.wikipedia.org/wiki/Anger Anger] is a simple negative emotion that occurs when you cannot achieve your goals and you blame someone or something else for it (Lelieveld et al., 2011). Anger can be the result of disappointment (van Dijk et al., 1999). '''Disillusionment''' : [https://en.wiktionary.org/wiki/disillusionment Disillusionment] is a complex negative emotion that occurs when you realise that what you believe or know is false (Maher et al., 2020). Disappointment is a key feature of disillusionment. </div> {{Robelbox/close}} Test yourself:<quiz display="simple"> {Mary's boss received a complaint from a customer about Mary. Mary was made aware of the complaint and then fired. Mary is likely to experience: |type="()"} + Person-related disappointment - Outcome-related disappointment {Alex is trying to get a snack from a vending machine. Alex put their money into the vending machine and typed in the code for lemonade. The vending machine did not give Alex lemonade, and took their money. Alex is likely to experience: |type="()"} - Person-related disappointment + Outcome-related disappointment </quiz> == What causes disappointment? == * Disappointment occurs when expectations are not met (Chua et al., 2009). * The two major theories of disappointment will be discussed. * Brain regions that have shown to be important in decision making/disappointment are the anterior insula, and various regions of the prefrontal cortex (Chua et al., 2009: Mohr et al., 2010: Kalat, 2019). === Theories of disappointment === * The two major theories of disappointment are Bell's theory of disappointment and Loomes and Sudgen's theory of disappointment (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). * Both theories are similar. Loomes and Sugden's theory came about due to their research and theory of regret (Loomes & Sugden, 1986). ==== Bell's theory of disappointment (1985) ==== * h * h ==== Loomes and Sugden's theory of disappointment (1986) ==== * j * h === Neuropsychology of disappointment === ==== Subcortical structure: Anterior insula ==== [[File:Insula structure.png|alt=Structure of the three sections of the insula|thumb|''Figure 2.'' Insula]] * When individuals experience disappointment their anterior insula is activated (Chua et al., 2009). See figure 2. * The anterior insula is active in the presence of potential loss, a part of the brain known for processing emotions such as disappointment (Mohr et al., 2010) ==== Cortical structure: Prefrontal cortex ==== [[File:Prefrontal cortex (left) animation.gif|alt=Rotating skull containing left Prefrontal cortex. The prefrontal cortex is highlighted |thumb|''Figure 3.'' Left hemisphere prefrontal cortex]] [[File:Cortical midline structures.png|thumb|''Figure 4.'' Brain image highlighting various cortical regions, including the ventromedial prefrontal cortex (VMPFC), dorsomedial prefrontal cortex (DMPFC), and the orbitofrontal cortex (OFC/MOPFC).]] * prefrontal cortex - middle 'zone': <u>emotional reactions</u>, cognitive control, and working memory (Kalat, 2019). * prefrontal cortex - anterior (front): decision making, evaluating which course will provide the best outcome, probability of achieving a good outcome (Kalat, 2019). * Prefrontal cortex - see figure 3. * The ventromedial prefrontal cortex [VMPFC] learns what choices are more beneficial for the individual and adjusts decision making accordingly (Kalat, 2019). It also monitors confidence in one's decisions (Kalat, 2019). Individuals with damage to their VMPFC tend to make impulsive decisions based on probability, rather than making considered decisions based on reality (Kalat, 2019). *Orbitofrontal cortex [OFC]- responds to information from the VMPFC based on how the expected reward compares to other possible choices (Kalat, 2019). The OFC changes and updates the expected outcomes of actions based on current circumstances (Kalat, 2019). *When individuals experience disappointment their dorsomedial prefrontal cortex [DMPFC] is activated (Chua et al., 2009) *See figure 4 for VMPFC, OFC, and DMPFC. Insert Quiz on brain regions and how they contribute to decision-making/disappointment OR insert table with brain region - what it does - how it contributes to disappointment ==How can disappointment be managed?== * Disappointment can be managed by lowering expectations (van Dijk et al., 2003). The ventromedial prefrontal cortex monitors how confident you should be about a decision (Kalat, 2019). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ... ^ therefore having lower expectations can improve wellbeing === The benefits of disappointment === * Expressing disappointment encourages higher levels of cooperation in others (Olekalns & Druckman, 2014). * Expressing disappointment can lead to helping behaviour in others (Johnson & Connelly, 2014). === The downsides to disappointment === * Disappointment can taint relationships, leading to blaming or [[wikipedia:Scapegoating|scapegoating]] (Clancy et al., 2012). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ==Conclusion== * Restate what disappointment is, the types of disappointment, and emotions similar to disappointment. * Restate the causes of disappointment * Restate how disappointment can be managed * Take home message: everybody experiences disappointment, but there are things you can do to minimise experiencing it and you can even take advantage of it. == See also == * [[wikipedia:Disappointment|Disappointment]] (Wikipedia) * [[Motivation and emotion/Book/2016/Regret|Regret]] (Book chapter, 2016) * [[Motivation and emotion/Book/2011/Anger|Anger]] (Book chapter, 2011) ==References== {{Hanging indent|1= Chua, Gonzalez, R., Taylor, S. F., Welsh, R. C., & Liberzon, I. (2009). Decision-related loss: Regret and disappointment. ''NeuroImage, 47''(4), 2031–2040. https://doi.org/10.1016/j.neuroimage.2009.06.006 Clancy, A., Vince, R., & Gabriel, Y. (2012). That Unwanted Feeling: A Psychodynamic Study of Disappointment in Organizations. ''British Journal of Management, 23''(4), 518–531. https://doi.org/10.1111/j.1467-8551.2011.00780.x de Meza, D., & Dawson, C. (2021). Neither an Optimist Nor a Pessimist Be: Mistaken Expectations Lower Well-Being. ''Personality & Social Psychology Bulletin'', ''47''(4), 540–550. https://doi.org/10.1177/0146167220934577 Izard, C. E. (2010). The Many Meanings/Aspects of Emotion: Definitions, Functions, Activation, and Regulation. ''Emotion Review, 2''(4), 363–370. https://doi.org/10.1177/1754073910374661 Johnson, G., & Connelly, S. (2014). Negative emotions in informal feedback: The benefits of disappointment and drawbacks of anger. ''Human Relations (New York), 67''(10), 1265–1290. https://doi.org/10.1177/0018726714532856 Kalat, J. W. (2019). ''Biological psychology'' (13th ed.). Boston, MA, USA: Cengage Lelieveld, G. J., Van Dijk, E., Van Beest, I., Steinel, W., & Van Kleef, G. A. (2011). Disappointed in you, angry about your offer: Distinct negative emotions induce concessions via different mechanisms. ''Journal of Experimental Social Psychology, 47''(3), 635–641. https://doi.org/10.1016/j.jesp.2010.12.015 Loomes, G., & Sugden, R. (1986). Disappointment and dynamic consistency in choice under uncertainty. ''The Review of Economic Studies, 53''(2), 271–282. https://doi.org/10.2307/2297651 Maher, P. J., Igou, E. R., & van Tilburg, W. A. P. (2020). Disillusionment: a prototype analysis. ''Cognition and Emotion, 34''(5), 947–959. https://doi.org/10.1080/02699931.2019.1705764 Olekalns, M., & Druckman, D. (2014). With Feeling: How Emotions Shape Negotiation. ''Negotiation Journal, 30''(4), 455–478. https://doi.org/10.1111/nejo.12071 Mohr, P. N. C., Biele, G., & Heekeren, H. R. (2010). Neural processing of risk. ''The Journal of Neuroscience, 30''(19), 6613–6619. https://doi.org/10.1523/JNEUROSCI.0003-10.2010 Ramachandran, V.S., & Jalal, B. (2017). The Evolutionary Psychology of Envy and Jealousy. ''Frontiers in Psychology, 8'', 1619–1619. https://doi.org/10.3389/fpsyg.2017.01619 Reeve, J. (2018). ''Understanding motivation and emotion'' (7th ed.). Hoboken, NJ: Wiley van Dijk, W. W., & Zeelenberg, M. (2002). What do we talk about when we talk about disappointment? Distinguishing outcome-related disappointment from person-related disappointment. ''Cognition and Emotion, 16''(6), 787–807. https://doi.org/10.1080/02699930143000563 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (1999). Not Having What You Want versus Having What You Do Not Want: The Impact of Type of Negative Outcome on the Experience of Disappointment and Related Emotions. ''Cognition and Emotion, 13''(2), 129–148. https://doi.org/10.1080/026999399379302 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (2003). Blessed are those who expect nothing: Lowering expectations as a way of avoiding disappointment. ''Journal of Economic Psychology, 24''(4), 505–516. https://doi.org/10.1016/S0167-4870(02)00211-8 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (1998a). The Experience of Regret and Disappointment. ''Cognition and Emotion, 12''(2), 221–230. https://doi.org/10.1080/026999398379727 Zeelenberg, M., van Dijk, W. W., van der Pligt, J., Manstead, A. S. ., van Empelen, P., & Reinderman, D. (1998b). Emotional Reactions to the Outcomes of Decisions: The Role of Counterfactual Thought in the Experience of Regret and Disappointment. ''Organizational Behavior and Human Decision Processes, 75''(2), 117–141. https://doi.org/10.1006/obhd.1998.2784 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (2000). On bad decisions and disconfirmed expectancies: The psychology of regret and disappointment. ''Cognition and Emotion, 14''(4), 521–541. https://doi.org/10.1080/026999300402781 }} ==External links== * [https://www.youtube.com/watch?v=8KgUFMN7aJQ The value of disappointment] (TEDxPCC) * [https://www.youtube.com/watch?v=gAMbkJk6gnE Feeling all the feels: Crash course psychology #25] (YouTube) 060ailtg0nof821o0hufz389je63uuq 2416424 2416423 2022-08-19T03:48:21Z U3216256 2942574 /* What is disappointment? */ wikitext text/x-wiki {{title|Disappointment<br>What is disappointment, what causes it, and how can it be managed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== [[File:Wayuu woman with sad face in the market buying.jpg|alt=Older woman with a disappointed look on her face in a busy market place.|thumb|''Figure 1.'' Person expressing disappointment through facial features]] * Example (case study) of disappointment - include statistics on frequency of occurrence (Van Dijk & Zeelenberg, 2002) * Define [[wikipedia:Emotion|emotions]], what emotions involve. (Reeve, 2018; Izard, 2010) Reference Figure 1 * Overview of what the chapter explores * Note on emotion research (Ramachandran & Jalal, 2017) {{RoundBoxTop|theme=13}} '''Key questions:''' * What is disappointment? * What causes disappointment? * How can disappointment be managed? {{RoundBoxBottom}} ==What is disappointment?== * Disappointment occurs when the outcome that has occurred is not as good as the outcome that could have occurred, or when your expectations are not met (Zeelenberg et al., 1998a). * Disappointment involves feeling powerless, a tendency to remove oneself from the situation, and a desire to do nothing (van Dijk et al., 1999) * Disappointment is a complex emotion (Ramachandran & Jalal, 2017) * Disappointment can be categorised as outcome-related or person-related; where outcome-related disappointment (ORD) occurs when the expected pleasurable outcome has not occurred, and person-related disappointment (PRD) occurs when you attribute the undesirable outcome to a\ person (you blame them for the 'bad' outcome) (Van Dijk & Zeelenberg, 2002). {{Robelbox|theme={{{theme|2}}}|title=Emotions similar to disappointment}} <div style="{{Robelbox/pad}}"> '''Regret''' : [https://en.wikipedia.org/wiki/Regret Regret] is a negative emotion that occurs when you know that the outcome that occurred could have been better if you made a different choice (Zeelenberg et al., 1998a). "Regret stems from bad decisions" (Zeelenberg et al., 1998a). '''Anger''' : [https://en.wikipedia.org/wiki/Anger Anger] is a simple negative emotion that occurs when you cannot achieve your goals and you blame someone or something else for it (Lelieveld et al., 2011). Anger can be the result of disappointment (van Dijk et al., 1999). '''Disillusionment''' : [https://en.wiktionary.org/wiki/disillusionment Disillusionment] is a complex negative emotion that occurs when you realise that what you believe or know is false (Maher et al., 2020). Disappointment is a key feature of disillusionment. </div> {{Robelbox/close}} Test yourself:<quiz display="simple"> {Mary's boss received a complaint from a customer about Mary. Mary was made aware of the complaint and then fired. Mary is likely to experience: |type="()"} + Person-related disappointment - Outcome-related disappointment {Alex is trying to get a snack from a vending machine. Alex put their money into the vending machine and typed in the code for lemonade. The vending machine did not give Alex lemonade, and took their money. Alex is likely to experience: |type="()"} - Person-related disappointment + Outcome-related disappointment </quiz> == What causes disappointment? == * Disappointment occurs when expectations are not met (Chua et al., 2009). * The two major theories of disappointment will be discussed. * Brain regions that have shown to be important in decision making/disappointment are the anterior insula, and various regions of the prefrontal cortex (Chua et al., 2009: Mohr et al., 2010: Kalat, 2019). === Theories of disappointment === * The two major theories of disappointment are Bell's theory of disappointment and Loomes and Sudgen's theory of disappointment (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). * Both theories are similar. Loomes and Sugden's theory came about due to their research and theory of regret (Loomes & Sugden, 1986). ==== Bell's theory of disappointment (1985) ==== * h * h ==== Loomes and Sugden's theory of disappointment (1986) ==== * j * h === Neuropsychology of disappointment === ==== Subcortical structure: Anterior insula ==== [[File:Insula structure.png|alt=Structure of the three sections of the insula|thumb|''Figure 2.'' Insula]] * When individuals experience disappointment their anterior insula is activated (Chua et al., 2009). See figure 2. * The anterior insula is active in the presence of potential loss, a part of the brain known for processing emotions such as disappointment (Mohr et al., 2010) ==== Cortical structure: Prefrontal cortex ==== [[File:Prefrontal cortex (left) animation.gif|alt=Rotating skull containing left Prefrontal cortex. The prefrontal cortex is highlighted |thumb|''Figure 3.'' Left hemisphere prefrontal cortex]] [[File:Cortical midline structures.png|thumb|''Figure 4.'' Brain image highlighting various cortical regions, including the ventromedial prefrontal cortex (VMPFC), dorsomedial prefrontal cortex (DMPFC), and the orbitofrontal cortex (OFC/MOPFC).]] * prefrontal cortex - middle 'zone': <u>emotional reactions</u>, cognitive control, and working memory (Kalat, 2019). * prefrontal cortex - anterior (front): decision making, evaluating which course will provide the best outcome, probability of achieving a good outcome (Kalat, 2019). * Prefrontal cortex - see figure 3. * The ventromedial prefrontal cortex [VMPFC] learns what choices are more beneficial for the individual and adjusts decision making accordingly (Kalat, 2019). It also monitors confidence in one's decisions (Kalat, 2019). Individuals with damage to their VMPFC tend to make impulsive decisions based on probability, rather than making considered decisions based on reality (Kalat, 2019). *Orbitofrontal cortex [OFC]- responds to information from the VMPFC based on how the expected reward compares to other possible choices (Kalat, 2019). The OFC changes and updates the expected outcomes of actions based on current circumstances (Kalat, 2019). *When individuals experience disappointment their dorsomedial prefrontal cortex [DMPFC] is activated (Chua et al., 2009) *See figure 4 for VMPFC, OFC, and DMPFC. Insert Quiz on brain regions and how they contribute to decision-making/disappointment OR insert table with brain region - what it does - how it contributes to disappointment ==How can disappointment be managed?== * Disappointment can be managed by lowering expectations (van Dijk et al., 2003). The ventromedial prefrontal cortex monitors how confident you should be about a decision (Kalat, 2019). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ... ^ therefore having lower expectations can improve wellbeing === The benefits of disappointment === * Expressing disappointment encourages higher levels of cooperation in others (Olekalns & Druckman, 2014). * Expressing disappointment can lead to helping behaviour in others (Johnson & Connelly, 2014). === The downsides to disappointment === * Disappointment can taint relationships, leading to blaming or [[wikipedia:Scapegoating|scapegoating]] (Clancy et al., 2012). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ==Conclusion== * Restate what disappointment is, the types of disappointment, and emotions similar to disappointment. * Restate the causes of disappointment * Restate how disappointment can be managed * Take home message: everybody experiences disappointment, but there are things you can do to minimise experiencing it and you can even take advantage of it. == See also == * [[wikipedia:Disappointment|Disappointment]] (Wikipedia) * [[Motivation and emotion/Book/2016/Regret|Regret]] (Book chapter, 2016) * [[Motivation and emotion/Book/2011/Anger|Anger]] (Book chapter, 2011) ==References== {{Hanging indent|1= Chua, Gonzalez, R., Taylor, S. F., Welsh, R. C., & Liberzon, I. (2009). Decision-related loss: Regret and disappointment. ''NeuroImage, 47''(4), 2031–2040. https://doi.org/10.1016/j.neuroimage.2009.06.006 Clancy, A., Vince, R., & Gabriel, Y. (2012). That Unwanted Feeling: A Psychodynamic Study of Disappointment in Organizations. ''British Journal of Management, 23''(4), 518–531. https://doi.org/10.1111/j.1467-8551.2011.00780.x de Meza, D., & Dawson, C. (2021). Neither an Optimist Nor a Pessimist Be: Mistaken Expectations Lower Well-Being. ''Personality & Social Psychology Bulletin'', ''47''(4), 540–550. https://doi.org/10.1177/0146167220934577 Izard, C. E. (2010). The Many Meanings/Aspects of Emotion: Definitions, Functions, Activation, and Regulation. ''Emotion Review, 2''(4), 363–370. https://doi.org/10.1177/1754073910374661 Johnson, G., & Connelly, S. (2014). Negative emotions in informal feedback: The benefits of disappointment and drawbacks of anger. ''Human Relations (New York), 67''(10), 1265–1290. https://doi.org/10.1177/0018726714532856 Kalat, J. W. (2019). ''Biological psychology'' (13th ed.). Boston, MA, USA: Cengage Lelieveld, G. J., Van Dijk, E., Van Beest, I., Steinel, W., & Van Kleef, G. A. (2011). Disappointed in you, angry about your offer: Distinct negative emotions induce concessions via different mechanisms. ''Journal of Experimental Social Psychology, 47''(3), 635–641. https://doi.org/10.1016/j.jesp.2010.12.015 Loomes, G., & Sugden, R. (1986). Disappointment and dynamic consistency in choice under uncertainty. ''The Review of Economic Studies, 53''(2), 271–282. https://doi.org/10.2307/2297651 Maher, P. J., Igou, E. R., & van Tilburg, W. A. P. (2020). Disillusionment: a prototype analysis. ''Cognition and Emotion, 34''(5), 947–959. https://doi.org/10.1080/02699931.2019.1705764 Olekalns, M., & Druckman, D. (2014). With Feeling: How Emotions Shape Negotiation. ''Negotiation Journal, 30''(4), 455–478. https://doi.org/10.1111/nejo.12071 Mohr, P. N. C., Biele, G., & Heekeren, H. R. (2010). Neural processing of risk. ''The Journal of Neuroscience, 30''(19), 6613–6619. https://doi.org/10.1523/JNEUROSCI.0003-10.2010 Ramachandran, V.S., & Jalal, B. (2017). The Evolutionary Psychology of Envy and Jealousy. ''Frontiers in Psychology, 8'', 1619–1619. https://doi.org/10.3389/fpsyg.2017.01619 Reeve, J. (2018). ''Understanding motivation and emotion'' (7th ed.). Hoboken, NJ: Wiley van Dijk, W. W., & Zeelenberg, M. (2002). What do we talk about when we talk about disappointment? Distinguishing outcome-related disappointment from person-related disappointment. ''Cognition and Emotion, 16''(6), 787–807. https://doi.org/10.1080/02699930143000563 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (1999). Not Having What You Want versus Having What You Do Not Want: The Impact of Type of Negative Outcome on the Experience of Disappointment and Related Emotions. ''Cognition and Emotion, 13''(2), 129–148. https://doi.org/10.1080/026999399379302 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (2003). Blessed are those who expect nothing: Lowering expectations as a way of avoiding disappointment. ''Journal of Economic Psychology, 24''(4), 505–516. https://doi.org/10.1016/S0167-4870(02)00211-8 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (1998a). The Experience of Regret and Disappointment. ''Cognition and Emotion, 12''(2), 221–230. https://doi.org/10.1080/026999398379727 Zeelenberg, M., van Dijk, W. W., van der Pligt, J., Manstead, A. S. ., van Empelen, P., & Reinderman, D. (1998b). Emotional Reactions to the Outcomes of Decisions: The Role of Counterfactual Thought in the Experience of Regret and Disappointment. ''Organizational Behavior and Human Decision Processes, 75''(2), 117–141. https://doi.org/10.1006/obhd.1998.2784 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (2000). On bad decisions and disconfirmed expectancies: The psychology of regret and disappointment. ''Cognition and Emotion, 14''(4), 521–541. https://doi.org/10.1080/026999300402781 }} ==External links== * [https://www.youtube.com/watch?v=8KgUFMN7aJQ The value of disappointment] (TEDxPCC) * [https://www.youtube.com/watch?v=gAMbkJk6gnE Feeling all the feels: Crash course psychology #25] (YouTube) 4p6y1bo8hbflrvnrwj987uzhorpndby 2416425 2416424 2022-08-19T03:50:31Z U3216256 2942574 /* What causes disappointment? */ wikitext text/x-wiki {{title|Disappointment<br>What is disappointment, what causes it, and how can it be managed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== [[File:Wayuu woman with sad face in the market buying.jpg|alt=Older woman with a disappointed look on her face in a busy market place.|thumb|''Figure 1.'' Person expressing disappointment through facial features]] * Example (case study) of disappointment - include statistics on frequency of occurrence (Van Dijk & Zeelenberg, 2002) * Define [[wikipedia:Emotion|emotions]], what emotions involve. (Reeve, 2018; Izard, 2010) Reference Figure 1 * Overview of what the chapter explores * Note on emotion research (Ramachandran & Jalal, 2017) {{RoundBoxTop|theme=13}} '''Key questions:''' * What is disappointment? * What causes disappointment? * How can disappointment be managed? {{RoundBoxBottom}} ==What is disappointment?== * Disappointment occurs when the outcome that has occurred is not as good as the outcome that could have occurred, or when your expectations are not met (Zeelenberg et al., 1998a). * Disappointment involves feeling powerless, a tendency to remove oneself from the situation, and a desire to do nothing (van Dijk et al., 1999) * Disappointment is a complex emotion (Ramachandran & Jalal, 2017) * Disappointment can be categorised as outcome-related or person-related; where outcome-related disappointment (ORD) occurs when the expected pleasurable outcome has not occurred, and person-related disappointment (PRD) occurs when you attribute the undesirable outcome to a\ person (you blame them for the 'bad' outcome) (Van Dijk & Zeelenberg, 2002). {{Robelbox|theme={{{theme|2}}}|title=Emotions similar to disappointment}} <div style="{{Robelbox/pad}}"> '''Regret''' : [https://en.wikipedia.org/wiki/Regret Regret] is a negative emotion that occurs when you know that the outcome that occurred could have been better if you made a different choice (Zeelenberg et al., 1998a). "Regret stems from bad decisions" (Zeelenberg et al., 1998a). '''Anger''' : [https://en.wikipedia.org/wiki/Anger Anger] is a simple negative emotion that occurs when you cannot achieve your goals and you blame someone or something else for it (Lelieveld et al., 2011). Anger can be the result of disappointment (van Dijk et al., 1999). '''Disillusionment''' : [https://en.wiktionary.org/wiki/disillusionment Disillusionment] is a complex negative emotion that occurs when you realise that what you believe or know is false (Maher et al., 2020). Disappointment is a key feature of disillusionment. </div> {{Robelbox/close}} Test yourself:<quiz display="simple"> {Mary's boss received a complaint from a customer about Mary. Mary was made aware of the complaint and then fired. Mary is likely to experience: |type="()"} + Person-related disappointment - Outcome-related disappointment {Alex is trying to get a snack from a vending machine. Alex put their money into the vending machine and typed in the code for lemonade. The vending machine did not give Alex lemonade, and took their money. Alex is likely to experience: |type="()"} - Person-related disappointment + Outcome-related disappointment </quiz> == What causes disappointment? == * Disappointment occurs when expectations are not met (Chua et al., 2009). * The two major theories of disappointment will be discussed. * Brain regions that have shown to be important in decision making/disappointment will be discussed (anterior insula, and various regions of the prefrontal cortex) (Chua et al., 2009: Mohr et al., 2010: Kalat, 2019). === Theories of disappointment === * The two major theories of disappointment are Bell's theory of disappointment and Loomes and Sudgen's theory of disappointment (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). * Both theories are similar. Loomes and Sugden's theory came about due to their research and theory of regret (Loomes & Sugden, 1986). ==== Bell's theory of disappointment (1985) ==== * h * h ==== Loomes and Sugden's theory of disappointment (1986) ==== * j * h === Neuropsychology of disappointment === ==== Subcortical structure: Anterior insula ==== [[File:Insula structure.png|alt=Structure of the three sections of the insula|thumb|''Figure 2.'' Insula]] * When individuals experience disappointment their anterior insula is activated (Chua et al., 2009). See figure 2. * The anterior insula is active in the presence of potential loss, a part of the brain known for processing emotions such as disappointment (Mohr et al., 2010) ==== Cortical structure: Prefrontal cortex ==== [[File:Prefrontal cortex (left) animation.gif|alt=Rotating skull containing left Prefrontal cortex. The prefrontal cortex is highlighted |thumb|''Figure 3.'' Left hemisphere prefrontal cortex]] [[File:Cortical midline structures.png|thumb|''Figure 4.'' Brain image highlighting various cortical regions, including the ventromedial prefrontal cortex (VMPFC), dorsomedial prefrontal cortex (DMPFC), and the orbitofrontal cortex (OFC/MOPFC).]] * prefrontal cortex - middle 'zone': <u>emotional reactions</u>, cognitive control, and working memory (Kalat, 2019). * prefrontal cortex - anterior (front): decision making, evaluating which course will provide the best outcome, probability of achieving a good outcome (Kalat, 2019). * Prefrontal cortex - see figure 3. * The ventromedial prefrontal cortex [VMPFC] learns what choices are more beneficial for the individual and adjusts decision making accordingly (Kalat, 2019). It also monitors confidence in one's decisions (Kalat, 2019). Individuals with damage to their VMPFC tend to make impulsive decisions based on probability, rather than making considered decisions based on reality (Kalat, 2019). *Orbitofrontal cortex [OFC]- responds to information from the VMPFC based on how the expected reward compares to other possible choices (Kalat, 2019). The OFC changes and updates the expected outcomes of actions based on current circumstances (Kalat, 2019). *When individuals experience disappointment their dorsomedial prefrontal cortex [DMPFC] is activated (Chua et al., 2009) *See figure 4 for VMPFC, OFC, and DMPFC. Insert Quiz on brain regions and how they contribute to decision-making/disappointment OR insert table with brain region - what it does - how it contributes to disappointment ==How can disappointment be managed?== * Disappointment can be managed by lowering expectations (van Dijk et al., 2003). The ventromedial prefrontal cortex monitors how confident you should be about a decision (Kalat, 2019). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]], therefore having lower expectations can improve wellbeing (de Meza & Dawson, 2021). === The benefits of disappointment === * Expressing disappointment encourages higher levels of cooperation in others (Olekalns & Druckman, 2014). * Expressing disappointment can lead to helping behaviour in others (Johnson & Connelly, 2014). === The downsides to disappointment === * Disappointment can taint relationships, leading to blaming or [[wikipedia:Scapegoating|scapegoating]] (Clancy et al., 2012). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ==Conclusion== * Restate what disappointment is, the types of disappointment, and emotions similar to disappointment. * Restate the causes of disappointment * Restate how disappointment can be managed * Take home message: everybody experiences disappointment, but there are things you can do to minimise experiencing it and you can even take advantage of it. == See also == * [[wikipedia:Disappointment|Disappointment]] (Wikipedia) * [[Motivation and emotion/Book/2016/Regret|Regret]] (Book chapter, 2016) * [[Motivation and emotion/Book/2011/Anger|Anger]] (Book chapter, 2011) ==References== {{Hanging indent|1= Chua, Gonzalez, R., Taylor, S. F., Welsh, R. C., & Liberzon, I. (2009). Decision-related loss: Regret and disappointment. ''NeuroImage, 47''(4), 2031–2040. https://doi.org/10.1016/j.neuroimage.2009.06.006 Clancy, A., Vince, R., & Gabriel, Y. (2012). That Unwanted Feeling: A Psychodynamic Study of Disappointment in Organizations. ''British Journal of Management, 23''(4), 518–531. https://doi.org/10.1111/j.1467-8551.2011.00780.x de Meza, D., & Dawson, C. (2021). Neither an Optimist Nor a Pessimist Be: Mistaken Expectations Lower Well-Being. ''Personality & Social Psychology Bulletin'', ''47''(4), 540–550. https://doi.org/10.1177/0146167220934577 Izard, C. E. (2010). The Many Meanings/Aspects of Emotion: Definitions, Functions, Activation, and Regulation. ''Emotion Review, 2''(4), 363–370. https://doi.org/10.1177/1754073910374661 Johnson, G., & Connelly, S. (2014). Negative emotions in informal feedback: The benefits of disappointment and drawbacks of anger. ''Human Relations (New York), 67''(10), 1265–1290. https://doi.org/10.1177/0018726714532856 Kalat, J. W. (2019). ''Biological psychology'' (13th ed.). Boston, MA, USA: Cengage Lelieveld, G. J., Van Dijk, E., Van Beest, I., Steinel, W., & Van Kleef, G. A. (2011). Disappointed in you, angry about your offer: Distinct negative emotions induce concessions via different mechanisms. ''Journal of Experimental Social Psychology, 47''(3), 635–641. https://doi.org/10.1016/j.jesp.2010.12.015 Loomes, G., & Sugden, R. (1986). Disappointment and dynamic consistency in choice under uncertainty. ''The Review of Economic Studies, 53''(2), 271–282. https://doi.org/10.2307/2297651 Maher, P. J., Igou, E. R., & van Tilburg, W. A. P. (2020). Disillusionment: a prototype analysis. ''Cognition and Emotion, 34''(5), 947–959. https://doi.org/10.1080/02699931.2019.1705764 Olekalns, M., & Druckman, D. (2014). With Feeling: How Emotions Shape Negotiation. ''Negotiation Journal, 30''(4), 455–478. https://doi.org/10.1111/nejo.12071 Mohr, P. N. C., Biele, G., & Heekeren, H. R. (2010). Neural processing of risk. ''The Journal of Neuroscience, 30''(19), 6613–6619. https://doi.org/10.1523/JNEUROSCI.0003-10.2010 Ramachandran, V.S., & Jalal, B. (2017). The Evolutionary Psychology of Envy and Jealousy. ''Frontiers in Psychology, 8'', 1619–1619. https://doi.org/10.3389/fpsyg.2017.01619 Reeve, J. (2018). ''Understanding motivation and emotion'' (7th ed.). Hoboken, NJ: Wiley van Dijk, W. W., & Zeelenberg, M. (2002). What do we talk about when we talk about disappointment? Distinguishing outcome-related disappointment from person-related disappointment. ''Cognition and Emotion, 16''(6), 787–807. https://doi.org/10.1080/02699930143000563 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (1999). Not Having What You Want versus Having What You Do Not Want: The Impact of Type of Negative Outcome on the Experience of Disappointment and Related Emotions. ''Cognition and Emotion, 13''(2), 129–148. https://doi.org/10.1080/026999399379302 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (2003). Blessed are those who expect nothing: Lowering expectations as a way of avoiding disappointment. ''Journal of Economic Psychology, 24''(4), 505–516. https://doi.org/10.1016/S0167-4870(02)00211-8 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (1998a). The Experience of Regret and Disappointment. ''Cognition and Emotion, 12''(2), 221–230. https://doi.org/10.1080/026999398379727 Zeelenberg, M., van Dijk, W. W., van der Pligt, J., Manstead, A. S. ., van Empelen, P., & Reinderman, D. (1998b). Emotional Reactions to the Outcomes of Decisions: The Role of Counterfactual Thought in the Experience of Regret and Disappointment. ''Organizational Behavior and Human Decision Processes, 75''(2), 117–141. https://doi.org/10.1006/obhd.1998.2784 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (2000). On bad decisions and disconfirmed expectancies: The psychology of regret and disappointment. ''Cognition and Emotion, 14''(4), 521–541. https://doi.org/10.1080/026999300402781 }} ==External links== * [https://www.youtube.com/watch?v=8KgUFMN7aJQ The value of disappointment] (TEDxPCC) * [https://www.youtube.com/watch?v=gAMbkJk6gnE Feeling all the feels: Crash course psychology #25] (YouTube) ekgllzssrsk1eas3usjp4lezxv3040j 2416427 2416425 2022-08-19T04:14:38Z U3216256 2942574 /* Theories of disappointment */ wikitext text/x-wiki {{title|Disappointment<br>What is disappointment, what causes it, and how can it be managed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== [[File:Wayuu woman with sad face in the market buying.jpg|alt=Older woman with a disappointed look on her face in a busy market place.|thumb|''Figure 1.'' Person expressing disappointment through facial features]] * Example (case study) of disappointment - include statistics on frequency of occurrence (Van Dijk & Zeelenberg, 2002) * Define [[wikipedia:Emotion|emotions]], what emotions involve. (Reeve, 2018; Izard, 2010) Reference Figure 1 * Overview of what the chapter explores * Note on emotion research (Ramachandran & Jalal, 2017) {{RoundBoxTop|theme=13}} '''Key questions:''' * What is disappointment? * What causes disappointment? * How can disappointment be managed? {{RoundBoxBottom}} ==What is disappointment?== * Disappointment occurs when the outcome that has occurred is not as good as the outcome that could have occurred, or when your expectations are not met (Zeelenberg et al., 1998a). * Disappointment involves feeling powerless, a tendency to remove oneself from the situation, and a desire to do nothing (van Dijk et al., 1999) * Disappointment is a complex emotion (Ramachandran & Jalal, 2017) * Disappointment can be categorised as outcome-related or person-related; where outcome-related disappointment (ORD) occurs when the expected pleasurable outcome has not occurred, and person-related disappointment (PRD) occurs when you attribute the undesirable outcome to a\ person (you blame them for the 'bad' outcome) (Van Dijk & Zeelenberg, 2002). {{Robelbox|theme={{{theme|2}}}|title=Emotions similar to disappointment}} <div style="{{Robelbox/pad}}"> '''Regret''' : [https://en.wikipedia.org/wiki/Regret Regret] is a negative emotion that occurs when you know that the outcome that occurred could have been better if you made a different choice (Zeelenberg et al., 1998a). "Regret stems from bad decisions" (Zeelenberg et al., 1998a). '''Anger''' : [https://en.wikipedia.org/wiki/Anger Anger] is a simple negative emotion that occurs when you cannot achieve your goals and you blame someone or something else for it (Lelieveld et al., 2011). Anger can be the result of disappointment (van Dijk et al., 1999). '''Disillusionment''' : [https://en.wiktionary.org/wiki/disillusionment Disillusionment] is a complex negative emotion that occurs when you realise that what you believe or know is false (Maher et al., 2020). Disappointment is a key feature of disillusionment. </div> {{Robelbox/close}} Test yourself:<quiz display="simple"> {Mary's boss received a complaint from a customer about Mary. Mary was made aware of the complaint and then fired. Mary is likely to experience: |type="()"} + Person-related disappointment - Outcome-related disappointment {Alex is trying to get a snack from a vending machine. Alex put their money into the vending machine and typed in the code for lemonade. The vending machine did not give Alex lemonade, and took their money. Alex is likely to experience: |type="()"} - Person-related disappointment + Outcome-related disappointment </quiz> == What causes disappointment? == * Disappointment occurs when expectations are not met (Chua et al., 2009). * The two major theories of disappointment will be discussed. * The brain regions that have shown to be important in decision making/disappointment will be discussed. === Theories of disappointment === * The two major theories of disappointment are Bell's theory of disappointment and Loomes and Sudgen's theory of disappointment (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). * Both theories are similar (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). Loomes and Sugden's theory came about due to their research and theory of regret (Loomes & Sugden, 1986). ==== Bell's theory of disappointment (1985) ==== * "Disappointment... is a psychological reaction to an outcome that does not match up with expectations" (Bell, 1985, p.1). * h ==== Loomes and Sugden's theory of disappointment (1986) ==== * "When considering any uncertain prospect, an individual forms some ''prior expectation''... if that consequence falls short of the prior expectation... the individual... experiences some degree of disappointment" (Loomes & Sugden, 1986, p.271). * h === Neuropsychology of disappointment === * Multiple brain regions have shown to be important to disappointment and decision making, namely the anterior insula, and various regions of the prefrontal cortex (Chua et al., 2009: Mohr et al., 2010: Kalat, 2019). * b ==== Subcortical structure: Anterior insula ==== [[File:Insula structure.png|alt=Structure of the three sections of the insula|thumb|''Figure 2.'' Insula]] * When individuals experience disappointment their anterior insula is activated (Chua et al., 2009). See figure 2. * The anterior insula is active in the presence of potential loss and is a part of the brain known for processing emotions such as disappointment (Mohr et al., 2010). ==== Cortical structure: Prefrontal cortex ==== [[File:Prefrontal cortex (left) animation.gif|alt=Rotating skull containing left Prefrontal cortex. The prefrontal cortex is highlighted |thumb|''Figure 3.'' Left hemisphere prefrontal cortex]] [[File:Cortical midline structures.png|thumb|''Figure 4.'' Brain image highlighting various cortical regions, including the ventromedial prefrontal cortex (VMPFC), dorsomedial prefrontal cortex (DMPFC), and the orbitofrontal cortex (OFC/MOPFC).]] * prefrontal cortex - middle 'zone': <u>emotional reactions</u>, cognitive control, and working memory (Kalat, 2019). * prefrontal cortex - anterior (front): decision making, evaluating which course will provide the best outcome, probability of achieving a good outcome (Kalat, 2019). * Prefrontal cortex - see figure 3. * The ventromedial prefrontal cortex [VMPFC] learns what choices are more beneficial for the individual and adjusts decision making accordingly (Kalat, 2019). It also monitors confidence in one's decisions (Kalat, 2019). Individuals with damage to their VMPFC tend to make impulsive decisions based on probability, rather than making considered decisions based on reality (Kalat, 2019). *Orbitofrontal cortex [OFC]- responds to information from the VMPFC based on how the expected reward compares to other possible choices (Kalat, 2019). The OFC changes and updates the expected outcomes of actions based on current circumstances (Kalat, 2019). *When individuals experience disappointment their dorsomedial prefrontal cortex [DMPFC] is activated (Chua et al., 2009) *See figure 4 for VMPFC, OFC, and DMPFC. Insert Quiz on brain regions and how they contribute to decision-making/disappointment OR insert table with brain region - what it does - how it contributes to disappointment ==How can disappointment be managed?== * Disappointment can be managed by lowering expectations (van Dijk et al., 2003). The ventromedial prefrontal cortex monitors how confident you should be about a decision (Kalat, 2019). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]], therefore having lower expectations can improve wellbeing (de Meza & Dawson, 2021). === The benefits of disappointment === * Expressing disappointment encourages higher levels of cooperation in others (Olekalns & Druckman, 2014). * Expressing disappointment can lead to helping behaviour in others (Johnson & Connelly, 2014). === The downsides to disappointment === * Disappointment can taint relationships, leading to blaming or [[wikipedia:Scapegoating|scapegoating]] (Clancy et al., 2012). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ==Conclusion== * Restate what disappointment is, the types of disappointment, and emotions similar to disappointment. * Restate the causes of disappointment * Restate how disappointment can be managed * Take home message: everybody experiences disappointment, but there are things you can do to minimise experiencing it and you can even take advantage of it. == See also == * [[wikipedia:Disappointment|Disappointment]] (Wikipedia) * [[Motivation and emotion/Book/2016/Regret|Regret]] (Book chapter, 2016) * [[Motivation and emotion/Book/2011/Anger|Anger]] (Book chapter, 2011) ==References== {{Hanging indent|1= Bell, D. E. (1985). Disappointment in Decision Making Under Uncertainty. ''Operations Research, 33''(1), 1–27. https://doi.org/10.1287/opre.33.1.1 Chua, Gonzalez, R., Taylor, S. F., Welsh, R. C., & Liberzon, I. (2009). Decision-related loss: Regret and disappointment. ''NeuroImage, 47''(4), 2031–2040. https://doi.org/10.1016/j.neuroimage.2009.06.006 Clancy, A., Vince, R., & Gabriel, Y. (2012). That Unwanted Feeling: A Psychodynamic Study of Disappointment in Organizations. ''British Journal of Management, 23''(4), 518–531. https://doi.org/10.1111/j.1467-8551.2011.00780.x de Meza, D., & Dawson, C. (2021). Neither an Optimist Nor a Pessimist Be: Mistaken Expectations Lower Well-Being. ''Personality & Social Psychology Bulletin'', ''47''(4), 540–550. https://doi.org/10.1177/0146167220934577 Izard, C. E. (2010). The Many Meanings/Aspects of Emotion: Definitions, Functions, Activation, and Regulation. ''Emotion Review, 2''(4), 363–370. https://doi.org/10.1177/1754073910374661 Johnson, G., & Connelly, S. (2014). Negative emotions in informal feedback: The benefits of disappointment and drawbacks of anger. ''Human Relations (New York), 67''(10), 1265–1290. https://doi.org/10.1177/0018726714532856 Kalat, J. W. (2019). ''Biological psychology'' (13th ed.). Boston, MA, USA: Cengage Lelieveld, G. J., Van Dijk, E., Van Beest, I., Steinel, W., & Van Kleef, G. A. (2011). Disappointed in you, angry about your offer: Distinct negative emotions induce concessions via different mechanisms. ''Journal of Experimental Social Psychology, 47''(3), 635–641. https://doi.org/10.1016/j.jesp.2010.12.015 Loomes, G., & Sugden, R. (1986). Disappointment and dynamic consistency in choice under uncertainty. ''The Review of Economic Studies, 53''(2), 271–282. https://doi.org/10.2307/2297651 Maher, P. J., Igou, E. R., & van Tilburg, W. A. P. (2020). Disillusionment: a prototype analysis. ''Cognition and Emotion, 34''(5), 947–959. https://doi.org/10.1080/02699931.2019.1705764 Olekalns, M., & Druckman, D. (2014). With Feeling: How Emotions Shape Negotiation. ''Negotiation Journal, 30''(4), 455–478. https://doi.org/10.1111/nejo.12071 Mohr, P. N. C., Biele, G., & Heekeren, H. R. (2010). Neural processing of risk. ''The Journal of Neuroscience, 30''(19), 6613–6619. https://doi.org/10.1523/JNEUROSCI.0003-10.2010 Ramachandran, V.S., & Jalal, B. (2017). The Evolutionary Psychology of Envy and Jealousy. ''Frontiers in Psychology, 8'', 1619–1619. https://doi.org/10.3389/fpsyg.2017.01619 Reeve, J. (2018). ''Understanding motivation and emotion'' (7th ed.). Hoboken, NJ: Wiley van Dijk, W. W., & Zeelenberg, M. (2002). What do we talk about when we talk about disappointment? Distinguishing outcome-related disappointment from person-related disappointment. ''Cognition and Emotion, 16''(6), 787–807. https://doi.org/10.1080/02699930143000563 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (1999). Not Having What You Want versus Having What You Do Not Want: The Impact of Type of Negative Outcome on the Experience of Disappointment and Related Emotions. ''Cognition and Emotion, 13''(2), 129–148. https://doi.org/10.1080/026999399379302 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (2003). Blessed are those who expect nothing: Lowering expectations as a way of avoiding disappointment. ''Journal of Economic Psychology, 24''(4), 505–516. https://doi.org/10.1016/S0167-4870(02)00211-8 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (1998a). The Experience of Regret and Disappointment. ''Cognition and Emotion, 12''(2), 221–230. https://doi.org/10.1080/026999398379727 Zeelenberg, M., van Dijk, W. W., van der Pligt, J., Manstead, A. S. ., van Empelen, P., & Reinderman, D. (1998b). Emotional Reactions to the Outcomes of Decisions: The Role of Counterfactual Thought in the Experience of Regret and Disappointment. ''Organizational Behavior and Human Decision Processes, 75''(2), 117–141. https://doi.org/10.1006/obhd.1998.2784 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (2000). On bad decisions and disconfirmed expectancies: The psychology of regret and disappointment. ''Cognition and Emotion, 14''(4), 521–541. https://doi.org/10.1080/026999300402781 }} ==External links== * [https://www.youtube.com/watch?v=8KgUFMN7aJQ The value of disappointment] (TEDxPCC) * [https://www.youtube.com/watch?v=gAMbkJk6gnE Feeling all the feels: Crash course psychology #25] (YouTube) jmks25x1apdg25ysixtferhc7emaq8q 2416429 2416427 2022-08-19T04:47:06Z U3216256 2942574 /* Bell's theory of disappointment (1985) */ wikitext text/x-wiki {{title|Disappointment<br>What is disappointment, what causes it, and how can it be managed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== [[File:Wayuu woman with sad face in the market buying.jpg|alt=Older woman with a disappointed look on her face in a busy market place.|thumb|''Figure 1.'' Person expressing disappointment through facial features]] * Example (case study) of disappointment - include statistics on frequency of occurrence (Van Dijk & Zeelenberg, 2002) * Define [[wikipedia:Emotion|emotions]], what emotions involve. (Reeve, 2018; Izard, 2010) Reference Figure 1 * Overview of what the chapter explores * Note on emotion research (Ramachandran & Jalal, 2017) {{RoundBoxTop|theme=13}} '''Key questions:''' * What is disappointment? * What causes disappointment? * How can disappointment be managed? {{RoundBoxBottom}} ==What is disappointment?== * Disappointment occurs when the outcome that has occurred is not as good as the outcome that could have occurred, or when your expectations are not met (Zeelenberg et al., 1998a). * Disappointment involves feeling powerless, a tendency to remove oneself from the situation, and a desire to do nothing (van Dijk et al., 1999) * Disappointment is a complex emotion (Ramachandran & Jalal, 2017) * Disappointment can be categorised as outcome-related or person-related; where outcome-related disappointment (ORD) occurs when the expected pleasurable outcome has not occurred, and person-related disappointment (PRD) occurs when you attribute the undesirable outcome to a\ person (you blame them for the 'bad' outcome) (Van Dijk & Zeelenberg, 2002). {{Robelbox|theme={{{theme|2}}}|title=Emotions similar to disappointment}} <div style="{{Robelbox/pad}}"> '''Regret''' : [https://en.wikipedia.org/wiki/Regret Regret] is a negative emotion that occurs when you know that the outcome that occurred could have been better if you made a different choice (Zeelenberg et al., 1998a). "Regret stems from bad decisions" (Zeelenberg et al., 1998a). '''Anger''' : [https://en.wikipedia.org/wiki/Anger Anger] is a simple negative emotion that occurs when you cannot achieve your goals and you blame someone or something else for it (Lelieveld et al., 2011). Anger can be the result of disappointment (van Dijk et al., 1999). '''Disillusionment''' : [https://en.wiktionary.org/wiki/disillusionment Disillusionment] is a complex negative emotion that occurs when you realise that what you believe or know is false (Maher et al., 2020). Disappointment is a key feature of disillusionment. </div> {{Robelbox/close}} Test yourself:<quiz display="simple"> {Mary's boss received a complaint from a customer about Mary. Mary was made aware of the complaint and then fired. Mary is likely to experience: |type="()"} + Person-related disappointment - Outcome-related disappointment {Alex is trying to get a snack from a vending machine. Alex put their money into the vending machine and typed in the code for lemonade. The vending machine did not give Alex lemonade, and took their money. Alex is likely to experience: |type="()"} - Person-related disappointment + Outcome-related disappointment </quiz> == What causes disappointment? == * Disappointment occurs when expectations are not met (Chua et al., 2009). * The two major theories of disappointment will be discussed. * The brain regions that have shown to be important in decision making/disappointment will be discussed. === Theories of disappointment === * The two major theories of disappointment are Bell's theory of disappointment and Loomes and Sudgen's theory of disappointment (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). * Both theories are similar (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). Loomes and Sugden's theory came about due to their research and theory of regret (Loomes & Sugden, 1986). ==== Bell's theory of disappointment (1985) ==== * "Disappointment... is a psychological reaction to an outcome that does not match up with expectations" (Bell, 1985, p.1). * van Dijk and van der Pligt (1997) conducted five experiments which confirmed Bell's theory; where the experience of disappointment is primarily determined by by the probability of the outcome. It is theorised that individuals who experience disappointment focus on the features of the situation which caused the disappointment - one of the features could be the probability with which the disappointing outcome occurred (van Dijk & van der Pligt, 1997). ==== Loomes and Sugden's theory of disappointment (1986) ==== * "When considering any uncertain prospect, an individual forms some ''prior expectation''... if that consequence falls short of the prior expectation... the individual... experiences some degree of disappointment" (Loomes & Sugden, 1986, p.271). * === Neuropsychology of disappointment === * Multiple brain regions have shown to be important to disappointment and decision making, namely the anterior insula, and various regions of the prefrontal cortex (Chua et al., 2009: Mohr et al., 2010: Kalat, 2019). * b ==== Subcortical structure: Anterior insula ==== [[File:Insula structure.png|alt=Structure of the three sections of the insula|thumb|''Figure 2.'' Insula]] * When individuals experience disappointment their anterior insula is activated (Chua et al., 2009). See figure 2. * The anterior insula is active in the presence of potential loss and is a part of the brain known for processing emotions such as disappointment (Mohr et al., 2010). ==== Cortical structure: Prefrontal cortex ==== [[File:Prefrontal cortex (left) animation.gif|alt=Rotating skull containing left Prefrontal cortex. The prefrontal cortex is highlighted |thumb|''Figure 3.'' Left hemisphere prefrontal cortex]] [[File:Cortical midline structures.png|thumb|''Figure 4.'' Brain image highlighting various cortical regions, including the ventromedial prefrontal cortex (VMPFC), dorsomedial prefrontal cortex (DMPFC), and the orbitofrontal cortex (OFC/MOPFC).]] * prefrontal cortex - middle 'zone': <u>emotional reactions</u>, cognitive control, and working memory (Kalat, 2019). * prefrontal cortex - anterior (front): decision making, evaluating which course will provide the best outcome, probability of achieving a good outcome (Kalat, 2019). * Prefrontal cortex - see figure 3. * The ventromedial prefrontal cortex [VMPFC] learns what choices are more beneficial for the individual and adjusts decision making accordingly (Kalat, 2019). It also monitors confidence in one's decisions (Kalat, 2019). Individuals with damage to their VMPFC tend to make impulsive decisions based on probability, rather than making considered decisions based on reality (Kalat, 2019). *Orbitofrontal cortex [OFC]- responds to information from the VMPFC based on how the expected reward compares to other possible choices (Kalat, 2019). The OFC changes and updates the expected outcomes of actions based on current circumstances (Kalat, 2019). *When individuals experience disappointment their dorsomedial prefrontal cortex [DMPFC] is activated (Chua et al., 2009) *See figure 4 for VMPFC, OFC, and DMPFC. Insert Quiz on brain regions and how they contribute to decision-making/disappointment OR insert table with brain region - what it does - how it contributes to disappointment ==How can disappointment be managed?== * Disappointment can be managed by lowering expectations (van Dijk et al., 2003). The ventromedial prefrontal cortex monitors how confident you should be about a decision (Kalat, 2019). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]], therefore having lower expectations can improve wellbeing (de Meza & Dawson, 2021). === The benefits of disappointment === * Expressing disappointment encourages higher levels of cooperation in others (Olekalns & Druckman, 2014). * Expressing disappointment can lead to helping behaviour in others (Johnson & Connelly, 2014). === The downsides to disappointment === * Disappointment can taint relationships, leading to blaming or [[wikipedia:Scapegoating|scapegoating]] (Clancy et al., 2012). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ==Conclusion== * Restate what disappointment is, the types of disappointment, and emotions similar to disappointment. * Restate the causes of disappointment * Restate how disappointment can be managed * Take home message: everybody experiences disappointment, but there are things you can do to minimise experiencing it and you can even take advantage of it. == See also == * [[wikipedia:Disappointment|Disappointment]] (Wikipedia) * [[Motivation and emotion/Book/2016/Regret|Regret]] (Book chapter, 2016) * [[Motivation and emotion/Book/2011/Anger|Anger]] (Book chapter, 2011) ==References== {{Hanging indent|1= Bell, D. E. (1985). Disappointment in Decision Making Under Uncertainty. ''Operations Research, 33''(1), 1–27. https://doi.org/10.1287/opre.33.1.1 Chua, Gonzalez, R., Taylor, S. F., Welsh, R. C., & Liberzon, I. (2009). Decision-related loss: Regret and disappointment. ''NeuroImage, 47''(4), 2031–2040. https://doi.org/10.1016/j.neuroimage.2009.06.006 Clancy, A., Vince, R., & Gabriel, Y. (2012). That Unwanted Feeling: A Psychodynamic Study of Disappointment in Organizations. ''British Journal of Management, 23''(4), 518–531. https://doi.org/10.1111/j.1467-8551.2011.00780.x de Meza, D., & Dawson, C. (2021). Neither an Optimist Nor a Pessimist Be: Mistaken Expectations Lower Well-Being. ''Personality & Social Psychology Bulletin'', ''47''(4), 540–550. https://doi.org/10.1177/0146167220934577 Izard, C. E. (2010). The Many Meanings/Aspects of Emotion: Definitions, Functions, Activation, and Regulation. ''Emotion Review, 2''(4), 363–370. https://doi.org/10.1177/1754073910374661 Johnson, G., & Connelly, S. (2014). Negative emotions in informal feedback: The benefits of disappointment and drawbacks of anger. ''Human Relations (New York), 67''(10), 1265–1290. https://doi.org/10.1177/0018726714532856 Kalat, J. W. (2019). ''Biological psychology'' (13th ed.). Boston, MA, USA: Cengage Lelieveld, G. J., Van Dijk, E., Van Beest, I., Steinel, W., & Van Kleef, G. A. (2011). Disappointed in you, angry about your offer: Distinct negative emotions induce concessions via different mechanisms. ''Journal of Experimental Social Psychology, 47''(3), 635–641. https://doi.org/10.1016/j.jesp.2010.12.015 Loomes, G., & Sugden, R. (1986). Disappointment and dynamic consistency in choice under uncertainty. ''The Review of Economic Studies, 53''(2), 271–282. https://doi.org/10.2307/2297651 Maher, P. J., Igou, E. R., & van Tilburg, W. A. P. (2020). Disillusionment: a prototype analysis. ''Cognition and Emotion, 34''(5), 947–959. https://doi.org/10.1080/02699931.2019.1705764 Olekalns, M., & Druckman, D. (2014). With Feeling: How Emotions Shape Negotiation. ''Negotiation Journal, 30''(4), 455–478. https://doi.org/10.1111/nejo.12071 Mohr, P. N. C., Biele, G., & Heekeren, H. R. (2010). Neural processing of risk. ''The Journal of Neuroscience, 30''(19), 6613–6619. https://doi.org/10.1523/JNEUROSCI.0003-10.2010 Ramachandran, V.S., & Jalal, B. (2017). The Evolutionary Psychology of Envy and Jealousy. ''Frontiers in Psychology, 8'', 1619–1619. https://doi.org/10.3389/fpsyg.2017.01619 Reeve, J. (2018). ''Understanding motivation and emotion'' (7th ed.). Hoboken, NJ: Wiley van Dijk, W. W., & van der Pligt, J. (1997). The Impact of Probability and Magnitude of Outcome on Disappointment and Elation. ''Organizational Behavior and Human Decision Processes, 69''(3), 277–284. https://doi.org/10.1006/obhd.1997.2688 van Dijk, W. W., & Zeelenberg, M. (2002). What do we talk about when we talk about disappointment? Distinguishing outcome-related disappointment from person-related disappointment. ''Cognition and Emotion, 16''(6), 787–807. https://doi.org/10.1080/02699930143000563 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (1999). Not Having What You Want versus Having What You Do Not Want: The Impact of Type of Negative Outcome on the Experience of Disappointment and Related Emotions. ''Cognition and Emotion, 13''(2), 129–148. https://doi.org/10.1080/026999399379302 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (2003). Blessed are those who expect nothing: Lowering expectations as a way of avoiding disappointment. ''Journal of Economic Psychology, 24''(4), 505–516. https://doi.org/10.1016/S0167-4870(02)00211-8 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (1998a). The Experience of Regret and Disappointment. ''Cognition and Emotion, 12''(2), 221–230. https://doi.org/10.1080/026999398379727 Zeelenberg, M., van Dijk, W. W., van der Pligt, J., Manstead, A. S. ., van Empelen, P., & Reinderman, D. (1998b). Emotional Reactions to the Outcomes of Decisions: The Role of Counterfactual Thought in the Experience of Regret and Disappointment. ''Organizational Behavior and Human Decision Processes, 75''(2), 117–141. https://doi.org/10.1006/obhd.1998.2784 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (2000). On bad decisions and disconfirmed expectancies: The psychology of regret and disappointment. ''Cognition and Emotion, 14''(4), 521–541. https://doi.org/10.1080/026999300402781 }} ==External links== * [https://www.youtube.com/watch?v=8KgUFMN7aJQ The value of disappointment] (TEDxPCC) * [https://www.youtube.com/watch?v=gAMbkJk6gnE Feeling all the feels: Crash course psychology #25] (YouTube) pqf7sbrv8gnoodt5y604gz5lcl8v8v7 2416432 2416429 2022-08-19T04:52:31Z U3216256 2942574 /* How can disappointment be managed? */ wikitext text/x-wiki {{title|Disappointment<br>What is disappointment, what causes it, and how can it be managed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== [[File:Wayuu woman with sad face in the market buying.jpg|alt=Older woman with a disappointed look on her face in a busy market place.|thumb|''Figure 1.'' Person expressing disappointment through facial features]] * Example (case study) of disappointment - include statistics on frequency of occurrence (Van Dijk & Zeelenberg, 2002) * Define [[wikipedia:Emotion|emotions]], what emotions involve. (Reeve, 2018; Izard, 2010) Reference Figure 1 * Overview of what the chapter explores * Note on emotion research (Ramachandran & Jalal, 2017) {{RoundBoxTop|theme=13}} '''Key questions:''' * What is disappointment? * What causes disappointment? * How can disappointment be managed? {{RoundBoxBottom}} ==What is disappointment?== * Disappointment occurs when the outcome that has occurred is not as good as the outcome that could have occurred, or when your expectations are not met (Zeelenberg et al., 1998a). * Disappointment involves feeling powerless, a tendency to remove oneself from the situation, and a desire to do nothing (van Dijk et al., 1999) * Disappointment is a complex emotion (Ramachandran & Jalal, 2017) * Disappointment can be categorised as outcome-related or person-related; where outcome-related disappointment (ORD) occurs when the expected pleasurable outcome has not occurred, and person-related disappointment (PRD) occurs when you attribute the undesirable outcome to a\ person (you blame them for the 'bad' outcome) (Van Dijk & Zeelenberg, 2002). {{Robelbox|theme={{{theme|2}}}|title=Emotions similar to disappointment}} <div style="{{Robelbox/pad}}"> '''Regret''' : [https://en.wikipedia.org/wiki/Regret Regret] is a negative emotion that occurs when you know that the outcome that occurred could have been better if you made a different choice (Zeelenberg et al., 1998a). "Regret stems from bad decisions" (Zeelenberg et al., 1998a). '''Anger''' : [https://en.wikipedia.org/wiki/Anger Anger] is a simple negative emotion that occurs when you cannot achieve your goals and you blame someone or something else for it (Lelieveld et al., 2011). Anger can be the result of disappointment (van Dijk et al., 1999). '''Disillusionment''' : [https://en.wiktionary.org/wiki/disillusionment Disillusionment] is a complex negative emotion that occurs when you realise that what you believe or know is false (Maher et al., 2020). Disappointment is a key feature of disillusionment. </div> {{Robelbox/close}} Test yourself:<quiz display="simple"> {Mary's boss received a complaint from a customer about Mary. Mary was made aware of the complaint and then fired. Mary is likely to experience: |type="()"} + Person-related disappointment - Outcome-related disappointment {Alex is trying to get a snack from a vending machine. Alex put their money into the vending machine and typed in the code for lemonade. The vending machine did not give Alex lemonade, and took their money. Alex is likely to experience: |type="()"} - Person-related disappointment + Outcome-related disappointment </quiz> == What causes disappointment? == * Disappointment occurs when expectations are not met (Chua et al., 2009). * The two major theories of disappointment will be discussed. * The brain regions that have shown to be important in decision making/disappointment will be discussed. === Theories of disappointment === * The two major theories of disappointment are Bell's theory of disappointment and Loomes and Sudgen's theory of disappointment (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). * Both theories are similar (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). Loomes and Sugden's theory came about due to their research and theory of regret (Loomes & Sugden, 1986). ==== Bell's theory of disappointment (1985) ==== * "Disappointment... is a psychological reaction to an outcome that does not match up with expectations" (Bell, 1985, p.1). * van Dijk and van der Pligt (1997) conducted five experiments which confirmed Bell's theory; where the experience of disappointment is primarily determined by by the probability of the outcome. It is theorised that individuals who experience disappointment focus on the features of the situation which caused the disappointment - one of the features could be the probability with which the disappointing outcome occurred (van Dijk & van der Pligt, 1997). ==== Loomes and Sugden's theory of disappointment (1986) ==== * "When considering any uncertain prospect, an individual forms some ''prior expectation''... if that consequence falls short of the prior expectation... the individual... experiences some degree of disappointment" (Loomes & Sugden, 1986, p.271). * === Neuropsychology of disappointment === * Multiple brain regions have shown to be important to disappointment and decision making, namely the anterior insula, and various regions of the prefrontal cortex (Chua et al., 2009: Mohr et al., 2010: Kalat, 2019). * b ==== Subcortical structure: Anterior insula ==== [[File:Insula structure.png|alt=Structure of the three sections of the insula|thumb|''Figure 2.'' Insula]] * When individuals experience disappointment their anterior insula is activated (Chua et al., 2009). See figure 2. * The anterior insula is active in the presence of potential loss and is a part of the brain known for processing emotions such as disappointment (Mohr et al., 2010). ==== Cortical structure: Prefrontal cortex ==== [[File:Prefrontal cortex (left) animation.gif|alt=Rotating skull containing left Prefrontal cortex. The prefrontal cortex is highlighted |thumb|''Figure 3.'' Left hemisphere prefrontal cortex]] [[File:Cortical midline structures.png|thumb|''Figure 4.'' Brain image highlighting various cortical regions, including the ventromedial prefrontal cortex (VMPFC), dorsomedial prefrontal cortex (DMPFC), and the orbitofrontal cortex (OFC/MOPFC).]] * prefrontal cortex - middle 'zone': <u>emotional reactions</u>, cognitive control, and working memory (Kalat, 2019). * prefrontal cortex - anterior (front): decision making, evaluating which course will provide the best outcome, probability of achieving a good outcome (Kalat, 2019). * Prefrontal cortex - see figure 3. * The ventromedial prefrontal cortex [VMPFC] learns what choices are more beneficial for the individual and adjusts decision making accordingly (Kalat, 2019). It also monitors confidence in one's decisions (Kalat, 2019). Individuals with damage to their VMPFC tend to make impulsive decisions based on probability, rather than making considered decisions based on reality (Kalat, 2019). *Orbitofrontal cortex [OFC]- responds to information from the VMPFC based on how the expected reward compares to other possible choices (Kalat, 2019). The OFC changes and updates the expected outcomes of actions based on current circumstances (Kalat, 2019). *When individuals experience disappointment their dorsomedial prefrontal cortex [DMPFC] is activated (Chua et al., 2009) *See figure 4 for VMPFC, OFC, and DMPFC. Insert Quiz on brain regions and how they contribute to decision-making/disappointment OR insert table with brain region - what it does - how it contributes to disappointment ==How can disappointment be managed?== * Disappointment can be managed by lowering expectations (van Dijk et al., 2003). The ventromedial prefrontal cortex monitors how confident you should be about a decision (Kalat, 2019). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]], therefore having lower expectations can improve wellbeing (de Meza & Dawson, 2021). * Disappointment can be managed by successfully living up to expectations; where individuals intensify their efforts to attain a certain outcome so as not to experience disappointment (Zeleenberg et al., 2000). === The benefits of disappointment === * Expressing disappointment encourages higher levels of cooperation in others (Olekalns & Druckman, 2014). * Expressing disappointment can lead to helping behaviour in others (Johnson & Connelly, 2014). === The downsides to disappointment === * Disappointment can taint relationships, leading to blaming or [[wikipedia:Scapegoating|scapegoating]] (Clancy et al., 2012). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ==Conclusion== * Restate what disappointment is, the types of disappointment, and emotions similar to disappointment. * Restate the causes of disappointment * Restate how disappointment can be managed * Take home message: everybody experiences disappointment, but there are things you can do to minimise experiencing it and you can even take advantage of it. == See also == * [[wikipedia:Disappointment|Disappointment]] (Wikipedia) * [[Motivation and emotion/Book/2016/Regret|Regret]] (Book chapter, 2016) * [[Motivation and emotion/Book/2011/Anger|Anger]] (Book chapter, 2011) ==References== {{Hanging indent|1= Bell, D. E. (1985). Disappointment in Decision Making Under Uncertainty. ''Operations Research, 33''(1), 1–27. https://doi.org/10.1287/opre.33.1.1 Chua, Gonzalez, R., Taylor, S. F., Welsh, R. C., & Liberzon, I. (2009). Decision-related loss: Regret and disappointment. ''NeuroImage, 47''(4), 2031–2040. https://doi.org/10.1016/j.neuroimage.2009.06.006 Clancy, A., Vince, R., & Gabriel, Y. (2012). That Unwanted Feeling: A Psychodynamic Study of Disappointment in Organizations. ''British Journal of Management, 23''(4), 518–531. https://doi.org/10.1111/j.1467-8551.2011.00780.x de Meza, D., & Dawson, C. (2021). Neither an Optimist Nor a Pessimist Be: Mistaken Expectations Lower Well-Being. ''Personality & Social Psychology Bulletin'', ''47''(4), 540–550. https://doi.org/10.1177/0146167220934577 Izard, C. E. (2010). The Many Meanings/Aspects of Emotion: Definitions, Functions, Activation, and Regulation. ''Emotion Review, 2''(4), 363–370. https://doi.org/10.1177/1754073910374661 Johnson, G., & Connelly, S. (2014). Negative emotions in informal feedback: The benefits of disappointment and drawbacks of anger. ''Human Relations (New York), 67''(10), 1265–1290. https://doi.org/10.1177/0018726714532856 Kalat, J. W. (2019). ''Biological psychology'' (13th ed.). Boston, MA, USA: Cengage Lelieveld, G. J., Van Dijk, E., Van Beest, I., Steinel, W., & Van Kleef, G. A. (2011). Disappointed in you, angry about your offer: Distinct negative emotions induce concessions via different mechanisms. ''Journal of Experimental Social Psychology, 47''(3), 635–641. https://doi.org/10.1016/j.jesp.2010.12.015 Loomes, G., & Sugden, R. (1986). Disappointment and dynamic consistency in choice under uncertainty. ''The Review of Economic Studies, 53''(2), 271–282. https://doi.org/10.2307/2297651 Maher, P. J., Igou, E. R., & van Tilburg, W. A. P. (2020). Disillusionment: a prototype analysis. ''Cognition and Emotion, 34''(5), 947–959. https://doi.org/10.1080/02699931.2019.1705764 Olekalns, M., & Druckman, D. (2014). With Feeling: How Emotions Shape Negotiation. ''Negotiation Journal, 30''(4), 455–478. https://doi.org/10.1111/nejo.12071 Mohr, P. N. C., Biele, G., & Heekeren, H. R. (2010). Neural processing of risk. ''The Journal of Neuroscience, 30''(19), 6613–6619. https://doi.org/10.1523/JNEUROSCI.0003-10.2010 Ramachandran, V.S., & Jalal, B. (2017). The Evolutionary Psychology of Envy and Jealousy. ''Frontiers in Psychology, 8'', 1619–1619. https://doi.org/10.3389/fpsyg.2017.01619 Reeve, J. (2018). ''Understanding motivation and emotion'' (7th ed.). Hoboken, NJ: Wiley van Dijk, W. W., & van der Pligt, J. (1997). The Impact of Probability and Magnitude of Outcome on Disappointment and Elation. ''Organizational Behavior and Human Decision Processes, 69''(3), 277–284. https://doi.org/10.1006/obhd.1997.2688 van Dijk, W. W., & Zeelenberg, M. (2002). What do we talk about when we talk about disappointment? Distinguishing outcome-related disappointment from person-related disappointment. ''Cognition and Emotion, 16''(6), 787–807. https://doi.org/10.1080/02699930143000563 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (1999). Not Having What You Want versus Having What You Do Not Want: The Impact of Type of Negative Outcome on the Experience of Disappointment and Related Emotions. ''Cognition and Emotion, 13''(2), 129–148. https://doi.org/10.1080/026999399379302 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (2003). Blessed are those who expect nothing: Lowering expectations as a way of avoiding disappointment. ''Journal of Economic Psychology, 24''(4), 505–516. https://doi.org/10.1016/S0167-4870(02)00211-8 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (1998a). The Experience of Regret and Disappointment. ''Cognition and Emotion, 12''(2), 221–230. https://doi.org/10.1080/026999398379727 Zeelenberg, M., van Dijk, W. W., van der Pligt, J., Manstead, A. S. ., van Empelen, P., & Reinderman, D. (1998b). Emotional Reactions to the Outcomes of Decisions: The Role of Counterfactual Thought in the Experience of Regret and Disappointment. ''Organizational Behavior and Human Decision Processes, 75''(2), 117–141. https://doi.org/10.1006/obhd.1998.2784 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (2000). On bad decisions and disconfirmed expectancies: The psychology of regret and disappointment. ''Cognition and Emotion, 14''(4), 521–541. https://doi.org/10.1080/026999300402781 }} ==External links== * [https://www.youtube.com/watch?v=8KgUFMN7aJQ The value of disappointment] (TEDxPCC) * [https://www.youtube.com/watch?v=gAMbkJk6gnE Feeling all the feels: Crash course psychology #25] (YouTube) tdhosa9dd84pjhsod8pz671vbtby5yz 2416435 2416432 2022-08-19T04:53:13Z U3216256 2942574 /* Loomes and Sugden's theory of disappointment (1986) */ wikitext text/x-wiki {{title|Disappointment<br>What is disappointment, what causes it, and how can it be managed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== [[File:Wayuu woman with sad face in the market buying.jpg|alt=Older woman with a disappointed look on her face in a busy market place.|thumb|''Figure 1.'' Person expressing disappointment through facial features]] * Example (case study) of disappointment - include statistics on frequency of occurrence (Van Dijk & Zeelenberg, 2002) * Define [[wikipedia:Emotion|emotions]], what emotions involve. (Reeve, 2018; Izard, 2010) Reference Figure 1 * Overview of what the chapter explores * Note on emotion research (Ramachandran & Jalal, 2017) {{RoundBoxTop|theme=13}} '''Key questions:''' * What is disappointment? * What causes disappointment? * How can disappointment be managed? {{RoundBoxBottom}} ==What is disappointment?== * Disappointment occurs when the outcome that has occurred is not as good as the outcome that could have occurred, or when your expectations are not met (Zeelenberg et al., 1998a). * Disappointment involves feeling powerless, a tendency to remove oneself from the situation, and a desire to do nothing (van Dijk et al., 1999) * Disappointment is a complex emotion (Ramachandran & Jalal, 2017) * Disappointment can be categorised as outcome-related or person-related; where outcome-related disappointment (ORD) occurs when the expected pleasurable outcome has not occurred, and person-related disappointment (PRD) occurs when you attribute the undesirable outcome to a\ person (you blame them for the 'bad' outcome) (Van Dijk & Zeelenberg, 2002). {{Robelbox|theme={{{theme|2}}}|title=Emotions similar to disappointment}} <div style="{{Robelbox/pad}}"> '''Regret''' : [https://en.wikipedia.org/wiki/Regret Regret] is a negative emotion that occurs when you know that the outcome that occurred could have been better if you made a different choice (Zeelenberg et al., 1998a). "Regret stems from bad decisions" (Zeelenberg et al., 1998a). '''Anger''' : [https://en.wikipedia.org/wiki/Anger Anger] is a simple negative emotion that occurs when you cannot achieve your goals and you blame someone or something else for it (Lelieveld et al., 2011). Anger can be the result of disappointment (van Dijk et al., 1999). '''Disillusionment''' : [https://en.wiktionary.org/wiki/disillusionment Disillusionment] is a complex negative emotion that occurs when you realise that what you believe or know is false (Maher et al., 2020). Disappointment is a key feature of disillusionment. </div> {{Robelbox/close}} Test yourself:<quiz display="simple"> {Mary's boss received a complaint from a customer about Mary. Mary was made aware of the complaint and then fired. Mary is likely to experience: |type="()"} + Person-related disappointment - Outcome-related disappointment {Alex is trying to get a snack from a vending machine. Alex put their money into the vending machine and typed in the code for lemonade. The vending machine did not give Alex lemonade, and took their money. Alex is likely to experience: |type="()"} - Person-related disappointment + Outcome-related disappointment </quiz> == What causes disappointment? == * Disappointment occurs when expectations are not met (Chua et al., 2009). * The two major theories of disappointment will be discussed. * The brain regions that have shown to be important in decision making/disappointment will be discussed. === Theories of disappointment === * The two major theories of disappointment are Bell's theory of disappointment and Loomes and Sudgen's theory of disappointment (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). * Both theories are similar (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). Loomes and Sugden's theory came about due to their research and theory of regret (Loomes & Sugden, 1986). ==== Bell's theory of disappointment (1985) ==== * "Disappointment... is a psychological reaction to an outcome that does not match up with expectations" (Bell, 1985, p.1). * van Dijk and van der Pligt (1997) conducted five experiments which confirmed Bell's theory; where the experience of disappointment is primarily determined by by the probability of the outcome. It is theorised that individuals who experience disappointment focus on the features of the situation which caused the disappointment - one of the features could be the probability with which the disappointing outcome occurred (van Dijk & van der Pligt, 1997). ==== Loomes and Sugden's theory of disappointment (1986) ==== * "When considering any uncertain prospect, an individual forms some ''prior expectation''... if that consequence falls short of the prior expectation... the individual... experiences some degree of disappointment" (Loomes & Sugden, 1986, p.271). *Key content (b) === Neuropsychology of disappointment === * Multiple brain regions have shown to be important to disappointment and decision making, namely the anterior insula, and various regions of the prefrontal cortex (Chua et al., 2009: Mohr et al., 2010: Kalat, 2019). * Key content (b) ==== Subcortical structure: Anterior insula ==== [[File:Insula structure.png|alt=Structure of the three sections of the insula|thumb|''Figure 2.'' Insula]] * When individuals experience disappointment their anterior insula is activated (Chua et al., 2009). See figure 2. * The anterior insula is active in the presence of potential loss and is a part of the brain known for processing emotions such as disappointment (Mohr et al., 2010). ==== Cortical structure: Prefrontal cortex ==== [[File:Prefrontal cortex (left) animation.gif|alt=Rotating skull containing left Prefrontal cortex. The prefrontal cortex is highlighted |thumb|''Figure 3.'' Left hemisphere prefrontal cortex]] [[File:Cortical midline structures.png|thumb|''Figure 4.'' Brain image highlighting various cortical regions, including the ventromedial prefrontal cortex (VMPFC), dorsomedial prefrontal cortex (DMPFC), and the orbitofrontal cortex (OFC/MOPFC).]] * prefrontal cortex - middle 'zone': <u>emotional reactions</u>, cognitive control, and working memory (Kalat, 2019). * prefrontal cortex - anterior (front): decision making, evaluating which course will provide the best outcome, probability of achieving a good outcome (Kalat, 2019). * Prefrontal cortex - see figure 3. * The ventromedial prefrontal cortex [VMPFC] learns what choices are more beneficial for the individual and adjusts decision making accordingly (Kalat, 2019). It also monitors confidence in one's decisions (Kalat, 2019). Individuals with damage to their VMPFC tend to make impulsive decisions based on probability, rather than making considered decisions based on reality (Kalat, 2019). *Orbitofrontal cortex [OFC]- responds to information from the VMPFC based on how the expected reward compares to other possible choices (Kalat, 2019). The OFC changes and updates the expected outcomes of actions based on current circumstances (Kalat, 2019). *When individuals experience disappointment their dorsomedial prefrontal cortex [DMPFC] is activated (Chua et al., 2009) *See figure 4 for VMPFC, OFC, and DMPFC. Insert Quiz on brain regions and how they contribute to decision-making/disappointment OR insert table with brain region - what it does - how it contributes to disappointment ==How can disappointment be managed?== * Disappointment can be managed by lowering expectations (van Dijk et al., 2003). The ventromedial prefrontal cortex monitors how confident you should be about a decision (Kalat, 2019). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]], therefore having lower expectations can improve wellbeing (de Meza & Dawson, 2021). * Disappointment can be managed by successfully living up to expectations; where individuals intensify their efforts to attain a certain outcome so as not to experience disappointment (Zeleenberg et al., 2000). === The benefits of disappointment === * Expressing disappointment encourages higher levels of cooperation in others (Olekalns & Druckman, 2014). * Expressing disappointment can lead to helping behaviour in others (Johnson & Connelly, 2014). === The downsides to disappointment === * Disappointment can taint relationships, leading to blaming or [[wikipedia:Scapegoating|scapegoating]] (Clancy et al., 2012). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ==Conclusion== * Restate what disappointment is, the types of disappointment, and emotions similar to disappointment. * Restate the causes of disappointment * Restate how disappointment can be managed * Take home message: everybody experiences disappointment, but there are things you can do to minimise experiencing it and you can even take advantage of it. == See also == * [[wikipedia:Disappointment|Disappointment]] (Wikipedia) * [[Motivation and emotion/Book/2016/Regret|Regret]] (Book chapter, 2016) * [[Motivation and emotion/Book/2011/Anger|Anger]] (Book chapter, 2011) ==References== {{Hanging indent|1= Bell, D. E. (1985). Disappointment in Decision Making Under Uncertainty. ''Operations Research, 33''(1), 1–27. https://doi.org/10.1287/opre.33.1.1 Chua, Gonzalez, R., Taylor, S. F., Welsh, R. C., & Liberzon, I. (2009). Decision-related loss: Regret and disappointment. ''NeuroImage, 47''(4), 2031–2040. https://doi.org/10.1016/j.neuroimage.2009.06.006 Clancy, A., Vince, R., & Gabriel, Y. (2012). That Unwanted Feeling: A Psychodynamic Study of Disappointment in Organizations. ''British Journal of Management, 23''(4), 518–531. https://doi.org/10.1111/j.1467-8551.2011.00780.x de Meza, D., & Dawson, C. (2021). Neither an Optimist Nor a Pessimist Be: Mistaken Expectations Lower Well-Being. ''Personality & Social Psychology Bulletin'', ''47''(4), 540–550. https://doi.org/10.1177/0146167220934577 Izard, C. E. (2010). The Many Meanings/Aspects of Emotion: Definitions, Functions, Activation, and Regulation. ''Emotion Review, 2''(4), 363–370. https://doi.org/10.1177/1754073910374661 Johnson, G., & Connelly, S. (2014). Negative emotions in informal feedback: The benefits of disappointment and drawbacks of anger. ''Human Relations (New York), 67''(10), 1265–1290. https://doi.org/10.1177/0018726714532856 Kalat, J. W. (2019). ''Biological psychology'' (13th ed.). Boston, MA, USA: Cengage Lelieveld, G. J., Van Dijk, E., Van Beest, I., Steinel, W., & Van Kleef, G. A. (2011). Disappointed in you, angry about your offer: Distinct negative emotions induce concessions via different mechanisms. ''Journal of Experimental Social Psychology, 47''(3), 635–641. https://doi.org/10.1016/j.jesp.2010.12.015 Loomes, G., & Sugden, R. (1986). Disappointment and dynamic consistency in choice under uncertainty. ''The Review of Economic Studies, 53''(2), 271–282. https://doi.org/10.2307/2297651 Maher, P. J., Igou, E. R., & van Tilburg, W. A. P. (2020). Disillusionment: a prototype analysis. ''Cognition and Emotion, 34''(5), 947–959. https://doi.org/10.1080/02699931.2019.1705764 Olekalns, M., & Druckman, D. (2014). With Feeling: How Emotions Shape Negotiation. ''Negotiation Journal, 30''(4), 455–478. https://doi.org/10.1111/nejo.12071 Mohr, P. N. C., Biele, G., & Heekeren, H. R. (2010). Neural processing of risk. ''The Journal of Neuroscience, 30''(19), 6613–6619. https://doi.org/10.1523/JNEUROSCI.0003-10.2010 Ramachandran, V.S., & Jalal, B. (2017). The Evolutionary Psychology of Envy and Jealousy. ''Frontiers in Psychology, 8'', 1619–1619. https://doi.org/10.3389/fpsyg.2017.01619 Reeve, J. (2018). ''Understanding motivation and emotion'' (7th ed.). Hoboken, NJ: Wiley van Dijk, W. W., & van der Pligt, J. (1997). The Impact of Probability and Magnitude of Outcome on Disappointment and Elation. ''Organizational Behavior and Human Decision Processes, 69''(3), 277–284. https://doi.org/10.1006/obhd.1997.2688 van Dijk, W. W., & Zeelenberg, M. (2002). What do we talk about when we talk about disappointment? Distinguishing outcome-related disappointment from person-related disappointment. ''Cognition and Emotion, 16''(6), 787–807. https://doi.org/10.1080/02699930143000563 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (1999). Not Having What You Want versus Having What You Do Not Want: The Impact of Type of Negative Outcome on the Experience of Disappointment and Related Emotions. ''Cognition and Emotion, 13''(2), 129–148. https://doi.org/10.1080/026999399379302 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (2003). Blessed are those who expect nothing: Lowering expectations as a way of avoiding disappointment. ''Journal of Economic Psychology, 24''(4), 505–516. https://doi.org/10.1016/S0167-4870(02)00211-8 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (1998a). The Experience of Regret and Disappointment. ''Cognition and Emotion, 12''(2), 221–230. https://doi.org/10.1080/026999398379727 Zeelenberg, M., van Dijk, W. W., van der Pligt, J., Manstead, A. S. ., van Empelen, P., & Reinderman, D. (1998b). Emotional Reactions to the Outcomes of Decisions: The Role of Counterfactual Thought in the Experience of Regret and Disappointment. ''Organizational Behavior and Human Decision Processes, 75''(2), 117–141. https://doi.org/10.1006/obhd.1998.2784 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (2000). On bad decisions and disconfirmed expectancies: The psychology of regret and disappointment. ''Cognition and Emotion, 14''(4), 521–541. https://doi.org/10.1080/026999300402781 }} ==External links== * [https://www.youtube.com/watch?v=8KgUFMN7aJQ The value of disappointment] (TEDxPCC) * [https://www.youtube.com/watch?v=gAMbkJk6gnE Feeling all the feels: Crash course psychology #25] (YouTube) gkq4pv5pfi64hqapfy2aiaj2rpefq2y 2416445 2416435 2022-08-19T06:07:56Z U3216256 2942574 /* Neuropsychology of disappointment */ wikitext text/x-wiki {{title|Disappointment<br>What is disappointment, what causes it, and how can it be managed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== [[File:Wayuu woman with sad face in the market buying.jpg|alt=Older woman with a disappointed look on her face in a busy market place.|thumb|''Figure 1.'' Person expressing disappointment through facial features]] * Example (case study) of disappointment - include statistics on frequency of occurrence (Van Dijk & Zeelenberg, 2002) * Define [[wikipedia:Emotion|emotions]], what emotions involve. (Reeve, 2018; Izard, 2010) Reference Figure 1 * Overview of what the chapter explores * Note on emotion research (Ramachandran & Jalal, 2017) {{RoundBoxTop|theme=13}} '''Key questions:''' * What is disappointment? * What causes disappointment? * How can disappointment be managed? {{RoundBoxBottom}} ==What is disappointment?== * Disappointment occurs when the outcome that has occurred is not as good as the outcome that could have occurred, or when your expectations are not met (Zeelenberg et al., 1998a). * Disappointment involves feeling powerless, a tendency to remove oneself from the situation, and a desire to do nothing (van Dijk et al., 1999) * Disappointment is a complex emotion (Ramachandran & Jalal, 2017) * Disappointment can be categorised as outcome-related or person-related; where outcome-related disappointment (ORD) occurs when the expected pleasurable outcome has not occurred, and person-related disappointment (PRD) occurs when you attribute the undesirable outcome to a\ person (you blame them for the 'bad' outcome) (Van Dijk & Zeelenberg, 2002). {{Robelbox|theme={{{theme|2}}}|title=Emotions similar to disappointment}} <div style="{{Robelbox/pad}}"> '''Regret''' : [https://en.wikipedia.org/wiki/Regret Regret] is a negative emotion that occurs when you know that the outcome that occurred could have been better if you made a different choice (Zeelenberg et al., 1998a). "Regret stems from bad decisions" (Zeelenberg et al., 1998a). '''Anger''' : [https://en.wikipedia.org/wiki/Anger Anger] is a simple negative emotion that occurs when you cannot achieve your goals and you blame someone or something else for it (Lelieveld et al., 2011). Anger can be the result of disappointment (van Dijk et al., 1999). '''Disillusionment''' : [https://en.wiktionary.org/wiki/disillusionment Disillusionment] is a complex negative emotion that occurs when you realise that what you believe or know is false (Maher et al., 2020). Disappointment is a key feature of disillusionment. </div> {{Robelbox/close}} Test yourself:<quiz display="simple"> {Mary's boss received a complaint from a customer about Mary. Mary was made aware of the complaint and then fired. Mary is likely to experience: |type="()"} + Person-related disappointment - Outcome-related disappointment {Alex is trying to get a snack from a vending machine. Alex put their money into the vending machine and typed in the code for lemonade. The vending machine did not give Alex lemonade, and took their money. Alex is likely to experience: |type="()"} - Person-related disappointment + Outcome-related disappointment </quiz> == What causes disappointment? == * Disappointment occurs when expectations are not met (Chua et al., 2009). * The two major theories of disappointment will be discussed. * The brain regions that have shown to be important in decision making/disappointment will be discussed. === Theories of disappointment === * The two major theories of disappointment are Bell's theory of disappointment and Loomes and Sudgen's theory of disappointment (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). * Both theories are similar (Zeelenberg et al., 2000; Zeelenberg et al., 1998b). Loomes and Sugden's theory came about due to their research and theory of regret (Loomes & Sugden, 1986). ==== Bell's theory of disappointment (1985) ==== * "Disappointment... is a psychological reaction to an outcome that does not match up with expectations" (Bell, 1985, p.1). * van Dijk and van der Pligt (1997) conducted five experiments which confirmed Bell's theory; where the experience of disappointment is primarily determined by by the probability of the outcome. It is theorised that individuals who experience disappointment focus on the features of the situation which caused the disappointment - one of the features could be the probability with which the disappointing outcome occurred (van Dijk & van der Pligt, 1997). ==== Loomes and Sugden's theory of disappointment (1986) ==== * "When considering any uncertain prospect, an individual forms some ''prior expectation''... if that consequence falls short of the prior expectation... the individual... experiences some degree of disappointment" (Loomes & Sugden, 1986, p.271). Key content (b) === Neuropsychology of disappointment === * Multiple brain regions have shown to be important to disappointment and decision making, namely the anterior insula, and various regions of the prefrontal cortex (Chua et al., 2009: Mohr et al., 2010: Kalat, 2019). Key content (b) ==== Subcortical structure: Anterior insula ==== [[File:Insula structure.png|alt=Structure of the three sections of the insula|thumb|''Figure 2.'' Insula]] * When individuals experience disappointment their anterior insula is activated (Chua et al., 2009). See figure 2. * The anterior insula is active in the presence of potential loss and is a part of the brain known for processing emotions such as disappointment (Mohr et al., 2010). ==== Cortical structure: Prefrontal cortex ==== [[File:Prefrontal cortex (left) animation.gif|alt=Rotating skull containing left Prefrontal cortex. The prefrontal cortex is highlighted |thumb|''Figure 3.'' Left hemisphere prefrontal cortex]] [[File:Cortical midline structures.png|thumb|''Figure 4.'' Brain image highlighting various cortical regions, including the ventromedial prefrontal cortex (VMPFC), dorsomedial prefrontal cortex (DMPFC), and the orbitofrontal cortex (OFC/MOPFC).]] * prefrontal cortex - middle 'zone': <u>emotional reactions</u>, cognitive control, and working memory (Kalat, 2019). * prefrontal cortex - anterior (front): decision making, evaluating which course will provide the best outcome, probability of achieving a good outcome (Kalat, 2019). * Prefrontal cortex - see figure 3. * The ventromedial prefrontal cortex [VMPFC] learns what choices are more beneficial for the individual and adjusts decision making accordingly (Kalat, 2019). It also monitors confidence in one's decisions (Kalat, 2019). Individuals with damage to their VMPFC tend to make impulsive decisions based on probability, rather than making considered decisions based on reality (Kalat, 2019). *Orbitofrontal cortex [OFC]- responds to information from the VMPFC based on how the expected reward compares to other possible choices (Kalat, 2019). The OFC changes and updates the expected outcomes of actions based on current circumstances (Kalat, 2019). *When individuals experience disappointment their dorsomedial prefrontal cortex [DMPFC] is activated (Chua et al., 2009) *See figure 4 for VMPFC, OFC, and DMPFC. Insert Quiz on brain regions and how they contribute to decision-making/disappointment OR insert table with brain region - what it does - how it contributes to disappointment ==How can disappointment be managed?== * Disappointment can be managed by lowering expectations (van Dijk et al., 2003). The ventromedial prefrontal cortex monitors how confident you should be about a decision (Kalat, 2019). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]], therefore having lower expectations can improve wellbeing (de Meza & Dawson, 2021). * Disappointment can be managed by successfully living up to expectations; where individuals intensify their efforts to attain a certain outcome so as not to experience disappointment (Zeleenberg et al., 2000). === The benefits of disappointment === * Expressing disappointment encourages higher levels of cooperation in others (Olekalns & Druckman, 2014). * Expressing disappointment can lead to helping behaviour in others (Johnson & Connelly, 2014). === The downsides to disappointment === * Disappointment can taint relationships, leading to blaming or [[wikipedia:Scapegoating|scapegoating]] (Clancy et al., 2012). * Mistaken expectations lower overall [[wikipedia:Well-being|wellbeing]] (de Meza & Dawson, 2021) ==Conclusion== * Restate what disappointment is, the types of disappointment, and emotions similar to disappointment. * Restate the causes of disappointment * Restate how disappointment can be managed * Take home message: everybody experiences disappointment, but there are things you can do to minimise experiencing it and you can even take advantage of it. == See also == * [[wikipedia:Disappointment|Disappointment]] (Wikipedia) * [[Motivation and emotion/Book/2016/Regret|Regret]] (Book chapter, 2016) * [[Motivation and emotion/Book/2011/Anger|Anger]] (Book chapter, 2011) ==References== {{Hanging indent|1= Bell, D. E. (1985). Disappointment in Decision Making Under Uncertainty. ''Operations Research, 33''(1), 1–27. https://doi.org/10.1287/opre.33.1.1 Chua, Gonzalez, R., Taylor, S. F., Welsh, R. C., & Liberzon, I. (2009). Decision-related loss: Regret and disappointment. ''NeuroImage, 47''(4), 2031–2040. https://doi.org/10.1016/j.neuroimage.2009.06.006 Clancy, A., Vince, R., & Gabriel, Y. (2012). That Unwanted Feeling: A Psychodynamic Study of Disappointment in Organizations. ''British Journal of Management, 23''(4), 518–531. https://doi.org/10.1111/j.1467-8551.2011.00780.x de Meza, D., & Dawson, C. (2021). Neither an Optimist Nor a Pessimist Be: Mistaken Expectations Lower Well-Being. ''Personality & Social Psychology Bulletin'', ''47''(4), 540–550. https://doi.org/10.1177/0146167220934577 Izard, C. E. (2010). The Many Meanings/Aspects of Emotion: Definitions, Functions, Activation, and Regulation. ''Emotion Review, 2''(4), 363–370. https://doi.org/10.1177/1754073910374661 Johnson, G., & Connelly, S. (2014). Negative emotions in informal feedback: The benefits of disappointment and drawbacks of anger. ''Human Relations (New York), 67''(10), 1265–1290. https://doi.org/10.1177/0018726714532856 Kalat, J. W. (2019). ''Biological psychology'' (13th ed.). Boston, MA, USA: Cengage Lelieveld, G. J., Van Dijk, E., Van Beest, I., Steinel, W., & Van Kleef, G. A. (2011). Disappointed in you, angry about your offer: Distinct negative emotions induce concessions via different mechanisms. ''Journal of Experimental Social Psychology, 47''(3), 635–641. https://doi.org/10.1016/j.jesp.2010.12.015 Loomes, G., & Sugden, R. (1986). Disappointment and dynamic consistency in choice under uncertainty. ''The Review of Economic Studies, 53''(2), 271–282. https://doi.org/10.2307/2297651 Maher, P. J., Igou, E. R., & van Tilburg, W. A. P. (2020). Disillusionment: a prototype analysis. ''Cognition and Emotion, 34''(5), 947–959. https://doi.org/10.1080/02699931.2019.1705764 Olekalns, M., & Druckman, D. (2014). With Feeling: How Emotions Shape Negotiation. ''Negotiation Journal, 30''(4), 455–478. https://doi.org/10.1111/nejo.12071 Mohr, P. N. C., Biele, G., & Heekeren, H. R. (2010). Neural processing of risk. ''The Journal of Neuroscience, 30''(19), 6613–6619. https://doi.org/10.1523/JNEUROSCI.0003-10.2010 Ramachandran, V.S., & Jalal, B. (2017). The Evolutionary Psychology of Envy and Jealousy. ''Frontiers in Psychology, 8'', 1619–1619. https://doi.org/10.3389/fpsyg.2017.01619 Reeve, J. (2018). ''Understanding motivation and emotion'' (7th ed.). Hoboken, NJ: Wiley van Dijk, W. W., & van der Pligt, J. (1997). The Impact of Probability and Magnitude of Outcome on Disappointment and Elation. ''Organizational Behavior and Human Decision Processes, 69''(3), 277–284. https://doi.org/10.1006/obhd.1997.2688 van Dijk, W. W., & Zeelenberg, M. (2002). What do we talk about when we talk about disappointment? Distinguishing outcome-related disappointment from person-related disappointment. ''Cognition and Emotion, 16''(6), 787–807. https://doi.org/10.1080/02699930143000563 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (1999). Not Having What You Want versus Having What You Do Not Want: The Impact of Type of Negative Outcome on the Experience of Disappointment and Related Emotions. ''Cognition and Emotion, 13''(2), 129–148. https://doi.org/10.1080/026999399379302 van Dijk, W. W., Zeelenberg, M., & van der Pligt, J. (2003). Blessed are those who expect nothing: Lowering expectations as a way of avoiding disappointment. ''Journal of Economic Psychology, 24''(4), 505–516. https://doi.org/10.1016/S0167-4870(02)00211-8 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (1998a). The Experience of Regret and Disappointment. ''Cognition and Emotion, 12''(2), 221–230. https://doi.org/10.1080/026999398379727 Zeelenberg, M., van Dijk, W. W., van der Pligt, J., Manstead, A. S. ., van Empelen, P., & Reinderman, D. (1998b). Emotional Reactions to the Outcomes of Decisions: The Role of Counterfactual Thought in the Experience of Regret and Disappointment. ''Organizational Behavior and Human Decision Processes, 75''(2), 117–141. https://doi.org/10.1006/obhd.1998.2784 Zeelenberg, M., van Dijk, W. W., Manstead, A. S. R., & van der Pligt, J. (2000). On bad decisions and disconfirmed expectancies: The psychology of regret and disappointment. ''Cognition and Emotion, 14''(4), 521–541. https://doi.org/10.1080/026999300402781 }} ==External links== * [https://www.youtube.com/watch?v=8KgUFMN7aJQ The value of disappointment] (TEDxPCC) * [https://www.youtube.com/watch?v=gAMbkJk6gnE Feeling all the feels: Crash course psychology #25] (YouTube) l18ly9zv2yj1r20wmm3kz5miw2u95lz 0 285912 2416342 2411253 2022-08-18T23:01:20Z U3200859 2947964 Overview update wikitext text/x-wiki {{title|Chapter title:<br>Subtitle?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Honesty Motivation== You are underway {{smile}}! What motivates honesty? This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] p2pu3ksqdxcrivx8pwl6futqvr8j3ed 0 285934 2416135 2416003 2022-08-18T12:10:35Z Ajeofula22 2947179 /* The role of GABA */ wikitext text/x-wiki {{title| Benzodiazepines and emotion:<br> What are the effects of benzodiazepines on emotion?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== Have you used something you believed at the time have no harm and later realised it has more risk (harm) than good? Well although many people believe benzodiazepines (BNZ) can do no harms, physician have serious concern about misuse and abuse of these drugs by patients. Many doctors and psychiatrics are still wondering by benzodiazepines are still prescribe for patients despite all risk associate the drugs . This book chapter will discuss the effects of BNZ on emotion, biological, cognitive, psychological aspects of BNZ. Additional, book chapter will touch on pathophysiology of anxiety, theories, and interventions. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What are benzodiazepines use for? * What roles does benzodiazepines play on emotions? * Can people management their emotions without benzodiazepines? {{RoundBoxBottom}} [[File:Benzodiazepine.svg|thumb|''Figure 1''. Description of benzodiazepines |169x169px]] ==What are benzodiazepines?== Benzodiazepines (BNZ) also known as CNS depressants- {| class="wikitable" |+ ! colspan="4" | |- | === Effects of benzodiazepines === | === Benzodiazepines commonly prescribed === | === Clinical aspect of benzodiazepines === | === Routes === |- | * Sedative effects * Hypnotic effects * Anxiolytic effects | * Alprazolam * Lorazepam * Temazepam * Diazepam * Midazolam and more | * Anxiety * Seizure * Insomnia * Panic attack * Before procedure (surgery) * Alcohol withdrawal | * IV - intravenous * IM - intramuscular * Rectal - through anus * Oral - through mouth |} {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} == What is emotion? == === Region in the brain affected by benzos === * Amygdala * Brain stem * Cortical region === The role of GABA === Gamma-aminobutyric acid (GABA) also known as neurotransmitters is the primary inhibitory in the brain (central nervous system). Benzodiazepines amplifies GABA effects by inhibitory neurotransmitter (Lydiard, 2003). == What is the effects of benzodiazepines on emotions == * Impaired threat processing * Generalised fear of threat * Reduce total alertness response * Lack of ability to recognised facial expressions ==== Effects of benzodiazepines on behavioural and neurophysiological aspects ==== * Impaired psychomotor activities and motor control * Impaired memory functions especially working and verbal memory * Decreased alertness on environmental threat * Increased reaction times as well as decrease reaction time when for BNZ * Driving impairments ( Stone, Corea, Brown, Spurgin, Stikic, Johnson, & Berka, 2015) ===== Effects of BNZ on social cognition ===== * Activities in amygdala reduce - this part of the brain is responsible in guiding behaviour and processes knowledge about vital stimuli for survival. Amygdala also contributes to area of cognitive processing such as attention, memory, decision making, and research found this part of the brain critical in facial expression especially negative emotions such as fear (Adolphs, 1999). Studies also found that patients with damaged amygdala are unlikely to recognized emotions associated with fear and anger. However, benzodiazepines play critical roles by producing calming effects during endless though in the brain which are mostly negative emotions (Zald,2003) Reduction in emotional processing ability - * Affect motivation and emotion * Deficit in social cognition due to emotional suppression * Alleviate anxiety symptoms and improve social cognitions impairment (Haime, Watson, Crellin, Marston, Joyce, & Moncrieff, 2021). === Issues related with withdrawal from benzodiazepines: === * Increased level of anxiety * Insomnia- difficulty with sleep or falling a sleep * Lack of appetite - lack of motivation to seek for food or eat food * Weight lose - losing weight without trying * Perception disturbance * Tremor * Irritability * Nausea * Panic attacks * Headache * Sweating * Problem with concentration (Petursson,1994). == Pathophysiology of anxiety and other emotional related disorders == == Theories == * Darwin biological and genetic theory on emotion * Psychological cognitive theory == Recommendation for interventions == * Cognitive therapy * Behavioural therapy * Psychological therapy * Life styles == conclusion == ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== # [[Motivation and emotion/Book/2020/Methamphetamine and emotion|Methamphetamine and emotion]] (Book chapter, 2020) # [[Motivation and emotion/Book/2019/Opioid system and human emotion|Opioid system and human emotion]] (Book chapter, 2019) # [[wikipedia:Benzodiazepine|Benzodiazepine]] (Wikipedia) Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]]" jf1rgjxhuyaqrek9z0u4b6x7dg6wu6t 2416140 2416135 2022-08-18T12:35:32Z Ajeofula22 2947179 /* What is emotion? */ wikitext text/x-wiki {{title| Benzodiazepines and emotion:<br> What are the effects of benzodiazepines on emotion?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== Have you used something you believed at the time have no harm and later realised it has more risk (harm) than good? Well although many people believe benzodiazepines (BNZ) can do no harms, physician have serious concern about misuse and abuse of these drugs by patients. Many doctors and psychiatrics are still wondering by benzodiazepines are still prescribe for patients despite all risk associate the drugs . This book chapter will discuss the effects of BNZ on emotion, biological, cognitive, psychological aspects of BNZ. Additional, book chapter will touch on pathophysiology of anxiety, theories, and interventions. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What are benzodiazepines use for? * What roles does benzodiazepines play on emotions? * Can people management their emotions without benzodiazepines? {{RoundBoxBottom}} [[File:Benzodiazepine.svg|thumb|''Figure 1''. Description of benzodiazepines |169x169px]] ==What are benzodiazepines?== Benzodiazepines (BNZ) also known as CNS depressants- {| class="wikitable" |+ ! colspan="4" | |- | === Effects of benzodiazepines === | === Benzodiazepines commonly prescribed === | === Clinical aspect of benzodiazepines === | === Routes === |- | * Sedative effects * Hypnotic effects * Anxiolytic effects | * Alprazolam * Lorazepam * Temazepam * Diazepam * Midazolam and more | * Anxiety * Seizure * Insomnia * Panic attack * Before procedure (surgery) * Alcohol withdrawal | * IV - intravenous * IM - intramuscular * Rectal - through anus * Oral - through mouth |} Physiological effects of benzodiazepines includes * Lethargy * Fatigue * Drowsiness * Impaired motor coordination * Slurred speech * Low mood (mood swings) * Hostility * Dizziness * Vertigo * Memory impairment * {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} == What is emotion? == === Region in the brain affected by benzos === * Amygdala * Brain stem * Cortical region === The role of GABA === Gamma-aminobutyric acid (GABA) also known as neurotransmitters is the primary inhibitory in the brain (central nervous system). Benzodiazepines amplifies GABA effects by inhibitory neurotransmitter (Lydiard, 2003). == What is the effects of benzodiazepines on emotions == * Impaired threat processing * Generalised fear of threat * Reduce total alertness response * Lack of ability to recognised facial expressions ==== Effects of benzodiazepines on behavioural and neurophysiological aspects ==== * Impaired psychomotor activities and motor control * Impaired memory functions especially working and verbal memory * Decreased alertness on environmental threat * Increased reaction times and while as decrease * Driving impairments ( Stone, Corea, Brown, Spurgin, Stikic, Johnson, & Berka, 2015) ===== Effects of BNZ on social cognition ===== * Activities in amygdala reduce * Reduction in emotional processing ability * Affect motivation and emotion * Deficit in social cognition due to emotional suppression * Alleviate anxiety symptoms and improve social cognitions impairment (Haime, Watson, Crellin, Marston, Joyce, & Moncrieff, 2021). === Issues related with withdrawal from benzodiazepines: === * Increased level of anxiety * Insomnia- difficulty with sleep or falling a sleep * Lack of appetite - lack of motivation to seek for food or eat food * Weight lose - losing weight without trying * Perception disturbance * Tremor * Irritability * Nausea * Panic attacks * Headache * Sweating * Problem with concentration (Petursson,1994). == Pathophysiology of anxiety and other emotional related disorders == == Theories == * Darwin biological and genetic theory on emotion * Psychological cognitive theory == Recommendation for interventions == * Cognitive therapy * Behavioural therapy * Psychological therapy * Life styles == conclusion == ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== # [[Motivation and emotion/Book/2020/Methamphetamine and emotion|Methamphetamine and emotion]] (Book chapter, 2020) # [[Motivation and emotion/Book/2019/Opioid system and human emotion|Opioid system and human emotion]] (Book chapter, 2019) # [[wikipedia:Benzodiazepine|Benzodiazepine]] (Wikipedia) Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]]" kr0h7fa2b0mv575o0cjtph4wrxo8fb7 2416142 2416140 2022-08-18T12:40:13Z Ajeofula22 2947179 /* Effects of BNZ on social cognition */ wikitext text/x-wiki {{title| Benzodiazepines and emotion:<br> What are the effects of benzodiazepines on emotion?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== Have you used something you believed at the time have no harm and later realised it has more risk (harm) than good? Well although many people believe benzodiazepines (BNZ) can do no harms, physician have serious concern about misuse and abuse of these drugs by patients. Many doctors and psychiatrics are still wondering by benzodiazepines are still prescribe for patients despite all risk associate the drugs . This book chapter will discuss the effects of BNZ on emotion, biological, cognitive, psychological aspects of BNZ. Additional, book chapter will touch on pathophysiology of anxiety, theories, and interventions. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What are benzodiazepines use for? * What roles does benzodiazepines play on emotions? * Can people management their emotions without benzodiazepines? {{RoundBoxBottom}} [[File:Benzodiazepine.svg|thumb|''Figure 1''. Description of benzodiazepines |169x169px]] ==What are benzodiazepines?== Benzodiazepines (BNZ) also known as CNS depressants- Table 1. summarising the effects, commonly prescribed BNZ, roles of BNZ in clinical aspects and form BNZ are administered. {| class="wikitable" |+ ! colspan="4" | |- | === Effects of benzodiazepines === | === Benzodiazepines commonly prescribed === | === Clinical aspect of benzodiazepines === | === Routes === |- | * Sedative effects * Hypnotic effects * Anxiolytic effects | * Alprazolam * Lorazepam * Temazepam * Diazepam * Midazolam and more | * Anxiety * Seizure * Insomnia * Panic attack * Before procedure (surgery) * Alcohol withdrawal | * IV - intravenous * IM - intramuscular * Rectal - through anus * Oral - through mouth |} {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} == What is emotion? == === Region in the brain affected by benzos === * Amygdala * Brain stem * Cortical region === The role of GABA === Gamma-aminobutyric acid (GABA) also known as neurotransmitters is the primary inhibitory in the brain (central nervous system). Benzodiazepines amplifies GABA effects by inhibitory neurotransmitter (Lydiard, 2003). == What is the effects of benzodiazepines on emotions == * Impaired threat processing * Generalised fear of threat * Reduce total alertness response * Lack of ability to recognised facial expressions ==== Effects of benzodiazepines on behavioural and neurophysiological aspects ==== * Impaired psychomotor activities and motor control * Impaired memory functions especially working and verbal memory * Decreased alertness on environmental threat * Increased reaction times as well as decrease reaction time when for BNZ * Driving impairments ( Stone, Corea, Brown, Spurgin, Stikic, Johnson, & Berka, 2015) ===== Effects of BNZ on social cognition ===== * Activities in amygdala reduce - this part of the brain is responsible in guiding behaviour and processes knowledge about vital stimuli for survival. Amygdala also contributes to area of cognitive processing such as attention, memory, decision making, and research found this part of the brain critical in facial expression especially negative emotions such as fear (Adolphs, 1999). Studies also found that patients with damaged amygdala are unlikely to recognized emotions associated with fear and anger. However, benzodiazepines play critical roles by producing calming effects during endless though in the brain which are mostly negative emotions (Zald,2003). * Reduction in emotional processing ability - * Affect motivation and emotion * Deficit in social cognition due to emotional suppression * Alleviate anxiety symptoms and improve social cognitions impairment (Haime, Watson, Crellin, Marston, Joyce, & Moncrieff, 2021). === Issues related with withdrawal from benzodiazepines: === * Increased level of anxiety * Insomnia- difficulty with sleep or falling a sleep * Lack of appetite - lack of motivation to seek for food or eat food * Weight lose - losing weight without trying * Perception disturbance * Tremor * Irritability * Nausea * Panic attacks * Headache * Sweating * Problem with concentration (Petursson,1994). == Pathophysiology of anxiety and other emotional related disorders == == Theories == * Darwin biological and genetic theory on emotion * Psychological cognitive theory == Recommendation for interventions == * Cognitive therapy * Behavioural therapy * Psychological therapy * Life styles == conclusion == ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== # [[Motivation and emotion/Book/2020/Methamphetamine and emotion|Methamphetamine and emotion]] (Book chapter, 2020) # [[Motivation and emotion/Book/2019/Opioid system and human emotion|Opioid system and human emotion]] (Book chapter, 2019) # [[wikipedia:Benzodiazepine|Benzodiazepine]] (Wikipedia) Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]]" e9nkbb9vvknt4gnck5sxkso0akiw2k1 2416143 2416142 2022-08-18T12:44:39Z Ajeofula22 2947179 /* What are benzodiazepines? */ link wikitext text/x-wiki {{title| Benzodiazepines and emotion:<br> What are the effects of benzodiazepines on emotion?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== Have you used something you believed at the time have no harm and later realised it has more risk (harm) than good? Well although many people believe benzodiazepines (BNZ) can do no harms, physician have serious concern about misuse and abuse of these drugs by patients. Many doctors and psychiatrics are still wondering by benzodiazepines are still prescribe for patients despite all risk associate the drugs . This book chapter will discuss the effects of BNZ on emotion, biological, cognitive, psychological aspects of BNZ. Additional, book chapter will touch on pathophysiology of anxiety, theories, and interventions. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What are benzodiazepines use for? * What roles does benzodiazepines play on emotions? * Can people management their emotions without benzodiazepines? {{RoundBoxBottom}} [[File:Benzodiazepine.svg|thumb|''Figure 1''. Description of benzodiazepines |169x169px]] ==What are benzodiazepines?== [[wikipedia:Benzodiazepine|Benzodiazepines]] (BNZ) also known as CNS depressants- Table 1. summarising the effects, commonly prescribed BNZ, roles of BNZ in clinical aspects and form BNZ are administered. {| class="wikitable" |+ ! colspan="4" | |- | === Effects of benzodiazepines === | === Benzodiazepines commonly prescribed === | === Clinical aspect of benzodiazepines === | === Routes === |- | * Sedative effects * Hypnotic effects * Anxiolytic effects | * Alprazolam * Lorazepam * Temazepam * Diazepam * Midazolam and more | * Anxiety * Seizure * Insomnia * Panic attack * Before procedure (surgery) * Alcohol withdrawal | * IV - intravenous * IM - intramuscular * Rectal - through anus * Oral - through mouth |} {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} == What is emotion? == === Region in the brain affected by benzos === * Amygdala * Brain stem * Cortical region === The role of GABA === Gamma-aminobutyric acid (GABA) also known as neurotransmitters is the primary inhibitory in the brain (central nervous system). Benzodiazepines amplifies GABA effects by inhibitory neurotransmitter (Lydiard, 2003). == What is the effects of benzodiazepines on emotions == * Impaired threat processing * Generalised fear of threat * Reduce total alertness response * Lack of ability to recognised facial expressions ==== Effects of benzodiazepines on behavioural and neurophysiological aspects ==== * Impaired psychomotor activities and motor control * Impaired memory functions especially working and verbal memory * Decreased alertness on environmental threat * Increased reaction times as well as decrease reaction time when for BNZ * Driving impairments ( Stone, Corea, Brown, Spurgin, Stikic, Johnson, & Berka, 2015) ===== Effects of BNZ on social cognition ===== * Activities in amygdala reduce - this part of the brain is responsible in guiding behaviour and processes knowledge about vital stimuli for survival. Amygdala also contributes to area of cognitive processing such as attention, memory, decision making, and research found this part of the brain critical in facial expression especially negative emotions such as fear (Adolphs, 1999). Studies also found that patients with damaged amygdala are unlikely to recognized emotions associated with fear and anger. However, benzodiazepines play critical roles by producing calming effects during endless though in the brain which are mostly negative emotions (Zald,2003). * Reduction in emotional processing ability - * Affect motivation and emotion * Deficit in social cognition due to emotional suppression * Alleviate anxiety symptoms and improve social cognitions impairment (Haime, Watson, Crellin, Marston, Joyce, & Moncrieff, 2021). === Issues related with withdrawal from benzodiazepines: === * Increased level of anxiety * Insomnia- difficulty with sleep or falling a sleep * Lack of appetite - lack of motivation to seek for food or eat food * Weight lose - losing weight without trying * Perception disturbance * Tremor * Irritability * Nausea * Panic attacks * Headache * Sweating * Problem with concentration (Petursson,1994). == Pathophysiology of anxiety and other emotional related disorders == == Theories == * Darwin biological and genetic theory on emotion * Psychological cognitive theory == Recommendation for interventions == * Cognitive therapy * Behavioural therapy * Psychological therapy * Life styles == conclusion == ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== # [[Motivation and emotion/Book/2020/Methamphetamine and emotion|Methamphetamine and emotion]] (Book chapter, 2020) # [[Motivation and emotion/Book/2019/Opioid system and human emotion|Opioid system and human emotion]] (Book chapter, 2019) # [[wikipedia:Benzodiazepine|Benzodiazepine]] (Wikipedia) Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]]" qt4p97eb6utj0x8fp6pmq64r8x81ef5 2416144 2416143 2022-08-18T12:55:32Z Ajeofula22 2947179 /* Issues related with withdrawal from benzodiazepines: */ wikitext text/x-wiki {{title| Benzodiazepines and emotion:<br> What are the effects of benzodiazepines on emotion?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== Have you used something you believed at the time have no harm and later realised it has more risk (harm) than good? Well although many people believe benzodiazepines (BZD) can do no harms, physician have serious concern about misuse and abuse of these drugs by patients. Many doctors and psychiatrics are still wondering by benzodiazepines are still prescribe for patients despite all risk associate the drugs . This book chapter will discuss the effects of BZD on emotion, biological, cognitive, psychological aspects of BZD. Additional, book chapter will touch on pathophysiology of anxiety, theories, and interventions. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What are benzodiazepines use for? * What roles does benzodiazepines play on emotions? * Can people management their emotions without benzodiazepines? {{RoundBoxBottom}} [[File:Benzodiazepine.svg|thumb|''Figure 1''. Description of benzodiazepines |169x169px]] ==What are benzodiazepines?== [[wikipedia:Benzodiazepine|Benzodiazepines]] (BZD) also known as CNS depressants- Table 1. summarising the effects, commonly prescribed BZD, roles of BZD in clinical aspects and form BZD are administered. {| class="wikitable" |+ ! colspan="4" | |- | === Effects of benzodiazepines === | === Benzodiazepines commonly prescribed === | === Clinical aspect of benzodiazepines === | === Routes === |- | * Sedative effects * Hypnotic effects * Anxiolytic effects | * Alprazolam * Lorazepam * Temazepam * Diazepam * Midazolam and more | * Anxiety * Seizure * Insomnia * Panic attack * Before procedure (surgery) * Alcohol withdrawal | * IV - intravenous * IM - intramuscular * Rectal - through anus * Oral - through mouth |} {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} == What is emotion? == * === pharmacological aspect of BZD === Gamma-aminobutyric acid (GABA) also known as neurotransmitters is the primary inhibitory in the brain (central nervous system). Benzodiazepines amplifies GABA effects by inhibitory neurotransmitter (Lydiard, 2003). == What is the effects of benzodiazepines on emotions == * Impaired threat processing * Generalised fear of threat * Reduce total alertness response * Lack of ability to recognised facial expressions ==== Effects of benzodiazepines on behavioural and neurophysiological aspects ==== * Impaired psychomotor activities and motor control * Impaired memory functions especially working and verbal memory * Decreased alertness on environmental threat * Increased reaction times as well as decrease reaction time when for BNZ * Driving impairments ( Stone, Corea, Brown, Spurgin, Stikic, Johnson, & Berka, 2015) ===== Effects of BZD on social cognition ===== * Activities in amygdala reduce - this part of the brain is responsible in guiding behaviour and processes knowledge about vital stimuli for survival. Amygdala also contributes to area of cognitive processing such as attention, memory, decision making, and research found this part of the brain critical in facial expression especially negative emotions such as fear (Adolphs, 1999). Studies also found that patients with damaged amygdala are unlikely to recognized emotions associated with fear and anger. However, benzodiazepines play critical roles by producing calming effects during endless though in the brain which are mostly negative emotions (Zald,2003). * Reduction in emotional processing ability - * Affect motivation and emotion * Deficit in social cognition due to emotional suppression * Alleviate anxiety symptoms and improve social cognitions impairment (Haime, Watson, Crellin, Marston, Joyce, & Moncrieff, 2021). === Issues related with dependence, withdrawal and tolerance from benzodiazepines: === * Increased level of anxiety * Insomnia- difficulty with sleep or falling a sleep * Lack of appetite - lack of motivation to seek for food or eat food * Weight lose - losing weight without trying * Perception disturbance * Tremor * Irritability * Nausea * Panic attacks * Headache * Sweating * Problem with concentration (Petursson,1994). == Pathophysiology of anxiety and other emotional related disorders == == Theories == * Darwin biological and genetic theory on emotion * Psychological cognitive theory == Recommendation for interventions == * Cognitive therapy * Behavioural therapy * Psychological therapy * Life styles == conclusion == ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== # [[Motivation and emotion/Book/2020/Methamphetamine and emotion|Methamphetamine and emotion]] (Book chapter, 2020) # [[Motivation and emotion/Book/2019/Opioid system and human emotion|Opioid system and human emotion]] (Book chapter, 2019) # [[wikipedia:Benzodiazepine|Benzodiazepine]] (Wikipedia) Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]]" 952dqo9xa14q67cw44jkh8zvl0mng31 2416153 2416144 2022-08-18T13:23:48Z Ajeofula22 2947179 /* What is emotion? */ wikitext text/x-wiki {{title| Benzodiazepines and emotion:<br> What are the effects of benzodiazepines on emotion?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== Have you used something you believed at the time have no harm and later realised it has more risk (harm) than good? Well although many people believe benzodiazepines (BZD) can do no harms, physician have serious concern about misuse and abuse of these drugs by patients. Many doctors and psychiatrics are still wondering by benzodiazepines are still prescribe for patients despite all risk associate the drugs . This book chapter will discuss the effects of BZD on emotion, biological, cognitive, psychological aspects of BZD. Additional, book chapter will touch on pathophysiology of anxiety, theories, and interventions. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What are benzodiazepines use for? * What roles does benzodiazepines play on emotions? * Can people management their emotions without benzodiazepines? {{RoundBoxBottom}} [[File:Benzodiazepine.svg|thumb|''Figure 1''. Description of benzodiazepines |169x169px]] ==What are benzodiazepines?== [[wikipedia:Benzodiazepine|Benzodiazepines]] (BZD) also known as CNS depressants- Table 1. summarising the effects, commonly prescribed BZD, roles of BZD in clinical aspects and form BZD are administered. {| class="wikitable" |+ ! colspan="4" | |- | === Effects of benzodiazepines === | === Benzodiazepines commonly prescribed === | === Clinical aspect of benzodiazepines === | === Routes === |- | * Sedative effects * Hypnotic effects * Anxiolytic effects | * Alprazolam * Lorazepam * Temazepam * Diazepam * Midazolam and more | * Anxiety * Seizure * Insomnia * Panic attack * Before procedure (surgery) * Alcohol withdrawal | * IV - intravenous * IM - intramuscular * Rectal - through anus * Oral - through mouth |} {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} == What is emotion? == The fact that emotions innate in all species according to evolutional theory, its critical to be aware of our emotions. Emotions play major roles in regulating behaviour and stimuli in the environment.  Although BZD provide calming effects, it also has devasting physiological effects when prolong the used and these includes dizziness, irritability, fatigue, lethargy, slurred speech, memory impairment and impaired motor- coordination. === Pharmacological aspect of BZD === Gamma-aminobutyric acid (GABA) also known as neurotransmitters is the primary inhibitory in the brain (central nervous system). Benzodiazepines amplifies GABA effects by inhibitory neurotransmitter (Lydiard, 2003). == What is the effects of benzodiazepines on emotions == * Impaired threat processing * Generalised fear of threat * Reduce total alertness response * Lack of ability to recognised facial expressions ==== Effects of benzodiazepines on behavioural and neurophysiological aspects ==== * Impaired psychomotor activities and motor control * Impaired memory functions especially working and verbal memory * Decreased alertness on environmental threat * Increased reaction times as well as decrease reaction time when for BNZ * Driving impairments ( Stone, Corea, Brown, Spurgin, Stikic, Johnson, & Berka, 2015) ===== Effects of BZD on social cognition ===== * Activities in amygdala reduce - this part of the brain is responsible in guiding behaviour and processes knowledge about vital stimuli for survival. Amygdala also contributes to area of cognitive processing such as attention, memory, decision making, and research found this part of the brain critical in facial expression especially negative emotions such as fear (Adolphs, 1999). Studies also found that patients with damaged amygdala are unlikely to recognized emotions associated with fear and anger. However, benzodiazepines play critical roles by producing calming effects during endless though in the brain which are mostly negative emotions (Zald,2003). * Reduction in emotional processing ability - * Affect motivation and emotion * Deficit in social cognition due to emotional suppression * Alleviate anxiety symptoms and improve social cognitions impairment (Haime, Watson, Crellin, Marston, Joyce, & Moncrieff, 2021). === Issues related with dependence, withdrawal and tolerance from benzodiazepines: === * Increased level of anxiety * Insomnia- difficulty with sleep or falling a sleep * Lack of appetite - lack of motivation to seek for food or eat food * Weight lose - losing weight without trying * Perception disturbance * Tremor * Irritability * Nausea * Panic attacks * Headache * Sweating * Problem with concentration (Petursson,1994). == Pathophysiology of anxiety and other emotional related disorders == == Theories == * Darwin biological and genetic theory on emotion * Psychological cognitive theory == Recommendation for interventions == * Cognitive therapy * Behavioural therapy * Psychological therapy * Life styles == conclusion == ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== # [[Motivation and emotion/Book/2020/Methamphetamine and emotion|Methamphetamine and emotion]] (Book chapter, 2020) # [[Motivation and emotion/Book/2019/Opioid system and human emotion|Opioid system and human emotion]] (Book chapter, 2019) # [[wikipedia:Benzodiazepine|Benzodiazepine]] (Wikipedia) Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]]" olyiah01c8z2194em4hc4r9ehpetok2 0 286014 2416412 2416022 2022-08-19T03:23:51Z Sebastian Armstrong 2947157 /* History */ added references into History wikitext text/x-wiki {{title|Fully functioning person <br>What is a FFP and how can full functioning be developed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ == Plan == === Overview === ''What is a FFP and How can full functioning be developed?'' # <u>A growing openness to experience</u> # <u>An increasingly existential lifestyle</u> # <u>Increasing organismic trust</u> # <u>Freedom of choice</u> # <u>Creativity</u> # <u>Reliability and constructiveness</u> # <u>A rich full life</u> #* Rogers, Carl (1961). ''On becoming a person: A therapist's view of psychotherapy''. London: Constable. ISBN <bdi>978-1-84529-057-3</bdi>. == [[Caregiving and dementia/Topics/Person centred care|person-centered theory of personality]] == == History == <u>Both Plato and Aristotle proposed theories about goodness in which functioning worked as the most central concept. Arguments about goodness and functioning keep developing in contemporary work on human goodness. The work of Sen, Staudinger, and Tomasello serves as renowned examples. (https://doi.org/10.1007/978-3-319-24612-3_1469)</u><blockquote>Client-centered therapy, also called the person-centered approach, describes Carl R. Rogers’ way of working with persons experiencing all types of personal disturbances or problems in living (Rogers, 1959; 1961; 1969; 1970; 1972; 1980a; 1986a). As early as 1939, Rogers developed his theory of psychotherapy with troubled children, and went on to expand his theoretical approach to include work with couples, families, and groups. His most comprehensive theoretical statement was published as a chapter in Sigmund Koch’s Psychology: A Study of a Science (Vol. III) in 1959, and includes his theory of motivation and personality development, as well as theory of group interaction and interpersonal relationships (Koch, 1959, 184–256). Over his long career, Rogers extrapolated client-centered values to the education, marriage, group encounter, personal power, and conflict resolution (Rogers, 1969, 1970, 1972). Today, the person-centered approach is practiced in the United Kingdom, Germany, France, Greece, Portugal, Demark, Poland, Hungary, The Netherlands, Italy, Japan, Brazil, Mexico, Australia, and South Africa, as well as here in the United States and Canada. A world association, which can be contacted online, was founded in Lisbon in 1997 that reflects the growth and vitality of the approach entitled the World Association for Person-Centered and Experiential Psychotherapy and Counseling (WAPCEPC). Another international organization comprised of a diverse membership—lay persons, educators, business consultants, therapists, artists, psychologists—the Association for the Development of the Person-Centered Approach (ADPCA), is also accessible on the internet. Witty, M. C. (2007). Client-centered therapy. In ''Handbook of homework assignments in psychotherapy'' (pp. 35-50). Springer, Boston, MA.</blockquote> === [[Motivation and emotion/Tutorials/Growth psychology|Growth psychology]] === === [[Talk:Carl Rogers]] === === [[Motivation and emotion/Textbook/Motivation/Self-actualisation|Self-actualisation]] === === [[wikipedia:Abraham_Maslow|Abraham Maslow]] === [[File:Carl Rogers.jpg|thumb|Figure 1. Carl Rogers|alt=Artist rendition of Carl Rogers]] == Clinical uses == * == Other uses == == Conclusion == === Marking rubric === ==== Overview ==== Easy to read and understand overview of the chapter. ==== Breadth ==== Theoretical framework for understanding the topic. ==== Depth ==== Clearly explain and integrate the theory(ies). ==== Key findings ==== Explain how key, peer-reviewed research findings apply to the problem. ==== Critical thinking ==== Critically analyse the research discussed. ==== Integration ==== Integrate discussion of theory and review of relevant research. ==== Conclusion ==== Emphasise the key points and take-home messages. ==== Written expression - Style ==== Readable for a layperson interested in psychological science. ==== Written expression - Learning features ==== Invite interactivity through features such as case studies, feature boxes, figures, links, tables, and quizzes. ==== Social contribution ==== Editing which enhances the quality of other book chapters. --- Add an image == Overview == You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] 85aea4u0mimb42qq69g6lugz7zntn8q 2416413 2416412 2022-08-19T03:24:17Z Sebastian Armstrong 2947157 /* History */ wikitext text/x-wiki {{title|Fully functioning person <br>What is a FFP and how can full functioning be developed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ == Plan == === Overview === ''What is a FFP and How can full functioning be developed?'' # <u>A growing openness to experience</u> # <u>An increasingly existential lifestyle</u> # <u>Increasing organismic trust</u> # <u>Freedom of choice</u> # <u>Creativity</u> # <u>Reliability and constructiveness</u> # <u>A rich full life</u> #* Rogers, Carl (1961). ''On becoming a person: A therapist's view of psychotherapy''. London: Constable. ISBN <bdi>978-1-84529-057-3</bdi>. == [[Caregiving and dementia/Topics/Person centred care|person-centered theory of personality]] == == History == <u>Both Plato and Aristotle proposed theories about goodness in which functioning worked as the most central concept. Arguments about goodness and functioning keep developing in contemporary work on human goodness. The work of Sen, Staudinger, and Tomasello serves as renowned examples. (https://doi.org/10.1007/978-3-319-24612-3_1469)</u><blockquote>Client-centered therapy, also called the person-centered approach, describes Carl R. Rogers’ way of working with persons experiencing all types of personal disturbances or problems in living (Rogers, 1959; 1961; 1969; 1970; 1972; 1980a; 1986a). As early as 1939, Rogers developed his theory of psychotherapy with troubled children, and went on to expand his theoretical approach to include work with couples, families, and groups. His most comprehensive theoretical statement was published as a chapter in Sigmund Koch’s Psychology: A Study of a Science (Vol. III) in 1959, and includes his theory of motivation and personality development, as well as theory of group interaction and interpersonal relationships (Koch, 1959, 184–256). Over his long career, Rogers extrapolated client-centered values to the education, marriage, group encounter, personal power, and conflict resolution (Rogers, 1969, 1970, 1972). Today, the person-centered approach is practiced in the United Kingdom, Germany, France, Greece, Portugal, Demark, Poland, Hungary, The Netherlands, Italy, Japan, Brazil, Mexico, Australia, and South Africa, as well as here in the United States and Canada. A world association, which can be contacted online, was founded in Lisbon in 1997 that reflects the growth and vitality of the approach entitled the World Association for Person-Centered and Experiential Psychotherapy and Counseling (WAPCEPC). Another international organization comprised of a diverse membership—lay persons, educators, business consultants, therapists, artists, psychologists—the Association for the Development of the Person-Centered Approach (ADPCA), is also accessible on the internet. Witty, M. C. (2007). Client-centered therapy. In ''Handbook of homework assignments in psychotherapy'' (pp. 35-50). Springer, Boston, MA.</blockquote> === [[Motivation and emotion/Tutorials/Growth psychology|Growth psychology]] === === [[Talk:Carl Rogers]] === === [[Motivation and emotion/Textbook/Motivation/Self-actualisation|Self-actualisation]] === === [[wikipedia:Abraham_Maslow|Abraham Maslow]] === [[File:Carl Rogers.jpg|thumb|Figure 1. Carl Rogers|alt=Artist rendition of Carl Rogers]] == Clinical uses == * == Other uses == == Conclusion == === Marking rubric === ==== Overview ==== Easy to read and understand overview of the chapter. ==== Breadth ==== Theoretical framework for understanding the topic. ==== Depth ==== Clearly explain and integrate the theory(ies). ==== Key findings ==== Explain how key, peer-reviewed research findings apply to the problem. ==== Critical thinking ==== Critically analyse the research discussed. ==== Integration ==== Integrate discussion of theory and review of relevant research. ==== Conclusion ==== Emphasise the key points and take-home messages. ==== Written expression - Style ==== Readable for a layperson interested in psychological science. ==== Written expression - Learning features ==== Invite interactivity through features such as case studies, feature boxes, figures, links, tables, and quizzes. ==== Social contribution ==== Editing which enhances the quality of other book chapters. --- Add an image == Overview == You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] 7qsekroghekulf8g4pxnbdzd8486zql 2416414 2416413 2022-08-19T03:29:06Z Sebastian Armstrong 2947157 /* Clinical uses */ Added references into Clinical wikitext text/x-wiki {{title|Fully functioning person <br>What is a FFP and how can full functioning be developed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ == Plan == === Overview === ''What is a FFP and How can full functioning be developed?'' # <u>A growing openness to experience</u> # <u>An increasingly existential lifestyle</u> # <u>Increasing organismic trust</u> # <u>Freedom of choice</u> # <u>Creativity</u> # <u>Reliability and constructiveness</u> # <u>A rich full life</u> #* Rogers, Carl (1961). ''On becoming a person: A therapist's view of psychotherapy''. London: Constable. ISBN <bdi>978-1-84529-057-3</bdi>. == [[Caregiving and dementia/Topics/Person centred care|person-centered theory of personality]] == == History == <u>Both Plato and Aristotle proposed theories about goodness in which functioning worked as the most central concept. Arguments about goodness and functioning keep developing in contemporary work on human goodness. The work of Sen, Staudinger, and Tomasello serves as renowned examples. (https://doi.org/10.1007/978-3-319-24612-3_1469)</u><blockquote>Client-centered therapy, also called the person-centered approach, describes Carl R. Rogers’ way of working with persons experiencing all types of personal disturbances or problems in living (Rogers, 1959; 1961; 1969; 1970; 1972; 1980a; 1986a). As early as 1939, Rogers developed his theory of psychotherapy with troubled children, and went on to expand his theoretical approach to include work with couples, families, and groups. His most comprehensive theoretical statement was published as a chapter in Sigmund Koch’s Psychology: A Study of a Science (Vol. III) in 1959, and includes his theory of motivation and personality development, as well as theory of group interaction and interpersonal relationships (Koch, 1959, 184–256). Over his long career, Rogers extrapolated client-centered values to the education, marriage, group encounter, personal power, and conflict resolution (Rogers, 1969, 1970, 1972). Today, the person-centered approach is practiced in the United Kingdom, Germany, France, Greece, Portugal, Demark, Poland, Hungary, The Netherlands, Italy, Japan, Brazil, Mexico, Australia, and South Africa, as well as here in the United States and Canada. A world association, which can be contacted online, was founded in Lisbon in 1997 that reflects the growth and vitality of the approach entitled the World Association for Person-Centered and Experiential Psychotherapy and Counseling (WAPCEPC). Another international organization comprised of a diverse membership—lay persons, educators, business consultants, therapists, artists, psychologists—the Association for the Development of the Person-Centered Approach (ADPCA), is also accessible on the internet. Witty, M. C. (2007). Client-centered therapy. In ''Handbook of homework assignments in psychotherapy'' (pp. 35-50). Springer, Boston, MA.</blockquote> === [[Motivation and emotion/Tutorials/Growth psychology|Growth psychology]] === === [[Talk:Carl Rogers]] === === [[Motivation and emotion/Textbook/Motivation/Self-actualisation|Self-actualisation]] === === [[wikipedia:Abraham_Maslow|Abraham Maslow]] === [[File:Carl Rogers.jpg|thumb|Figure 1. Carl Rogers|alt=Artist rendition of Carl Rogers]] == Clinical uses == *<blockquote>This book explores, in depth, the link between modern psychiatric practice and the person-centred approach. It promotes an open dialogue between traditional rivals – counsellors and psychiatrists within the NHS – to assist greater understanding and improve practice. Easy to read and comprehend, it explains complex issues in a clear and accessible manner. The author is a full-time psychiatrist and qualified counsellor who offers a unique perspective drawing on personal experience. Humanising Psychiatry and Mental Health Care will be of significant interest and help to all mental health professionals including psychiatrists and psychiatric nurses, social care workers, occupational therapists, psychologists, person-centred counsellors and therapists. Health and social care policy makers and shapers, including patient groups, will also find it helpful and informative. Freeth, R., Thorne, B., & Shooter, M. (2007). Humanising Psychiatry and Mental Health Care: The challenge of the person-centred approach (1st ed.). CRC Press. <nowiki>https://doi.org/10.1201/9781315385051</nowiki></blockquote> == Other uses == == Conclusion == === Marking rubric === ==== Overview ==== Easy to read and understand overview of the chapter. ==== Breadth ==== Theoretical framework for understanding the topic. ==== Depth ==== Clearly explain and integrate the theory(ies). ==== Key findings ==== Explain how key, peer-reviewed research findings apply to the problem. ==== Critical thinking ==== Critically analyse the research discussed. ==== Integration ==== Integrate discussion of theory and review of relevant research. ==== Conclusion ==== Emphasise the key points and take-home messages. ==== Written expression - Style ==== Readable for a layperson interested in psychological science. ==== Written expression - Learning features ==== Invite interactivity through features such as case studies, feature boxes, figures, links, tables, and quizzes. ==== Social contribution ==== Editing which enhances the quality of other book chapters. --- Add an image == Overview == You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] faseasdqus1aklfo070ac3zyu65z0dc 2416428 2416414 2022-08-19T04:42:09Z Sebastian Armstrong 2947157 /* Clinical uses */ wikitext text/x-wiki {{title|Fully functioning person <br>What is a FFP and how can full functioning be developed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ == Plan == === Overview === ''What is a FFP and How can full functioning be developed?'' # <u>A growing openness to experience</u> # <u>An increasingly existential lifestyle</u> # <u>Increasing organismic trust</u> # <u>Freedom of choice</u> # <u>Creativity</u> # <u>Reliability and constructiveness</u> # <u>A rich full life</u> #* Rogers, Carl (1961). ''On becoming a person: A therapist's view of psychotherapy''. London: Constable. ISBN <bdi>978-1-84529-057-3</bdi>. == [[Caregiving and dementia/Topics/Person centred care|person-centered theory of personality]] == == History == <u>Both Plato and Aristotle proposed theories about goodness in which functioning worked as the most central concept. Arguments about goodness and functioning keep developing in contemporary work on human goodness. The work of Sen, Staudinger, and Tomasello serves as renowned examples. (https://doi.org/10.1007/978-3-319-24612-3_1469)</u><blockquote>Client-centered therapy, also called the person-centered approach, describes Carl R. Rogers’ way of working with persons experiencing all types of personal disturbances or problems in living (Rogers, 1959; 1961; 1969; 1970; 1972; 1980a; 1986a). As early as 1939, Rogers developed his theory of psychotherapy with troubled children, and went on to expand his theoretical approach to include work with couples, families, and groups. His most comprehensive theoretical statement was published as a chapter in Sigmund Koch’s Psychology: A Study of a Science (Vol. III) in 1959, and includes his theory of motivation and personality development, as well as theory of group interaction and interpersonal relationships (Koch, 1959, 184–256). Over his long career, Rogers extrapolated client-centered values to the education, marriage, group encounter, personal power, and conflict resolution (Rogers, 1969, 1970, 1972). Today, the person-centered approach is practiced in the United Kingdom, Germany, France, Greece, Portugal, Demark, Poland, Hungary, The Netherlands, Italy, Japan, Brazil, Mexico, Australia, and South Africa, as well as here in the United States and Canada. A world association, which can be contacted online, was founded in Lisbon in 1997 that reflects the growth and vitality of the approach entitled the World Association for Person-Centered and Experiential Psychotherapy and Counseling (WAPCEPC). Another international organization comprised of a diverse membership—lay persons, educators, business consultants, therapists, artists, psychologists—the Association for the Development of the Person-Centered Approach (ADPCA), is also accessible on the internet. Witty, M. C. (2007). Client-centered therapy. In ''Handbook of homework assignments in psychotherapy'' (pp. 35-50). Springer, Boston, MA.</blockquote> === [[Motivation and emotion/Tutorials/Growth psychology|Growth psychology]] === === [[Talk:Carl Rogers]] === === [[Motivation and emotion/Textbook/Motivation/Self-actualisation|Self-actualisation]] === === [[wikipedia:Abraham_Maslow|Abraham Maslow]] === [[File:Carl Rogers.jpg|thumb|Figure 1. Carl Rogers|alt=Artist rendition of Carl Rogers]] == Clinical uses == <blockquote>This book explores, in depth, the link between modern psychiatric practice and the person-centred approach. It promotes an open dialogue between traditional rivals – counsellors and psychiatrists within the NHS – to assist greater understanding and improve practice. Easy to read and comprehend, it explains complex issues in a clear and accessible manner. The author is a full-time psychiatrist and qualified counsellor who offers a unique perspective drawing on personal experience. Humanising Psychiatry and Mental Health Care will be of significant interest and help to all mental health professionals including psychiatrists and psychiatric nurses, social care workers, occupational therapists, psychologists, person-centred counsellors and therapists. Health and social care policy makers and shapers, including patient groups, will also find it helpful and informative. Freeth, R., Thorne, B., & Shooter, M. (2007). Humanising Psychiatry and Mental Health Care: The challenge of the person-centred approach (1st ed.). CRC Press. <nowiki>https://doi.org/10.1201/9781315385051</nowiki></blockquote> == Other uses == == Conclusion == === Marking rubric === ==== Overview ==== Easy to read and understand overview of the chapter. ==== Breadth ==== Theoretical framework for understanding the topic. ==== Depth ==== Clearly explain and integrate the theory(ies). ==== Key findings ==== Explain how key, peer-reviewed research findings apply to the problem. ==== Critical thinking ==== Critically analyse the research discussed. ==== Integration ==== Integrate discussion of theory and review of relevant research. ==== Conclusion ==== Emphasise the key points and take-home messages. ==== Written expression - Style ==== Readable for a layperson interested in psychological science. ==== Written expression - Learning features ==== Invite interactivity through features such as case studies, feature boxes, figures, links, tables, and quizzes. ==== Social contribution ==== Editing which enhances the quality of other book chapters. --- Add an image == Overview == You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] jprlwn7w8zh597pcptx34q7tbs6dtdx 2416431 2416428 2022-08-19T04:52:08Z Sebastian Armstrong 2947157 /* History */ added references into History wikitext text/x-wiki {{title|Fully functioning person <br>What is a FFP and how can full functioning be developed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ == Plan == === Overview === ''What is a FFP and How can full functioning be developed?'' # <u>A growing openness to experience</u> # <u>An increasingly existential lifestyle</u> # <u>Increasing organismic trust</u> # <u>Freedom of choice</u> # <u>Creativity</u> # <u>Reliability and constructiveness</u> # <u>A rich full life</u> #* Rogers, Carl (1961). ''On becoming a person: A therapist's view of psychotherapy''. London: Constable. ISBN <bdi>978-1-84529-057-3</bdi>. == [[Caregiving and dementia/Topics/Person centred care|person-centered theory of personality]] == == History == <u>Both Plato and Aristotle proposed theories about goodness in which functioning worked as the most central concept. Arguments about goodness and functioning keep developing in contemporary work on human goodness. The work of Sen, Staudinger, and Tomasello serves as renowned examples. (https://doi.org/10.1007/978-3-319-24612-3_1469)</u><blockquote>Client-centered therapy, also called the person-centered approach, describes Carl R. Rogers’ way of working with persons experiencing all types of personal disturbances or problems in living (Rogers, 1959; 1961; 1969; 1970; 1972; 1980a; 1986a). As early as 1939, Rogers developed his theory of psychotherapy with troubled children, and went on to expand his theoretical approach to include work with couples, families, and groups. His most comprehensive theoretical statement was published as a chapter in Sigmund Koch’s Psychology: A Study of a Science (Vol. III) in 1959, and includes his theory of motivation and personality development, as well as theory of group interaction and interpersonal relationships (Koch, 1959, 184–256). Over his long career, Rogers extrapolated client-centered values to the education, marriage, group encounter, personal power, and conflict resolution (Rogers, 1969, 1970, 1972). Today, the person-centered approach is practiced in the United Kingdom, Germany, France, Greece, Portugal, Demark, Poland, Hungary, The Netherlands, Italy, Japan, Brazil, Mexico, Australia, and South Africa, as well as here in the United States and Canada. A world association, which can be contacted online, was founded in Lisbon in 1997 that reflects the growth and vitality of the approach entitled the World Association for Person-Centered and Experiential Psychotherapy and Counseling (WAPCEPC). Another international organization comprised of a diverse membership—lay persons, educators, business consultants, therapists, artists, psychologists—the Association for the Development of the Person-Centered Approach (ADPCA), is also accessible on the internet. Witty, M. C. (2007). Client-centered therapy. In ''Handbook of homework assignments in psychotherapy'' (pp. 35-50). Springer, Boston, MA.</blockquote><blockquote>According to Rogers, when fully functioning, the individual lives in close and confident relationship to the organismic valuing process, trusting that inner direction. Congruence is a constant companion. Furthermore, the fully functioning individual spontaneously communicates inner impulses both verbally and nonverbally. They are open to experience, accepts experiences as they are, and expresses those experiences in an unedited manner. The fully functioning individual is authentic. To characterize the moment-to-moment experience of the fully functioning individual, Figure 15.4 illustrates the sequential process of a motive’s emergence, acceptance, and unedited expression. Reeve, J. (2018). Understanding motivation and emotion. John Wiley & Sons.</blockquote> === [[Motivation and emotion/Tutorials/Growth psychology|Growth psychology]] === === [[Talk:Carl Rogers]] === === [[Motivation and emotion/Textbook/Motivation/Self-actualisation|Self-actualisation]] === === [[wikipedia:Abraham_Maslow|Abraham Maslow]] === [[File:Carl Rogers.jpg|thumb|Figure 1. Carl Rogers|alt=Artist rendition of Carl Rogers]] == Clinical uses == <blockquote>This book explores, in depth, the link between modern psychiatric practice and the person-centred approach. It promotes an open dialogue between traditional rivals – counsellors and psychiatrists within the NHS – to assist greater understanding and improve practice. Easy to read and comprehend, it explains complex issues in a clear and accessible manner. The author is a full-time psychiatrist and qualified counsellor who offers a unique perspective drawing on personal experience. Humanising Psychiatry and Mental Health Care will be of significant interest and help to all mental health professionals including psychiatrists and psychiatric nurses, social care workers, occupational therapists, psychologists, person-centred counsellors and therapists. Health and social care policy makers and shapers, including patient groups, will also find it helpful and informative. Freeth, R., Thorne, B., & Shooter, M. (2007). Humanising Psychiatry and Mental Health Care: The challenge of the person-centred approach (1st ed.). CRC Press. <nowiki>https://doi.org/10.1201/9781315385051</nowiki></blockquote> == Other uses == == Conclusion == === Marking rubric === ==== Overview ==== Easy to read and understand overview of the chapter. ==== Breadth ==== Theoretical framework for understanding the topic. ==== Depth ==== Clearly explain and integrate the theory(ies). ==== Key findings ==== Explain how key, peer-reviewed research findings apply to the problem. ==== Critical thinking ==== Critically analyse the research discussed. ==== Integration ==== Integrate discussion of theory and review of relevant research. ==== Conclusion ==== Emphasise the key points and take-home messages. ==== Written expression - Style ==== Readable for a layperson interested in psychological science. ==== Written expression - Learning features ==== Invite interactivity through features such as case studies, feature boxes, figures, links, tables, and quizzes. ==== Social contribution ==== Editing which enhances the quality of other book chapters. --- Add an image == Overview == You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] 7ptkabikhvsynci0qllfbpvw6bzprx1 2416438 2416431 2022-08-19T05:01:34Z Sebastian Armstrong 2947157 /* Growth psychology */ Adding more to history wikitext text/x-wiki {{title|Fully functioning person <br>What is a FFP and how can full functioning be developed?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ == Plan == === Overview === ''What is a FFP and How can full functioning be developed?'' # <u>A growing openness to experience</u> # <u>An increasingly existential lifestyle</u> # <u>Increasing organismic trust</u> # <u>Freedom of choice</u> # <u>Creativity</u> # <u>Reliability and constructiveness</u> # <u>A rich full life</u> #* Rogers, Carl (1961). ''On becoming a person: A therapist's view of psychotherapy''. London: Constable. ISBN <bdi>978-1-84529-057-3</bdi>. == [[Caregiving and dementia/Topics/Person centred care|person-centered theory of personality]] == == History == <u>Both Plato and Aristotle proposed theories about goodness in which functioning worked as the most central concept. Arguments about goodness and functioning keep developing in contemporary work on human goodness. The work of Sen, Staudinger, and Tomasello serves as renowned examples. (https://doi.org/10.1007/978-3-319-24612-3_1469)</u><blockquote>Client-centered therapy, also called the person-centered approach, describes Carl R. Rogers’ way of working with persons experiencing all types of personal disturbances or problems in living (Rogers, 1959; 1961; 1969; 1970; 1972; 1980a; 1986a). As early as 1939, Rogers developed his theory of psychotherapy with troubled children, and went on to expand his theoretical approach to include work with couples, families, and groups. His most comprehensive theoretical statement was published as a chapter in Sigmund Koch’s Psychology: A Study of a Science (Vol. III) in 1959, and includes his theory of motivation and personality development, as well as theory of group interaction and interpersonal relationships (Koch, 1959, 184–256). Over his long career, Rogers extrapolated client-centered values to the education, marriage, group encounter, personal power, and conflict resolution (Rogers, 1969, 1970, 1972). Today, the person-centered approach is practiced in the United Kingdom, Germany, France, Greece, Portugal, Demark, Poland, Hungary, The Netherlands, Italy, Japan, Brazil, Mexico, Australia, and South Africa, as well as here in the United States and Canada. A world association, which can be contacted online, was founded in Lisbon in 1997 that reflects the growth and vitality of the approach entitled the World Association for Person-Centered and Experiential Psychotherapy and Counseling (WAPCEPC). Another international organization comprised of a diverse membership—lay persons, educators, business consultants, therapists, artists, psychologists—the Association for the Development of the Person-Centered Approach (ADPCA), is also accessible on the internet. Witty, M. C. (2007). Client-centered therapy. In ''Handbook of homework assignments in psychotherapy'' (pp. 35-50). Springer, Boston, MA.</blockquote><blockquote>According to Rogers, when fully functioning, the individual lives in close and confident relationship to the organismic valuing process, trusting that inner direction. Congruence is a constant companion. Furthermore, the fully functioning individual spontaneously communicates inner impulses both verbally and nonverbally. They are open to experience, accepts experiences as they are, and expresses those experiences in an unedited manner. The fully functioning individual is authentic. To characterize the moment-to-moment experience of the fully functioning individual, Figure 15.4 illustrates the sequential process of a motive’s emergence, acceptance, and unedited expression. The congruent, fully functioning individual lives in close proximity to the actualizing tendency and therefore experiences a marked sense of autonomy, openness to experience,and personal growth. Reeve, J. (2018). Understanding motivation and emotion. John Wiley & Sons.</blockquote> === [[Motivation and emotion/Tutorials/Growth psychology|Growth psychology]] === <blockquote>A second criticism is that humanistic theorists use a number of vague and ill-defined constructs. It is difficult to pinpoint precisely what an “organismic valuing process” and a “fully functioning individual” are, for example. Any theoretical construct that evades a precise operational definition must remain scientifically dubious. For this reason, humanistic views on motivation have been harshly criticized (Daniels,1988; Neher, 1991), and these criticisms were instrumental to the rise and eventual popularity of the more empirically driven (evidence-based) positive psychology. Reeve, J. (2018). Understanding motivation and emotion. John Wiley & Sons.</blockquote> === [[Talk:Carl Rogers]] === <blockquote></blockquote> === [[Motivation and emotion/Textbook/Motivation/Self-actualisation|Self-actualisation]] === === [[wikipedia:Abraham_Maslow|Abraham Maslow]] === [[File:Carl Rogers.jpg|thumb|Figure 1. Carl Rogers|alt=Artist rendition of Carl Rogers]] == Clinical uses == <blockquote>This book explores, in depth, the link between modern psychiatric practice and the person-centred approach. It promotes an open dialogue between traditional rivals – counsellors and psychiatrists within the NHS – to assist greater understanding and improve practice. Easy to read and comprehend, it explains complex issues in a clear and accessible manner. The author is a full-time psychiatrist and qualified counsellor who offers a unique perspective drawing on personal experience. Humanising Psychiatry and Mental Health Care will be of significant interest and help to all mental health professionals including psychiatrists and psychiatric nurses, social care workers, occupational therapists, psychologists, person-centred counsellors and therapists. Health and social care policy makers and shapers, including patient groups, will also find it helpful and informative. Freeth, R., Thorne, B., & Shooter, M. (2007). Humanising Psychiatry and Mental Health Care: The challenge of the person-centred approach (1st ed.). CRC Press. <nowiki>https://doi.org/10.1201/9781315385051</nowiki></blockquote> == Other uses == == Conclusion == === Marking rubric === ==== Overview ==== Easy to read and understand overview of the chapter. ==== Breadth ==== Theoretical framework for understanding the topic. ==== Depth ==== Clearly explain and integrate the theory(ies). ==== Key findings ==== Explain how key, peer-reviewed research findings apply to the problem. ==== Critical thinking ==== Critically analyse the research discussed. ==== Integration ==== Integrate discussion of theory and review of relevant research. ==== Conclusion ==== Emphasise the key points and take-home messages. ==== Written expression - Style ==== Readable for a layperson interested in psychological science. ==== Written expression - Learning features ==== Invite interactivity through features such as case studies, feature boxes, figures, links, tables, and quizzes. ==== Social contribution ==== Editing which enhances the quality of other book chapters. --- Add an image == Overview == You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] bsndk0k7eyd8yyj6ght4b5hccvabq01 0 286021 2416392 2411722 2022-08-19T01:29:52Z U3215603 2947641 wikitext text/x-wiki {{title|Chapter title:<br>Subtitle?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== Work breaks, well being and productivity … You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main heading 1== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Main heading 2== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] b0wvr99tz3l448zrxbtyx459erxnyv8 0 286028 2416169 2416133 2022-08-18T15:06:10Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright" |+ <math>2>p>q>1</math> |+ <math>f_{pq}>f_p>f_q>f_0</math> |+ <math>T_0<T_p<T_q<T_{pq}</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- | colspan=3 |<math>T_{pq}= pqT_0</math> |- | colspan=3 ||<math>x</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- j6z02jl86kaei3xxl2k2wp4o4qvkrrj 2416170 2416169 2022-08-18T15:15:09Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>2>p>q>1</math> |+ <math>f_{pq}>f_p>f_q>f_0</math> |+ <math>T_0<T_p<T_q<T_{pq}</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- | colspan=3 |<math>T_{pq}= pqT_0</math> |- | colspan=3 ||<math>x</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- c5xxmk0g7b0dyj9rhb7oei1nfh3i2c4 2416173 2416170 2022-08-18T16:20:38Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>2>p>q>1</math> |+ <math>f_{pq}>f_p>f_q>f_0</math> |+ <math>T_0<T_p<T_q<T_{pq}</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- | colspan=3 ||<math>T_{pq}= pqT_0</math> |- | colspan=3 ||<math>x</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- ph9b1roqv3babhhay30mdcfeolrztg2 2416174 2416173 2022-08-18T16:21:47Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>2>p>q>1</math> |+ <math>f_{pq}>f_p>f_q>f_0</math> |+ <math>T_0<T_p<T_q<T_{pq}</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- | colspan=3 style="text-align:right; ||<math>T_{pq}= pqT_0</math> |- | colspan=3 ||<math>x</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 2oe2tng5mihtg3lddxs621lhybpir1l 2416175 2416174 2022-08-18T16:22:35Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>2>p>q>1</math> |+ <math>f_{pq}>f_p>f_q>f_0</math> |+ <math>T_0<T_p<T_q<T_{pq}</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- | "colspan=3" style="text-align:center;" ||<math>T_{pq}= pqT_0</math> |- | colspan=3 ||<math>x</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- rj6y7oiyebsyqphf31jwbide4ydgd9g 2416177 2416175 2022-08-18T16:27:24Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>2>p>q>1</math> |+ <math>f_{pq}>f_p>f_q>f_0</math> |+ <math>T_0<T_p<T_q<T_{pq}</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align:center;" ||<math>T_{pq}= pqT_0</math> |- | colspan=3 style="text-align:center;" ||<math>x</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- p7wqdqtkax7iw7z6ua1oqfwsga9ms4g 2416178 2416177 2022-08-18T16:28:06Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>2>p>q>1</math> |+ <math>f_{pq}>f_p>f_q>f_0</math> |+ <math>T_0<T_p<T_q<T_{pq}</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |rowspan=3 style="text-align:center;" ||<math>T_{pq}= pqT_0</math> |- | rowspan=3 style="text-align:center;" ||<math>x</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- t77u1eaajk72dnlkxrpf9axcbiv5bh4 2416179 2416178 2022-08-18T16:29:49Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>2>p>q>1</math> |+ <math>f_{pq}>f_p>f_q>f_0</math> |+ <math>T_0<T_p<T_q<T_{pq}</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=2 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=2 style="text-align: center;" |<math>x</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- qh3i4ll5672jeu6a1t06ngjrcblutvm 2416180 2416179 2022-08-18T16:30:14Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>2>p>q>1</math> |+ <math>f_{pq}>f_p>f_q>f_0</math> |+ <math>T_0<T_p<T_q<T_{pq}</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>x</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- h5cl94n3lcgkwpgnlnkhqgiwj5csv9c 2416181 2416180 2022-08-18T16:31:54Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>2>p>q>1</math> |+ <math>f_{pq}>f_p>f_q>f_0</math> |+ <math>T_0<T_p<T_q<T_{pq}</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>pqf_0=f_{pq}</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- g6ec4ch49hzex03eye849x3m73mvcs1 2416182 2416181 2022-08-18T16:35:34Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>pqf_0=f_{pq}</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- fza7g66u2clncsbymn6oym0jxidbn1i 2416183 2416182 2022-08-18T16:36:59Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q/math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=2 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |||<math>pqf_0=f_{pq}</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 03yfdiwzo1bcga1k6dw8f964orjgcld 2416184 2416183 2022-08-18T16:37:23Z Guy vandegrift 813252 Undo revision 2416183 by [[Special:Contributions/Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|talk]]) wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>pqf_0=f_{pq}</math> |} Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- fza7g66u2clncsbymn6oym0jxidbn1i 2416185 2416184 2022-08-18T16:38:09Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>pqf_0=f_{pq}</math> |} ===back=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>pqf_0=f_{pq}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- af3m26gxq71s2ss7bb7m7qz1y3tl350 2416186 2416185 2022-08-18T16:38:29Z Guy vandegrift 813252 /* Helmholtz model */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>pqf_0=f_{pq}</math> |} ===back=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>pqf_0=f_{pq}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: {{clear}} ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- n2iltznyats617ok2dds516ca08bxag 2416187 2416186 2022-08-18T16:38:54Z Guy vandegrift 813252 /* a */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>pqf_0=f_{pq}</math> |} ===back=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>pqf_0=f_{pq}</math> |} {{clear}} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 38q8dcbqzzsgd2hvs7vgppx9xcjcpg9 2416188 2416187 2022-08-18T16:39:39Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>pqf_0=f_{pq}</math> |} ===back=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>pqf_0=f_{pq}</math> |} {{clear}} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- ee2k0qi3s3c0l6pe7silx35k7lztbrx 2416189 2416188 2022-08-18T16:40:29Z Guy vandegrift 813252 /* back */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> |- |colspan=3 style="text-align: center;" |<math>pqf_0=f_{pq}</math> |} ===back=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> and <math>pqf_0=f_{pq}</math> |} {{clear}} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- e0hn7prgxq1dpe0wck27u75pz5qhszo 2416190 2416189 2022-08-18T16:41:24Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> &nbsp;and&nbsp; <math>pqf_0=f_{pq}</math> |} ===back=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> || |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> and <math>pqf_0=f_{pq}</math> |} {{clear}} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- njx0zkzru87880ki1r6vq2ykznnkmk3 2416191 2416190 2022-08-18T16:41:42Z Guy vandegrift 813252 /* back */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0</math> &nbsp;and&nbsp; <math>pqf_0=f_{pq}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- ohrooz3ar8f2v91yuusljjj6jag5pyh 2416192 2416191 2022-08-18T16:42:14Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0<</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad pqf_0=f_{pq}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- q7ktir0oe9r54atxvqdih2yknqo9v1c 2416193 2416192 2022-08-18T16:43:41Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>T_q=qT_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad pqf_0=f_{pq}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- ttoqcpxzrx9jpmy9wju6mxtqrb764u0 2416194 2416193 2022-08-18T16:44:43Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |+ <math>1<q<p<2</math> |+ <math>f_0<f_q<f_p<f_{pq}</math> |+ <math>T_{pq}<T_p<T_q<T_0</math> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad pqf_0=f_{pq}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- kfw338ywaddmeyxqp0l0moqf6jw02eh 2416195 2416194 2022-08-18T16:49:20Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad pqf_0=f_{pq}</math> |} An understanding of phase beating between two different frequencies requires the introduction of two additional frequencies. The introduction period (as well as frequency) makes the algebra difficult to follow. These inequalities might help. :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_{pq}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 6sy3rqgkyp2pvno9ira623m3muo5fml 2416196 2416195 2022-08-18T16:50:04Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad\quad pqf_0=f_{pq}</math> |} An understanding of phase beating between two different frequencies requires the introduction of two additional frequencies. The introduction period (as well as frequency) makes the algebra difficult to follow. These inequalities might help. :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_{pq}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- somf4czjser99w5ni24syafbbu44k5d 2416197 2416196 2022-08-18T16:51:15Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad\quad pqf_0=f_{pq}</math> |} An understanding of phase beating between two different frequencies requires the introduction of two additional frequencies. The introduction period (as well as frequency) renders the algebra difficult to follow. These inequalities might help. :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_{pq}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- eh87hog992f10jiwm5j1yoftwwd0uu4 2416199 2416197 2022-08-18T16:52:36Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable style=text-align:center"<!--"wikitable floatright style=text-align:center" --? |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad\quad pqf_0=f_{pq}</math> |} An understanding of phase beating between two different frequencies requires the introduction of two additional frequencies. The introduction period (as well as frequency) renders the algebra difficult to follow. These inequalities might help. :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_{pq}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- pcuaktvelow4zkxnev44x4j3avxd0sa 2416200 2416199 2022-08-18T16:52:56Z Guy vandegrift 813252 /* Consonance and musical intervals */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad\quad pqf_0=f_{pq}</math> |} An understanding of phase beating between two different frequencies requires the introduction of two additional frequencies. The introduction period (as well as frequency) renders the algebra difficult to follow. These inequalities might help. :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_{pq}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- pvvvme12n28tnj8l9nrcte30jeyeg6l 2416201 2416200 2022-08-18T16:53:17Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatleft style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad\quad pqf_0=f_{pq}</math> |} An understanding of phase beating between two different frequencies requires the introduction of two additional frequencies. The introduction period (as well as frequency) renders the algebra difficult to follow. These inequalities might help. :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_{pq}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 3o0zbw2eq6mhch0ch5fx09lo4e8wuhk 2416202 2416201 2022-08-18T16:53:46Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatleft style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad\quad pqf_0=f_{pq}</math> |} An understanding of phase beating between two different frequencies requires the introduction of two additional frequencies. The introduction period (as well as frequency) renders the algebra difficult to follow. These inequalities might help.{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_{pq}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- kl5axypptkyjqoft0x5gsn6pwll6cn2 2416203 2416202 2022-08-18T17:00:19Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatleft style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad\quad pqf_0=f_\text x}</math> |} An understanding of phase beating between two different frequencies requires the introduction of two additional frequencies. The introduction period (as well as frequency) renders the algebra difficult to follow. These inequalities might help.{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_{pq}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- nbr2c573v4p02xv8k51oz38w199i8qx 2416204 2416203 2022-08-18T17:00:40Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatleft style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad\quad pqf_0=f_\text x</math> |} An understanding of phase beating between two different frequencies requires the introduction of two additional frequencies. The introduction period (as well as frequency) renders the algebra difficult to follow. These inequalities might help.{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_{pq}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- hc2e3y0gebq8wtoslv4bwln03r76hnc 2416205 2416204 2022-08-18T17:00:58Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatleft style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_{pq}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} An understanding of phase beating between two different frequencies requires the introduction of two additional frequencies. The introduction period (as well as frequency) renders the algebra difficult to follow. These inequalities might help.{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_{pq}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- pu6lq8fyws2ht0w0ehw3pf27euk1o1s 2416206 2416205 2022-08-18T17:02:11Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatleft style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} An understanding of phase beating between two different frequencies requires the introduction of two additional frequencies. The introduction period (as well as frequency) renders the algebra difficult to follow. These inequalities might help.{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_{pq}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- t1ndhiu0tdwya4urbcgq0cyaaxl974k 2416207 2416206 2022-08-18T17:02:42Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatleft style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} An understanding of phase beating between two different frequencies requires the introduction of two additional frequencies. The introduction period (as well as frequency) renders the algebra difficult to follow. These inequalities might help.{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 200vge0vzuk4kch99uo6mpwsfh58zsa 2416209 2416207 2022-08-18T17:12:06Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatleft style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</mat> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math> as follows:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- djn0wm0suy1hytp8ptj3vr7i8pzwzri 2416210 2416209 2022-08-18T17:13:46Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatleft style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 07tcl2h2fv8s3ouv7bgmlygcylzcl9w 2416211 2416210 2022-08-18T17:14:09Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatleft style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- lrn1e2gvondcymluatre3z88kfja3fk 2416212 2416211 2022-08-18T17:15:08Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 7w57ntjzz4ro2hnbx0krv4m3tf9biiv 2416214 2416212 2022-08-18T17:18:51Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| border=1 style="border-collapse: collapse;" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- qq1qb56ts5zms6eex9qep88xzq9w9e5 2416215 2416214 2022-08-18T17:19:14Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| border=.5 style="border-collapse: collapse;" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- t53v2ybtdkryo9fg0kcc6hgjrouztyj 2416216 2416215 2022-08-18T17:19:35Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| border=.1 style="border-collapse: collapse;" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- hit6xtcmsasxusstumfrvsf8ncvem68 2416217 2416216 2022-08-18T17:20:40Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| style="border: 1px, border-collapse: collapse;" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- re69lma0m2eb61w39k3i3gl7843vru3 2416218 2416217 2022-08-18T17:21:49Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| style="border: 1px"<!--border-collapse: collapse;"--> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- cm8x6o7cmyep15zgykmi1mnwv8kq7em 2416219 2416218 2022-08-18T17:22:28Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| style="border: 4px"<!--border-collapse: collapse;"--> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- kqi3evz2apz5ee94kiyewuq3sl2mgqr 2416220 2416219 2022-08-18T17:23:03Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| "wikitable floatright style=text-align:center" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- kmcohy1elje87gsdko1wvmkkeued7ww 2416221 2416220 2022-08-18T17:23:30Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| "wikitable floatleft style=text-align:center" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- lond3hr9uv7jov6hh93mkgxvz6y6iur 2416222 2416221 2022-08-18T17:24:08Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| "wikitable floatright style=text-align:center" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 7xhpbvd4tb7fyql9pyc8m7im82s0qyo 2416223 2416222 2022-08-18T17:24:38Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- l7idg8wsr92mocyislwr14tp7oro0so 2416224 2416223 2022-08-18T17:26:05Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center bgcolor =white"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- l4i6dexcjx5giolqr62lmhbldds0ywk 2416225 2416224 2022-08-18T17:26:53Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center"<!--"wikitable floatright style=text-align:center" --> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 930njj3w7jnoam7iwmdu9c52obwk01g 2416226 2416225 2022-08-18T17:38:08Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- ka5zw559ljuleqoo9uugdzyo0ipbp8z 2416227 2416226 2022-08-18T17:40:09Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {|"padding-left: 2em" "style=text-align:center" "bgcolor ="white"" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- ebma0c119iurua0p10nui9un62uc1qz 2416228 2416227 2022-08-18T17:41:07Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {|"padding: 2em"; "style=text-align:center"; "bgcolor ="white"; "border=1" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- hgp1syyaczwsoi2hylzh5l35gsanz2f 2416229 2416228 2022-08-18T17:41:21Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {|"padding: 6em"; "style=text-align:center"; "bgcolor ="white"; "border=1" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- kqopg1ceekedaks3qqnwgw8mmzd8z53 2416230 2416229 2022-08-18T17:42:32Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center bgcolor="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- aaaiwhmu7ll0j18jlf5tr6noeclrgjw 2416231 2416230 2022-08-18T17:43:18Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center background=white |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 8la1wamkh4fk61xxj3ga8cs7h74951r 2416232 2416231 2022-08-18T17:43:39Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center background:white |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 929wl0zhcqsig0lq17xgm5wvn631tvx 2416233 2416232 2022-08-18T17:44:03Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {| class="wikitable floatright style=text-align:center background:white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- g1lmdgrbbgqgvu4yvw7zvq2vgr2ngil 2416239 2416233 2022-08-18T17:54:50Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {|{| border=1 style="border-collapse: collapse;" |- | A || B || C |- | D || E || F |} |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- thqaoq0yuf42tjzip6ygvp9zwyp4rge 2416240 2416239 2022-08-18T17:55:04Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {|{| border=1 style="border-collapse: collapse;" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- ctvqd6vhb5ng0b1fj01mnidcroc703y 2416241 2416240 2022-08-18T17:55:37Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {|{| border=1 style="border-collapse: collapse; text-align: center" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 6qus4r138e5tafuap6x3lzc6cfwer76 2416242 2416241 2022-08-18T17:56:53Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {|{| border=1 style="padding: 6em; border-collapse: collapse; text-align: center" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 55snukumzlrznwh2hyuuz70a1dkb8p5 2416244 2416242 2022-08-18T17:57:46Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {|{| border=1 style="padding: 6em;"<!-- border-collapse: collapse; text-align: center"--> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 45qmlqb4loz8j5qdvdtmufqnbh0w5jd 2416248 2416244 2022-08-18T17:59:22Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {|{| border=1 style="padding: 6em; text-align: center"--> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- mzg5tnh1uvw4fxikicthyysmoi9rvlp 2416251 2416248 2022-08-18T18:04:29Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== {|border=1 style="padding: 6em; text-align: center"--> |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 53j6g98e61n3vy9goj5tbv4v8vqskij 2416252 2416251 2022-08-18T18:05:07Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 6i9l8vltva0q4osdv2s4z94urzh6ipd 2416253 2416252 2022-08-18T18:08:08Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- ! <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- go2q36tk3w7jj4tunzyk35ant2kpz0t 2416254 2416253 2022-08-18T18:08:30Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- ! <math>f_p=pf_0</math> !! <math>f_q=qf_0</math> !!<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- tcqlhwkq0u2yxft7y0ztyudkaf61n94 2416255 2416254 2022-08-18T18:15:49Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 8aelbartqkzqhm8izw0em7rw8lm977c 2416256 2416255 2022-08-18T18:16:22Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatright style=text-align:center background-color:white |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 7b0x2x8lhew34yw996xix3xl4mzmlvc 2416257 2416256 2022-08-18T18:20:46Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} :<math>1<q<p<2</math> :<math>f_0<f_q<f_p<f_{pq}</math> :<math>T_\text{x}<T_p<T_q<T_0</math> ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- j4e92x5ok04ftxyt8gwpf2v1t2vqh1o 2416258 2416257 2022-08-18T18:26:51Z Guy vandegrift 813252 /* back */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} ====back==== {| class="wikitable floatright style=text-align:center bgcolor ="white" |- | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |- | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |- |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 8tzd0ha5we2daub153c1ofzlctm8h93 2416281 2416258 2022-08-18T20:55:34Z Guy vandegrift 813252 /* back */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} ==a== Notation will always be confusing if we mix periods and frequencies in the same discussion. Here we adopt the following convention: ===Helmholtz model=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. It is not difficult to calculate the beat frequency associated with this model: Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 4wl6ajvxovftzroahag4mobaqaus55w 2416284 2416281 2022-08-18T21:00:34Z Guy vandegrift 813252 /* a */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} ===Helmholtz argument=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> matches the {{math|p}}-harmonic of <math>f_q</math>. Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- sn20y9r195eo09d6bb7wc67us8g3h3j 2416285 2416284 2022-08-18T21:09:14Z Guy vandegrift 813252 /* Helmholtz argument */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} ===Helmholtz argument=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_beat = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- qsxjtrzfba4h6g2c4v5syvd9o4gljbf 2416286 2416285 2022-08-18T21:10:10Z Guy vandegrift 813252 /* Helmholtz argument */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} ===Helmholtz argument=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\beat{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- hb9zcegfnlpxnml7dnx9mrflybx95pv 2416288 2416286 2022-08-18T21:10:43Z Guy vandegrift 813252 /* Helmholtz argument */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} ===Helmholtz argument=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> Define interval ratio, <math>1<p/q<2</math>, where: <math>\omega_p=p\omega_0</math> and <math>\omega_q=q\omega_0</math> The the lowest common harmonic of <math>\omega_p</math> and <math>\omega_q</math> is <math>\omega_h=pq\omega_0</math> Replace <math>p\to p+\Delta p </math> and <math>q\to q+\Delta q </math> :<math>2\pi / T_b = \omega_\text{beat}=\left| (p+\Delta p)q\omega_0 - p(q+\Delta q)\omega_0 \right| = \left|p\Delta\omega_q - q\Delta\omega_p\right| \implies</math> :<math>f_b=\left|\frac{p\Delta\omega_q - q\Delta\omega_p}{2\pi}\right|= \left| p\Delta f_q - q\Delta f_p\right|</math> Use <math>q\omega_p=p\omega_q \equiv qf_p=pf_q \equiv pT_p = qT_q</math> where <math>T=1/f</math> denotes period. It is easy to show<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> that :<math>\frac{\Delta f}{f}=-\frac{\Delta T}{T}</math><math>\implies \Delta f = -\frac{\Delta T}{T^2}</math>. Hence we can also express the beat frequency in terms of beat periods: :<math>f_b=\frac{1}{T_b}=\left| \frac{q\Delta T_p}{T_p^2} - \frac{p\Delta T_q}{T_q^2} \right|= \left| \frac{p\Delta T_p-q\Delta T_q}{T_pT_q}\right|</math> These formulas should be trusted only in the limit that <math>\Delta f/f</math> is small. '''It is also essential to understand both formulas might be wrong!''' ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- q5fqaq7scqtcxtaz5qd2o7dqfmfu48k 2416289 2416288 2022-08-18T21:12:43Z Guy vandegrift 813252 /* Helmholtz argument */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} ===Helmholtz argument=== [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- tu66xwa6a2bm938m1dw9tldj9z0amji 2416290 2416289 2022-08-18T21:13:11Z Guy vandegrift 813252 /* Helmholtz argument */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Notation=== ====back==== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 9dh3aww4aq5piied17tfsj9hwc3yk8w 2416291 2416290 2022-08-18T21:14:39Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Cycle repetition== ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- kc45fgohq3qu84mn8i2fj540lfcgfsw 2416292 2416291 2022-08-18T21:15:21Z Guy vandegrift 813252 /* Cycle repetition */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. It also helps to keep in mind the following inequalities shown below:{{clear}} ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- nqm6molyfvyostk7x5os73jmvlgsudl 2416293 2416292 2022-08-18T21:18:59Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- e5re26dx56vzugujxzykm8fo6ufd0tu 2416294 2416293 2022-08-18T21:20:43Z Guy vandegrift 813252 /* Phase-shift beating */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white; text-align:center" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- kkn9tcrvnzw9ncg8lxvvhjnmeiaqhrh 2416295 2416294 2022-08-18T21:21:29Z Guy vandegrift 813252 /* Phase-shift beating */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white text-align:center" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- b720u9l5bvx67473ql2v5uv4a1a7xkd 2416296 2416295 2022-08-18T21:22:09Z Guy vandegrift 813252 Undo revision 2416295 by [[Special:Contributions/Guy vandegrift|Guy vandegrift]] ([[User talk:Guy vandegrift|talk]]) wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white" | <math>1<q<p<2</math> |-style="background-color:white; text-align:center" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- kkn9tcrvnzw9ncg8lxvvhjnmeiaqhrh 2416297 2416296 2022-08-18T21:22:47Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft style=text-align:center |-style="background-color:white" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white; text-align:center" | <math>1<q<p<2</math> |-style="background-color:white; text-align:center" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- f4c9cto7ad19c749qgnyjyldqme3r04 2416298 2416297 2022-08-18T21:23:44Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft |-style="background-color:white"| style=text-align:center | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white; text-align:center" | <math>1<q<p<2</math> |-style="background-color:white; text-align:center" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 2jlp86zwo26agb97oc3u50pfcb2xvfo 2416299 2416298 2022-08-18T21:24:44Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft |-style="background-color:white; style=text-align:center" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white; text-align:center" | <math>1<q<p<2</math> |-style="background-color:white; text-align:center" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- lxza5qk57s8529c9s3ba79ehwiz9s6h 2416300 2416299 2022-08-18T21:25:28Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft |-style="background-color:white; style=text-align:center" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white; style=text-align:center" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white; style=text-align:center" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white; text-align:center" | <math>1<q<p<2</math> |-style="background-color:white; text-align:center" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white; style=text-align:center" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 8iv1jsis6d2bq6vgfoc7ygu5d5kpb7y 2416301 2416300 2022-08-18T21:28:57Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft |-style="background-color:white; style=text-align:center" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white; style=text-align:center" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white; style=text-align:center" |colspan=3 style="text-align: center;"|<math>T_\text{x}= pqT_0 \quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white; text-align:center" | <math>1<q<p<2</math> |-style="background-color:white; text-align:center" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white; style=text-align:center" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 96ny1110gv4s6o7isq2pxro00wnjv23 2416302 2416301 2022-08-18T21:29:24Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft |-style="background-color:white; style=text-align:center" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white; style=text-align:center" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white; style=text-align:center" |colspan=3 |<math>T_\text{x}= pqT_0 \quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white; text-align:center" | <math>1<q<p<2</math> |-style="background-color:white; text-align:center" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white; style=text-align:center" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 604hag3j7r2eql1t6cox205c0z8z65f 2416303 2416302 2022-08-18T21:30:01Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft |-style="background-color:white; style=text-align:center" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white; style=text-align:center" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white; " |colspan=3 style=text-align:center|<math>T_\text{x}= pqT_0 \quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white; text-align:center" | <math>1<q<p<2</math> |-style="background-color:white; text-align:center" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white; style=text-align:center" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- g6bxxtd9uztmehbucgak8en2xlwk9rj 2416304 2416303 2022-08-18T21:31:25Z Guy vandegrift 813252 /* Notation */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft |-style="background-color:white; style=text-align:center" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white; style=text-align:center" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white; " |colspan=3 style=text-align:center|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white; text-align:center" | <math>1<q<p<2</math> |-style="background-color:white; text-align:center" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white; style=text-align:center" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|1400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 4o9488xnv8npna4idmv7n2orjciacu5 2416305 2416304 2022-08-18T21:32:34Z Guy vandegrift 813252 /* Images */ wikitext text/x-wiki __TOC__ ==Consonance and musical intervals== {{center|''The reader should know that this section is poorly understood. Look elsewhere if you want simple answers or a full discussion. <br>The intent here is to create an excuse for [https://www.pinterest.com/pin/164662930098107644/ "messing around"] with mathematical and computational methods.''}} [[Image:Beating Frequency.svg|thumb|300px|Diagram of beat frequency]] [[w:special:permalink/1102305761|Wikipedia has defined]] a '''beat''' as an [[w:Interference (wave propagation)|interference]] pattern between two [[w:sound|sounds]] of slightly different [[w:frequency|frequencies]], ''perceived'' as a periodic variation in [[w:amplitude (music)|volume]] whose rate is the [[w:Difference (mathematics)|difference]] of the two frequencies. [[w:Special:Permalink/1102305761#Mathematics_and_physics_of_beat_tones|The mathematics of beating]] can be summarized by the figure to the left, which leads to the following: ::<math> f_\text{beat} = f_2 - f_1 = \frac{1}{T_b} = \frac{\left|\omega_2-\omega_1\right|}{2\pi}</math> ::where, ::<math>fT =1</math>, and <math>\omega T = 2\pi,</math> ::are easy ways to remember the relation between frequency <math>f</math>, period <math>T</math>, and angular frequency, <math>\omega</math>. ===Helmholtz argument=== Footnote.<ref>Let <math>f(t)=T^{-1}</math>, and take the derivative to get, <math>df/dt=-T^{-2}</math>, which leads to: <math>df/f=-dT/T</math></ref> [[w:special:permalink/1101504020#Consonant_and_dissonant|Helmholtz]] proposed that beats produced by the consonant musical intervals are caused by beating between pairs of harmonics of the two pitches which have nearly the same frequency. The the {{math|q}}-th harmonic of <math>f_p</math> and the {{math|p}}-harmonic of <math>f_q</math> both equal <math>pqf_0</math>. Using the formula for beats between signals of the same frequency: <math>f_\text{beat} = pq\Delta f_0 = p\Delta f_q = q\Delta f_p</math> ==Phase-shift beating== ===Notation=== {| class="wikitable floatleft |-style="background-color:white; style=text-align:center" | <math>f_p=pf_0</math> || <math>f_q=qf_0</math> ||<math>qf_p=pf_q</math> |-style="background-color:white; style=text-align:center" | <math>pT_p=T_0</math> || <math>qT_q=T_0</math> ||<math>pT_p=qT_q</math> |-style="background-color:white; " |colspan=3 style=text-align:center|<math>T_\text{x}= pqT_0 \quad\quad\quad pqf_0=f_\text{x}</math> |} {| class="wikitable floatright style=text-align:center |-style="background-color:white; text-align:center" | <math>1<q<p<2</math> |-style="background-color:white; text-align:center" | <math>f_0<f_q<f_p<f_{pq}</math> |-style="background-color:white; style=text-align:center" |<math>T_\text{x}<T_p<T_q<T_0</math> |} Let <math>p</math> and <math>q</math> be relative prime numbers that establish a musical interval between frequencies <math>f_q</math> and <math>f_p</math> that are less than one octave apart. It helps to also define a low frequency <math>f_0</math> and a high frequency <math>f_\text{x}</math>. All these frequencies (and associated periods) are shown in the table to the left. The table to the right might help reader sort out these frequencies and periods. ===Images=== {{wide image|Beat pattern for fifth.svg|1600px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} {{wide image|File:Beat pattern for tritone.svg|2400px|Beat pattern for a 3:2 ratio (perfect fifth) that is out of tune by 57 cents.}} [[File:Beat pattern for tritone.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] [[File:Perfect fifth phase shift.svg|thumb|220px|Range of phase shifts for a just perfect fifth musical interval]] ===Discussion=== Consider a just interval between the two periods, <math>T_p=pT_0</math>, and <math>T_q=qT_0</math>, where <math>p=3</math> and <math>q=2</math> as shown in the figure. Because <math>T_p</math> and <math>T_q</math> make an exactly just interval, we have: :<math>q T_p =p T_q</math> ( <math>3 T_p = 2T_q</math> for a P5<ref>P5 denotes a perfect fifth, a 3:2 frequency ratio.</ref>) ALSO DEFINE A LONG PERIOD :<math>T_L=pqT_0=qT_p=pT_q</math> CHANGE EITHER <math>T_p\rightarrow T_p+\Delta T_p</math> or <math>T_q\rightarrow T_q+\Delta T_q</math> to change number of long periods by one: <math>(N+1)</math> ===Code=== ‎<syntaxhighlight lang="python"> triton=True if tritone: p , q = 7, 4 f_0=50 Df_p=2 else: p , q = 3, 2 f_0=100 Df_p=2 Df_q=0 f_p= p*f_0 f_q= q*f_0 Df_q=0 f_b=abs(p*Df_q - q*Df_p)#--------------------Helmholtz (checked) f_c=p*f_b/q#new T_c=1/f_c#new cents=1200*(np.log2(1+Df_p/f_p)-np.log2(1+Df_q/f_q))#error (cents) tbeat=1/f_b#-----------------------------------calculate beat period topi=2*np.pi om_p,om_q=(f_p+Df_p)*topi,(f_q+Df_q)*topi#-----define two omegas yp=Amplitude*np.cos(om_p*t)/2 yq=Amplitude*np.cos(om_q*t)/2 y=yp+yq </syntaxhighlight> ==Fourier analysis== {{cot|Not yet needed}} ''See also'' [[w:Kramers–Kronig relations]], [[w:Cauchy principal value]], [https://wiki.seg.org/wiki/Dictionary:Hilbert_transform]and [[w:Sokhotski–Plemelj theorem]] <math> \int_\infty^\infty e^{i\omega t}d\omega=2\pi\delta(t)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - i \pi \delta(X)</math> <math>\frac{1}{X+i\epsilon} = \frac {X}{X^2+ \epsilon^2} - \frac {i\epsilon}{X^2+ \epsilon^2}</math> <math>\frac 1 X = \text{pp} \frac 1 X -i\pi \delta(X)</math> {{cob}} ==Links== '''Basic''' * <u>[[Wikipedia:Beat (acoustics)]]</u>: The fact that Wikipedia only covers the basic ideas supports my contention that material beyond these well known topics will always be murky.<ref>{{cite journal |last= Weinberger |first= Norman |date= September 2006 |title= Music And The Brain |url= https://www.scientificamerican.com/article/music-and-the-brain-2006-09/ |journal= Scientific American |volume= 16 |issue= 3 |pages= 36-43 |doi= 10.1038/scientificamerican0906-36sp |accessdate=2022-08-04 }}</ref><ref>Note the inserted footnote templates regarding "verification" and "original research?" at [[w:special:permalink/1102305761]]. </ref> * <u>[http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/beat.html Hyperphysics: Sound/beat:]</u> parallels the Wikipedia article. * [https://www.omnicalculator.com/physics/beat-frequency <u>Omnicalculator's beat frequency demonstration</u>] effectively how the 3:2 rhythm pattern is just an ultra slow version of the consonant perfect fifth in music. '''Advanced (or otherwise different)''' * <u>[https://www.violinist.com/discussion/archive/18551/ Violinist.com]</u> discussion on using beats to tune a violin ----- 20cswl1hm6mr68ydurno0lk2cowi1t6 0 286127 2416275 2415957 2022-08-18T19:50:12Z Ougurlu 2947730 wikitext text/x-wiki <!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --> <!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DAF3EF; padding: 1em; overflow:hidden;"> <div style="overflow:hidden;"> <div style="width:50%; float:left;"> {{Eventmath lesson plan overview <!--EDIT BELOW THIS LINE ONLY--> <!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--> |assumed-knowledge = <!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--> Students are expected to have basic knowledge of combinatorics; basics of counting, permutations and combinations. |activities = <!--Below, please briefly summarize what students will do during this lesson and what they will learn. Using inclusion-exclusion to understand COVID reinfection--> Students will read an NPR article discussing the reinfection rate of COVID-19, and then they will create a combinatorial model. |class-time = <!--Below, please type one of the following, exactly as it appears: 00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes. Your best guess is fine! It can be changed later.--> 30-45 minutes <!--Next, you'll provide all citation data for the source (article/social media post/video). Article example: | source-website = NPR | source-title = The future of the pandemic is looking clearer as we learn more about infection | source-date = 2022-02-07 | source-url = https://www.npr.org/sections/goatsandsoda/2022/02/07/1057245449/the-future-of-the-pandemic-is-looking-clearer-as-we-learn-more-about-infection--> | source-website = NPR | source-title = The future of the pandemic is looking clearer as we learn more about infection | source-date = 2022-02-07 | source-url = https://www.npr.org/sections/goatsandsoda/2022/02/07/1057245449/the-future-of-the-pandemic-is-looking-clearer-as-we-learn-more-about-infection }} </div> <div style="float:left; width:20%;"> [[File:Eventmath_logo_bitmap_graphic.png|250px]] </div> </div> <div style="width:70%; float:left;"> <!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN 1. Note that "==My heading==" is a level 2 section heading, "===My subheading===" is level 3, etc. Level 1 is reserved for the title. 2. After you fill in a section, please delete the ''italicized instructions.'' 3. Check out the "Advanced" and "Help" menus at the top of the editor if you don't know how to format lists, etc. --> ==Activities== '''1. Reading:''' Students will be reading the article [https://www.npr.org/sections/goatsandsoda/2022/02/07/1057245449/the-future-of-the-pandemic-is-looking-clearer-as-we-learn-more-about-infection The future of the pandemic is looking clearer as we learn more about infection] before the class meeting. '''2. Discussion:'''Students will be discussing the stated opinions and will be asked to point out the mathematical evidence to support those opinions in the article. ==Assignments== ''You're welcome to suggest any exercises, activities, assignments, or projects based on the material of this lesson.'' ==Resources== ''(Include these sections at your discretion, keeping in mind that you and others can always edit and add more to these sections later.)'' ===Background=== ''You're welcome to share links to openly-accessible content (e.g. from Khan Academy, YouTube, Wikiversity, etc.) about domain knowledge or math skills students should have. Likewise, you may wish to include contextual information for the instructor.'' ===Explorations=== ''You're welcome to share references for additional learning and exploration, such as links to other articles, videos, spreadsheets, or computer code. When an open-access substitute is unavailable, links to paywalled sites are acceptable in this section.'' ==Feedback== <!--PLEASE DO NOT EDIT THIS SECTION DIRECTLY.--> Have you found this lesson plan helpful? Tell us about it! Just click ''Endorse'' below to open up an editor and type your comments. When you're ready, they'll appear at the bottom of this section to help other educators look for good lesson plans. (Alternatively, if you see a way to improve this lesson plan, be bold and make an edit! You're also welcome to discuss the lesson plan or provide constructive feedback on its [[{{TALKPAGENAME}}|Discussion page]].) <div style="float:left;"> <inputbox> type=commenttitle hidden=yes page={{FULLPAGENAME}} buttonlabel=Endorse arialabel=Endorse the lesson plan editintro=Eventmath/Lesson plan endorsement instructions preload=Template:Eventmath lesson plan endorsement </inputbox> </div> </div> <div style="width:10%; padding:4em; float:left;"> __TOC__ </div> </div> <!--CATEGORY TAG INSTRUCTIONS Please follow these instructions so that people can find this lesson plan! 1. Select a primary event type from the following list: government, health 2. Select a primary math type from the following list: combinatorics 3. Type your category tags directly underneath "[[Category: Eventmath lesson plans]]," at the bottom of this editor. 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For example, if you have a lesson plan for differential equations, you might decide to include that under the calculus category. It's up to you. --> [[Category: Eventmath lesson plans]] c20zwkcpi80kk9kts0tlaub1nb9rpyz 2 286188 2416404 2413036 2022-08-19T02:44:34Z U3215976 2947554 /* Social Contributions */ wikitext text/x-wiki Hello, I am a student of '''[[w:Psychology|Psychology]]''' at the ''[https://www.canberra.edu.au/future-students/?ef_id=CjwKCAjwi8iXBhBeEiwAKbUofV30BVaJPsDkgQ0xdtyJNdLiFDAd6pFhk9AOU1wCVoYTOdQP-gRGDxoCp4AQAvD_BwE:G:s&s_kwcid=AL!10441!3!589066594450!e!!g!!university%20of%20canberra!12340014388!118571458020&gclid=CjwKCAjwi8iXBhBeEiwAKbUofV30BVaJPsDkgQ0xdtyJNdLiFDAd6pFhk9AOU1wCVoYTOdQP-gRGDxoCp4AQAvD_BwE University of Canberra]''. I am currently studying a bachelor of science in psychology and I am in my second year of study. I am studying [[motivation and emotion]]. == About me: == I love skiing, snowboarding and riding mountain bikes. I have a keen interest in psychology. and I am keen to pursue post-graduate study after finishing my undergrad studies. == Book Chapter in progress == == Social Contributions == Book Chapter Edit - Social Cure 19/08/2022 == Hobbies == [[File:Klammer Franz 049.jpg|alt=Skiing|thumb|Figure 1. Downhill ski racing]] * [[wikipedia:Skiing|Skiing]] * Snowboarding * Downhill * Soccer 301o4hz81p3zhsqp2uylsn15uevjeob 2 286189 2416447 2413050 2022-08-19T06:40:26Z U3203265 2947925 /* About me */ wikitext text/x-wiki == About me == I am a university student at the [https://www.canberra.edu.au/ University of Canberra], studying [[w:Psychology|psychology]]. I have been studying at University of Canberra for four years. I am currently studying [[motivation and emotion]]. == Hobbies == * Basketball * Gaming * Writing * Bowling == Book chapter I'm working on == Interoception and Emotion. == Social Contributions == 5fswyi3wzand3e0nug6rhmnfu0yhk3g 0 286394 2416459 2416026 2022-08-19T08:55:35Z Hamish84 1362807 Adding text wikitext text/x-wiki Aristotle: “Give me the child until he is seven, and I will give you the man” The presumption is that all humans at seven years of age have embedded their so-called' “individual” personality. We develop and mature with personality intact. Living with the accepted concept of evolution, can we presume also that we evolve accordingly to our “individual” lifestyle. Human consciousness exerts its own definitive laws that are inescapable – never to destroy. The greatest form of communication that has evolved with human beings is speech. No matter what language you speak, it is a gift you have acquired along the way. You will pay a price, or gain a dividend, depending on how you use the consciousness gift we all have acquired. Which begs the question. Is there any such thing as “free speech”. My initial intention in searching for some clues to delay the onset of dementia arrived at the concept of semantics. There seemed to be some ancient wisdom that has already explored their existence. Namely one Heraclitus (The Weeping Philosopher) and his “Unity of Opposites”. [[File:Heraclitus, Johannes Moreelse.jpg|center|thumb]] Also the ancient Chinese symbol of the unity of opposites Yin and Yang: [[File:Traditional yin and yang with dots.png|center|thumb]] Both images above are repeats to earlier documents but quite appropriate to use again. Such unity applies to a reality which is absolute and confined only to understanding, beyond question. This is where I found contemporary scientific wisdom coming into play, and the development of The Semantic Template. The mistaken concept that any form of free speech provides anyone the right to use it to harm innocent people. So, “Never send to know for whom the bell tolls; it tolls for thee.” John Donne. Not censorship, but civilized restraint. Any form of civilized dialogue has its own benefits for both the recipient and the messenger. There is a moribund stupidity about so-called free speech. Stupidiy is at the bottom of the scale where Intelligence prevails. Perhaps with evolution it will die a natural death. Dung beetle and free speech retribution if by any chance you qualify for reincarnation! Given that we are supposed to be what we eat, can you imagine what these two “dung” beetles are calling each other over whose property it belongs to? [[File:Large Copper Dung Beetles (Kheper nigroaeneus) (6040799657).jpg|center|thumb]] The founder’s of social media communication on the Internet may well pay due regard to their office. No censorship, but the easy simplicity of civilized constructive debate, preferably sourced from the product of each semantic template. “The greatest form of knowledge you can ever have is your own” Whatever construction our brain imposes, its determinate need is to express its functional being. Everything that is awaits the arrival of its past. Everything that ever was, and always is, is always proclaimed, by its existence. Nothing is ever lost in the universal unity of Space-Time – Energy – Matter. Historically we have evolved towards the Jungian concept of collective human consciousness. Principles are the transparent manifestation of Nature and properties that are constantly evolving. They are ethical imperatives, and we have developed the positive properties of language to establish them for our use. [[File:Couple in Centennail Park 001.jpg|center|thumb]] We can only be defined through relationship principles for they offer us the best hope to recognize the factors that lead to complementation. Realationship = We depend on each others love to provide an honest and productive life for everyones future. There is a fundamental need to grasp simple common-sense (a sense of what is common) essentials. The Here and Now is not a temporary transitional time phase that we move in and out of. It is a constant certainty that is essential to recognize, so that our focus of attention has a foundation. Centrism can imply a fixation, which also implies vulnerability, which can be perfectly true if it does not lead to extension. To understand who we are, it is essential that we recognize and become aware of the very principles that we operate from. They encapsulate all the measure of any human societies ethics, morals, and laws, which is a continuous evolutionary educational process within which the realization of its total essence is always available. To use the doctrine that reason is a reliable tool to discover Truth – therefore ‘mutual agreement’ in the context of the traffic analogy where drivers at traffic lights ( to an observer) obey the rules of the road is ‘correct information’ which translates to the Truth, which translates in a circular way back to reason! Evolution is a constant dynamic process. The human phenomena ‘who we are’ is only understood in our union with each other, and ‘what is’. The paradox again is that there never is any separation. Separation is a mythical non-existent. The principles that are our necessities have continuous expansion properties that as humans we are privileged to assist their propagation. The human constellation in its evolutionary march must use these fundamental principles to ensure continuity. To maintain coherence and consistency our source is centered in the principles and factors that we have interpreted from our association with Nature, and Reality. marxy9dtxpxfs05aru612f1nmkq43nj 3 286408 2416137 2414729 2022-08-18T12:15:59Z WUser1307 2948261 wikitext text/x-wiki ----</noinclude>{{title|ERG Theory What is Alderfer's ERG theory?}}{{MECR3|https://yourlinkgoeshere.com}} == Overview == The ERG model was created by Clayton Alderfer and was based on Maslow's Hierarchy of Needs. This model stands for Existence, Relatedness and Growth, It is a three factor model based on the theory of motivation. Alderfer's ERG theory explains that individuals can be motivated by multiple levels of need at the same time, and that the level the level which is most important to them can change over time. These levels of motivation can move upwards or downwards at anytime making the individuals priorities fluid. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What are the three components to the ERG theory? * How can this theory be used? * How does this theory affect motivation? {{RoundBoxBottom}}{{tip|Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter.}} [[File:Alderfer's ERG Theory.svg|thumb|Figure 1. Alderfer's ERG Theory]] == Main headings == '''What are the three tiers of the ERG theory?''' Similar to Maslows Heirarch of Needs, the ERG theory suggests that a person has three main core groups of needs. As with life, these needs shift and change over depending on priorities and life in general. The first level is Existence and is categorised as a physiological need, also refered to as existence need. This entails categories such as food, water, shelter, and feeling safe. If you are unable to satisfy your existence needs then you are unable to move forward and reach your higher needs. For example, if you do not have enough food and don't have access to source any, then your life is in immediate peril and sourcing food will occupy most of your thoughts, at the expense of almost all of your other needs. The second level is Relatedness and refers to our need to relate to other people and develop relationships. While it is not a survival need, human needs good relationships and interactions in order to be happy and content. The third level is Growth and refers to our need of personal development, to be creative and perform meaningful work. This growth allows us to explore our day to day lives to see what potential may be in our environment. '''What is Frustration-Regression Principle?''' The ERG theory proposes that if a higher level need is not satisfied, a person may regress backwards to a lower level in order to fulfil their lower needs first. For example, if an ambitious employee is not given growth opportunities or duties which utilise their knowledge and skillset, then their motivation will drop. This will result in them becoming frustrated and focusing on their relatedness goals instead, which would involve forming relationships with other colleagues and socialising with the team more. Implementing the ERG theory in the workplace {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading.}} == Learning features == What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. === Case studies === Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[User:WUser1307#Feature boxes|feature boxes]]. === Boxes === Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} === Figures === Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[c:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). === Links === Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[wikipedia:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[wikipedia:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). === Tables === Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. === Quizzes === Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit":<quiz display="simple"> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz>To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. == Conclusion == The Conclusion is arguably the most important section. It should be possible for someone to read the [[User:WUser1307#Overview|Overview]] and the Conclusion and still get a good idea of the topic.{{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages?}} == See also == Provide up to half-a-dozen [[Help:Contents/Links#Interwiki links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[wikipedia:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[wikipedia:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[wikipedia:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses.}} == References == List the cited references in [[wikipedia:APA style|APA style]] (7th ed.) or [[wikipedia:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x}}https://worldofwork.io/2019/02/alderfers-erg-theory-of-motivation/<nowiki/>{{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted}} == External links == Provide up to half-a-dozen [[Help:Contents/Links#External links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link}}<includeonly> [[Category:{{#titleparts:{{PAGENAME}}|3}}]]</includeonly><noinclude> </noinclude> 6jak46retqcrc54pvcdlhcqq7ravbdl 0 286463 2416430 2415770 2022-08-19T04:49:59Z Hanarose123 2946863 wikitext text/x-wiki {{title|Courage: How is it motivated and how can it be enhanced?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== #* Define courage from an emotion perspective #* Define different types of courage #* Explain how courage can be expressed and how this differs between certain individuals #* Explain how courage can be motivated #* Explain how courage can be enhanced - physiologically and psychologically {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is courage? * What are the different types of courage? * How can courage be expressed? * How can courage be motivated? * How can courage be enhanced? {{RoundBoxBottom}} ==What is courage == * The three dimensions of courage (Gould, 2011) * Implicit theories of courage (Rate, Clarke, Lindsay, & Sternberg, 2007) ==What are the different types of courage?== * General v Personal courage (Pury, Kowalski, & Spearman, 2007) * Civil courage (Greitemeyer, Osswald, Fischer, & Frey, 2007) * Physical v Moral courage (Olsthoorn, 2007) * Courage in the military (Olsthoorn, 2007) == How can courage be expressed? == * The subjective act and experience of courage (Hannah, Sweeney, & Lester, 2007) * Theoretical and empirical evaluation of courage measure (Howard & Alipour, 2014) * Courage in nursing practices (Lindh, Barbosa da Silva, Berg, & Severinsson, 2010 * Courage expression through Locke and Latham’s Legacy (Oettingen, 2019) == What motivates courage? == * Having the courage to make the right decision in medical ethics (Malik, Conroy, & Turner, 2020) * Determinants of courage (Ozkaptan, 1994) == How can courage be enhanced? == * Skills to develop courage (Gould, 2011) * How to build courage (DeSimone, 2019) * Courageous actions and how to improve them (Pury & Kowalski, 2007) ==Conclusion == * Summarising key points - answer focus question; what is courage, what motivates it and what enhances it * Final message 1 - everyone has different levels of courage in different situations - do not be discouraged if you feel limited. * Final message 2 - courage can be enhanced, as shown above so do not panic or feel helpless if you currently have little courage ==See also== * [[wikipedia:Courage|Courage]] (Wikipedia) * ==References== DeSimone, B. B. (2019). Curriculum Redesign to Build the Moral Courage Values of Accelerated Bachelor's Degree Nursing Students. ''Journal Indexing & Metrics'', 1-10. <nowiki>https://doi.org/10.1177/2377960819827086</nowiki> Gould, N. H. (2011). Courage: Its Nature and Development. ''The Journal of Humanistic Counseling, Education and Development /'', 102-116. <nowiki>https://doi.org/10.1002/j.2164-490X.2005.tb00060.x</nowiki> Greitemeyer, T., Osswald, S., Fischer, P., & Frey, D. (2007). Civil courage: Implicit theories, related concepts, and measurement. ''The Journal of Positive Psychology'', 115-119. <nowiki>https://doi.org/10.1080/17439760701228789</nowiki> Hannah, S. T., Sweeney, P. J., & Lester, P. B. (2007). Toward a courageous mindset: The subjective act and experience of courage. ''The Journal of Positive Psychology'', 129-135. <nowiki>https://doi.org/10.1080/17439760701228854</nowiki> Howard, M. C., & Alipour, K. K. (2014). Does the courage measure really measure courage? A theoretical and empirical evaluation. ''The Journal of Positive Psychology'', 449-459. <nowiki>https://doi.org/10.1080/17439760.2014.910828</nowiki> Lindh, I.-B., Barbosa da Silva, A., Berg, A., & Severinsson, E. (2010). Courgae and nursing practice: A theoretical analysis. ''Nursing Ethics'', 551-565. <nowiki>https://doi.org/10.1177/0969733010369475</nowiki> Malik, A., Conroy, M., & Turner, C. (2020). Phronesis in Medical Ethics: Courage and Motivation to Keep on the Track of Rightness in Decision-Making. ''Health Care Analysis'', 158-175. <nowiki>https://doi.org/10.1007/s10728-020-00398-7</nowiki> Oettingen, G. (2019). Locke and Latham's Legacy Provides a Model of Courage. ''Motivation Science'', 108-109. <nowiki>http://dx.doi.org/10.1037/mot0000131</nowiki> Olsthoorn, P. (2007). Courage in the military: Physical and moral. ''Journal of Military Ethics'', 270-279. <nowiki>https://doi.org/10.1080/1502757070155471</nowiki> Ozkaptan, H. (1994). Determinants of Courage. ''Determinants of Effective Unit Performance: Research on Measuring and Managing Unit Training Readiness'', 233-245. retrieved from: <nowiki>https://books.google.com.au/books?hl=en&lr=&id=mM-pluC0FCwC&oi=fnd&pg=PA233&dq=does+fear+motivate+courage&ots=3GVZj3PhXF&sig=IjjyqlG2qVR6L_03tMTkgzX09Mk#v=onepage&q=does%20fear%20motivate%20courage&f=false</nowiki> Pury, C. L., & Kowalski, R. M. (2007). Human strengths, courageous actions, general and personal courage. ''The Journal of Positive Psychology'', 120-128. <nowiki>https://doi.org/10.1080/17439760701228813</nowiki> Pury, C. L., Kowalski, R. M., & Spearman, J. (2007). Distinctions between general and personal courage. ''The Journal of Positive Psychology'', 99-114. <nowiki>https://doi.org/10.1080/17439760701237962</nowiki> Rate, C. R., Clarke, J. A., Lindsay, D. R., & Sternberg, R. J. (2007). Implicit theories of courage. ''The Journal of Positive Psychology'', 80-98. <nowiki>https://doi.org/10.1080/17439760701228755</nowiki> ==External links== * https://www.youtube.com/watch?v=BuLBmE0OvuY<nowiki/>(Aristotle's Courage - Youtube) * https://www.youtube.com/watch?v=P5H0rDWgEeU (How to Build Your Courage - TEDtalk) * https://www.merriam-webster.com/dictionary/courage<nowiki/>(Courage - Merriam-Webster) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] rxe4ob4k0g05x8t1kknt4wflle7dj7c 2416433 2416430 2022-08-19T04:52:40Z Hanarose123 2946863 /* References */ wikitext text/x-wiki {{title|Courage: How is it motivated and how can it be enhanced?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== #* Define courage from an emotion perspective #* Define different types of courage #* Explain how courage can be expressed and how this differs between certain individuals #* Explain how courage can be motivated #* Explain how courage can be enhanced - physiologically and psychologically {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is courage? * What are the different types of courage? * How can courage be expressed? * How can courage be motivated? * How can courage be enhanced? {{RoundBoxBottom}} ==What is courage == * The three dimensions of courage (Gould, 2011) * Implicit theories of courage (Rate, Clarke, Lindsay, & Sternberg, 2007) ==What are the different types of courage?== * General v Personal courage (Pury, Kowalski, & Spearman, 2007) * Civil courage (Greitemeyer, Osswald, Fischer, & Frey, 2007) * Physical v Moral courage (Olsthoorn, 2007) * Courage in the military (Olsthoorn, 2007) == How can courage be expressed? == * The subjective act and experience of courage (Hannah, Sweeney, & Lester, 2007) * Theoretical and empirical evaluation of courage measure (Howard & Alipour, 2014) * Courage in nursing practices (Lindh, Barbosa da Silva, Berg, & Severinsson, 2010 * Courage expression through Locke and Latham’s Legacy (Oettingen, 2019) == What motivates courage? == * Having the courage to make the right decision in medical ethics (Malik, Conroy, & Turner, 2020) * Determinants of courage (Ozkaptan, 1994) == How can courage be enhanced? == * Skills to develop courage (Gould, 2011) * How to build courage (DeSimone, 2019) * Courageous actions and how to improve them (Pury & Kowalski, 2007) ==Conclusion == * Summarising key points - answer focus question; what is courage, what motivates it and what enhances it * Final message 1 - everyone has different levels of courage in different situations - do not be discouraged if you feel limited. * Final message 2 - courage can be enhanced, as shown above so do not panic or feel helpless if you currently have little courage ==See also== * [[wikipedia:Courage|Courage]] (Wikipedia) ==References== DeSimone, B. B. (2019). Curriculum Redesign to Build the Moral Courage Values of Accelerated Bachelor's Degree Nursing Students. ''Journal Indexing & Metrics'', 1-10. https://doi.org/10.1177/2377960819827086 Gould, N. H. (2011). Courage: Its Nature and Development. ''The Journal of Humanistic Counseling, Education and Development /'', 102-116. <nowiki>https://doi.org/10.1002/j.2164-490X.2005.tb00060.x</nowiki> Greitemeyer, T., Osswald, S., Fischer, P., & Frey, D. (2007). Civil courage: Implicit theories, related concepts, and measurement. ''The Journal of Positive Psychology'', 115-119. https://doi.org/10.1080/17439760701228789 Hannah, S. T., Sweeney, P. J., & Lester, P. B. (2007). Toward a courageous mindset: The subjective act and experience of courage. ''The Journal of Positive Psychology'', 129-135. https://doi.org/10.1080/17439760701228854 Howard, M. C., & Alipour, K. K. (2014). Does the courage measure really measure courage? A theoretical and empirical evaluation. ''The Journal of Positive Psychology'', 449-459. https://doi.org/10.1080/17439760.2014.910828 Lindh, I.-B., Barbosa da Silva, A., Berg, A., & Severinsson, E. (2010). Courgae and nursing practice: A theoretical analysis. ''Nursing Ethics'', 551-565. https://doi.org/10.1177/0969733010369475 Malik, A., Conroy, M., & Turner, C. (2020). Phronesis in Medical Ethics: Courage and Motivation to Keep on the Track of Rightness in Decision-Making. ''Health Care Analysis'', 158-175. https://doi.org/10.1007/s10728-020-00398-7 Oettingen, G. (2019). Locke and Latham's Legacy Provides a Model of Courage. ''Motivation Science'', 108-109. http://dx.doi.org/10.1037/mot0000131 Olsthoorn, P. (2007). Courage in the military: Physical and moral. ''Journal of Military Ethics'', 270-279. https://doi.org/10.1080/1502757070155471 Ozkaptan, H. (1994). Determinants of Courage. ''Determinants of Effective Unit Performance: Research on Measuring and Managing Unit Training Readiness'', 233-245. retrieved from: https://books.google.com.au/books?hl=en&lr=&id=mM-pluC0FCwC&oi=fnd&pg=PA233&dq=does+fear+motivate+courage&ots=3GVZj3PhXF&sig=IjjyqlG2qVR6L_03tMTkgzX09Mk#v=onepage&q=does%20fear%20motivate%20courage&f=false Pury, C. L., & Kowalski, R. M. (2007). Human strengths, courageous actions, general and personal courage. ''The Journal of Positive Psychology'', 120-128. https://doi.org/10.1080/17439760701228813 Pury, C. L., Kowalski, R. M., & Spearman, J. (2007). Distinctions between general and personal courage. ''The Journal of Positive Psychology'', 99-114. https://doi.org/10.1080/17439760701237962 Rate, C. R., Clarke, J. A., Lindsay, D. R., & Sternberg, R. J. (2007). Implicit theories of courage. ''The Journal of Positive Psychology'', 80-98. https://doi.org/10.1080/17439760701228755 ==External links== * https://www.youtube.com/watch?v=BuLBmE0OvuY<nowiki/>(Aristotle's Courage - Youtube) * https://www.youtube.com/watch?v=P5H0rDWgEeU (How to Build Your Courage - TEDtalk) * https://www.merriam-webster.com/dictionary/courage<nowiki/>(Courage - Merriam-Webster) [[Category:{{#titleparts:{{PAGENAME}}|3}}]] k4tqor3yrgwbxkxckw2wz27bex1dhjt 5 286502 2416172 2415958 2022-08-18T15:57:32Z Margob28 2943256 I have added a response. wikitext text/x-wiki Hello, I am Mariam. I am new on wikiversity. My field is humanities with a specialisation in English Language. I need someone who can sensitise me on the wikiversity projects and how to participate in those projects that are related to language development. {{Unsigned|Margob28|17 August 2022‎}} :{{At|Margob28}} Welcome! You'll find the [[Wikiversity:Colloquium]] is a better place to ask general Wikiversity questions. There are many resources on languages. See [[:Category:Languages]] for a starting point. See [[Help:Guides]] for resources to help you get started. I recommend reviewing several of the [[Wikiversity:Featured]] resources to give you an idea of how others have used Wikiversity effectively. As for how to participate, [[be bold]]! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:29, 18 August 2022 (UTC) :I appreciate your timely response. It is quite helpful. I will explore the recommended pages. tiedmevp5nbd1oc8m8wazgvt446vr3o 2416341 2416172 2022-08-18T23:01:09Z Dave Braunschweig 426084 Unsigned wikitext text/x-wiki Hello, I am Mariam. I am new on wikiversity. My field is humanities with a specialisation in English Language. I need someone who can sensitise me on the wikiversity projects and how to participate in those projects that are related to language development. {{Unsigned|Margob28|17 August 2022‎}} :{{At|Margob28}} Welcome! You'll find the [[Wikiversity:Colloquium]] is a better place to ask general Wikiversity questions. There are many resources on languages. See [[:Category:Languages]] for a starting point. See [[Help:Guides]] for resources to help you get started. I recommend reviewing several of the [[Wikiversity:Featured]] resources to give you an idea of how others have used Wikiversity effectively. As for how to participate, [[be bold]]! -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:29, 18 August 2022 (UTC) :I appreciate your timely response. It is quite helpful. I will explore the recommended pages. {{Unsigned|Margob28|18 August 2022‎}} ec7m7vh1nlv2ifflp2dmoa3pvhi3jtd 2 286519 2416134 2416120 2022-08-18T12:09:13Z WUser1307 2948261 /* Main headings */ wikitext text/x-wiki ----</noinclude>{{title|ERG Theory What is Alderfer's ERG theory?}}{{MECR3|https://yourlinkgoeshere.com}} == Overview == The ERG model was created by Clayton Alderfer and was based on Maslow's Hierarchy of Needs. This model stands for Existence, Relatedness and Growth, It is a three factor model based on the theory of motivation. Alderfer's ERG theory explains that individuals can be motivated by multiple levels of need at the same time, and that the level the level which is most important to them can change over time. These levels of motivation can move upwards or downwards at anytime making the individuals priorities fluid. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What are the three components to the ERG theory? * How can this theory be used? * How does this theory affect motivation? {{RoundBoxBottom}}{{tip|Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter.}} [[File:Alderfer's ERG Theory.svg|thumb|Figure 1. Alderfer's ERG Theory]] == Main headings == '''What are the three tiers of the ERG theory?''' Similar to Maslows Heirarch of Needs, the ERG theory suggests that a person has three main core groups of needs. As with life, these needs shift and change over depending on priorities and life in general. The first level is Existence and is categorised as a physiological need, also refered to as existence need. This entails categories such as food, water, shelter, and feeling safe. If you are unable to satisfy your existence needs then you are unable to move forward and reach your higher needs. For example, if you do not have enough food and don't have access to source any, then your life is in immediate peril and sourcing food will occupy most of your thoughts, at the expense of almost all of your other needs. The second level is Relatedness and refers to our need to relate to other people and develop relationships. While it is not a survival need, human needs good relationships and interactions in order to be happy and content. The third level is Growth and refers to our need of personal development, to be creative and perform meaningful work. This growth allows us to explore our day to day lives to see what potential may be in our environment. '''What is Frustration-Regression Principle?''' The ERG theory proposes that if a higher level need is not satisfied, a person may regress backwards to a lower level in order to fulfil their lower needs first. For example, if an ambitious employee is not given growth opportunities or duties which utilise their knowledge and skillset, then their motivation will drop. This will result in them becoming frustrated and focusing on their relatedness goals instead, which would involve forming relationships with other colleagues and socialising with the team more. Implementing the ERG theory in the workplace {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading.}} == Learning features == What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. === Case studies === Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[User:WUser1307#Feature boxes|feature boxes]]. === Boxes === Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} === Figures === Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[c:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). === Links === Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[wikipedia:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[wikipedia:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). === Tables === Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. === Quizzes === Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit":<quiz display="simple"> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz>To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. == Conclusion == The Conclusion is arguably the most important section. It should be possible for someone to read the [[User:WUser1307#Overview|Overview]] and the Conclusion and still get a good idea of the topic.{{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages?}} == See also == Provide up to half-a-dozen [[Help:Contents/Links#Interwiki links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[wikipedia:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[wikipedia:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[wikipedia:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses.}} == References == List the cited references in [[wikipedia:APA style|APA style]] (7th ed.) or [[wikipedia:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x}}https://worldofwork.io/2019/02/alderfers-erg-theory-of-motivation/<nowiki/>{{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted}} == External links == Provide up to half-a-dozen [[Help:Contents/Links#External links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link}}<includeonly> [[Category:{{#titleparts:{{PAGENAME}}|3}}]]</includeonly><noinclude> </noinclude> 6jak46retqcrc54pvcdlhcqq7ravbdl 2416138 2416134 2022-08-18T12:18:26Z WUser1307 2948261 Replaced content with "== About me == == Hobbies == == Book Chapter == [[Motivation and emotion/Book/2022/Conspiracy theory motivation|Conspiracy theory motivation]] - What motivates people to believe in conspiracy theories? == Social Contribtion ==" wikitext text/x-wiki == About me == == Hobbies == == Book Chapter == [[Motivation and emotion/Book/2022/Conspiracy theory motivation|Conspiracy theory motivation]] - What motivates people to believe in conspiracy theories? == Social Contribtion == o5hxhekwd54ygggn08c02k67ge51ghn 2416139 2416138 2022-08-18T12:19:51Z WUser1307 2948261 /* About me */ wikitext text/x-wiki == About me == I am studying a psychology degree at the University of Canberra. == Hobbies == My dog Theo Drinking Wine Making candles == Book Chapter == ERG Theory == Social Contribtion == 6yt2hziepjkrtn6j1sfuoa5qe3lrr30 6 286525 2416157 2022-08-18T13:26:23Z Young1lim 21186 {{Information |Description=Laurent.5: Applications 6A (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=Laurent.5: Applications 6A (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} f43j4bm5o7xj413rj81g9iuktmnabkt 6 286526 2416158 2022-08-18T13:27:08Z Young1lim 21186 {{Information |Description=C04.Series.1: Arrays 1A (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=C04.Series.1: Arrays 1A (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 7smtp2tk3c4y985jxkah6j4mfo0jjjn 6 286527 2416159 2022-08-18T13:27:53Z Young1lim 21186 {{Information |Description=C04.Series.3: Array Pointers 1A (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=C04.Series.3: Array Pointers 1A (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 6mkxaubcn89kj8vlnpnounra5ac37l7 6 286528 2416160 2022-08-18T13:28:35Z Young1lim 21186 {{Information |Description=C04.Series.1App: Applications of Arrays 1A (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=C04.Series.1App: Applications of Arrays 1A (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} gffg6v9dk33xjgk3sns4cfktnpz4cos 6 286529 2416161 2022-08-18T13:29:28Z Young1lim 21186 {{Information |Description=VLSI.Arith.1.A: Variable Block Adders (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=VLSI.Arith.1.A: Variable Block Adders (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 4xve51g32zr6ryuart92cvm7xwmcd8h 0 286530 2416234 2022-08-18T17:44:21Z 204.193.7.106 New resource with "Baton Rouge" wikitext text/x-wiki Baton Rouge 8mdxrs6n3684kts84rfcggrhnjfwcow 0 286531 2416259 2022-08-18T19:08:42Z Professorbrendan 2917898 New resource with "<!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --><!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DA..." wikitext text/x-wiki <!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --><!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DAF3EF; padding: 1em; overflow:hidden;"> <div style="overflow:hidden;"> <div style="width:50%; margin-right:4em; float:left;"> {{Eventmath lesson plan overview <!--EDIT BELOW THIS LINE ONLY--> <!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--> |assumed-knowledge = <!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--> Students only need basic arithmetic skills. |activities = <!--Below, please briefly summarize what students will do during this lesson and what they will learn.--> Students will read a tweet and perform some calculations.... |class-time = <!--Below, please type one of the following, exactly as it appears: 00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes. Your best guess is fine! It can be changed later.--> 00-15 minutes <!--Next, you'll provide all citation data for the source (article/social media post/video). Article example: | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2021-10-28 Post example: | source-website = Twitter | source-title = Remember: Germany is supposed to be a SUCCESS STORY. | source-date = 2020-12-22 | source-url = https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-url = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-date = 2021-02-16 Notes: 1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD, including leading zeros: January is 01, not 1, for example. 2. If you cannot find a date, use n.d. as the date (no date). 3. To ensure that the article/post will remain accessible for educators and students... a. Archive it at https://archive.org/web/ (use the "Save Page Now" form to generate a permanent URL). b. Paste that URL here as the source-archive-url. c. Enter the date you archived the source as the source-archive date. 4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may avoid spreading disinformation by omitting the url of the original site entirely. In that case, use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date. --> | source-website = | source-title = | source-date = | source-url = | source-archive-url = | source-archive-date = }} </div> <div style="float:left; width:20%;"> [[File:Eventmath_logo_bitmap_graphic.png|250px]] </div> </div> <div style="width:70%; float:left;"> <!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN 1. Note that "==My heading==" is a level 2 section heading, "===My subheading===" is level 3, etc. Level 1 is reserved for the title. 2. After you fill in a section, please delete the ''italicized instructions.'' 3. Check out the "Advanced" and "Help" menus at the top of the editor if you don't know how to format lists, etc. --> ==Activities== ''This is the main content of the lesson plan. Write as much or as little as needed to convey the main objective, the activities and tasks, and any other essential information. You may organize this content however you'd like.'' ''You can use LaTeX to render mathematical symbols, such as <math>\sum_{k=1}^n k = \frac{1}{2}n(n+1)</math>. See the [https://en.wikiversity.org/wiki/Help:Formula Help:Formula] page for more details.'' ''You may also wish to include helpful materials for an instructor, such as:'' * ''solutions to problems posed by the lesson,'' * ''anticipated points of confusion and suggestions,'' * ''discussion topics that may arise when working with students, or'' * ''any other relevant ideas or materials.'' ==Assignments== ''You're welcome to suggest exercises, activities, assignments, or projects based on the material of this lesson.'' ==Resources== ''(Include these sections at your discretion, keeping in mind that you and others can always edit and add more to these sections later.)'' ===Background=== ''You're welcome to share links to openly-accessible content (e.g. from Khan Academy, YouTube, Wikiversity, etc.) about domain knowledge or math skills students should have. Likewise, you may wish to include contextual information for the instructor.'' ===Explorations=== ''You're welcome to share references for additional learning and exploration, such as links to other articles, videos, spreadsheets, or computer code. When an open-access substitute is unavailable, links to paywalled sites are acceptable in this section.'' ==Feedback== <!--PLEASE DO NOT EDIT THIS SECTION DIRECTLY.--> Have you found this lesson plan helpful? Tell us about it! Just click ''Endorse'' below to open up an editor and type your comments. When you're ready, they'll appear at the bottom of this section to help other educators looking for good lesson plans. (Alternatively, if you see a way to improve this lesson plan, be bold and make an edit! You're also welcome to discuss the lesson plan or provide constructive feedback on its [[{{TALKPAGENAME}}|Discussion page]].) <div style="float:left;"> <inputbox> type=commenttitle hidden=yes page={{FULLPAGENAME}} buttonlabel=Endorse arialabel=Endorse the lesson plan editintro=Eventmath/Lesson plan endorsement instructions preload=Template:Eventmath lesson plan endorsement </inputbox> </div> </div> <div style="width:10%; padding:4em; float:left;"> __TOC__ </div> </div> <!--CATEGORY TAG INSTRUCTIONS Please follow these instructions so that people can find this lesson plan! 1. Select a primary event type from the following list: business, culture, economics, education, finance, government, health, politics, science, technology 2. Select a primary math type from the following list: algebra, arithmetic, calculus, combinatorics, geometry, logic, probability, statistics 3. Type your category tags directly underneath "[[Category: Eventmath lesson plans]]," at the bottom of this editor. For example, if you choose calculus and economics, you would have the following category tags on consecutive lines: [[Category: Eventmath lesson plans]] [[Category: Eventmath calculus]] [[Category: Eventmath economics]] NOTE: Be sure to use sentence casing for the category names (e.g. "Eventmath calculus" not "Eventmath Calculus"). FAQ Q: Can I apply multiple event or math tags? A: Yes! For example, you might want both a probability tag and a statistics tag. Q: Can I make a new category? A: Yes! Just type it in the same format. Example: [[Category: Eventmath number theory]] However, this category will not automatically have its own display box on the Lesson plans page. If you think there should be one, you can add one. If you don't know how, leave a comment on the Lesson plans "Discuss" tab. Alternatively, you might interpret an existing category more broadly. For example, if you have a lesson plan for differential equations, you might decide to include that under the calculus category. It's up to you. --> [[Category: Eventmath lesson plans]] 839iqb9eaw9z5kyrh64lehjsi37p4vo 0 286532 2416270 2022-08-18T19:28:08Z Davidejehle 2948441 New resource with "<!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --><!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DA..." wikitext text/x-wiki <!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --><!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DAF3EF; padding: 1em; overflow:hidden;"> <div style="overflow:hidden;"> <div style="width:50%; margin-right:4em; float:left;"> {{Eventmath lesson plan overview <!--EDIT BELOW THIS LINE ONLY--> <!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--> |assumed-knowledge = <!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--> Basic knowledge of percents and simple calculations of a weighted average |activities = <!--Below, please briefly summarize what students will do during this lesson and what they will learn.--> Based on a contingency table (cross-tabulation) of 2020 Presidential voting choices and voter age categories, calculate the percentage of voters that voted for Biden without knowing the actual number of votes cast. |class-time = <!--Below, please type one of the following, exactly as it appears: 00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes. Your best guess is fine! It can be changed later.--> 00-15 minutes <!--Next, you'll provide all citation data for the source (article/social media post/video). Article example: | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2021-10-28 Post example: | source-website = Twitter | source-title = Remember: Germany is supposed to be a SUCCESS STORY. | source-date = 2020-12-22 | source-url = https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-url = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-date = 2021-02-16 Notes: 1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD, including leading zeros: January is 01, not 1, for example. 2. If you cannot find a date, use n.d. as the date (no date). 3. To ensure that the article/post will remain accessible for educators and students... a. Archive it at https://archive.org/web/ (use the "Save Page Now" form to generate a permanent URL). b. Paste that URL here as the source-archive-url. c. Enter the date you archived the source as the source-archive date. 4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may avoid spreading disinformation by omitting the url of the original site entirely. In that case, use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date. --> | source-website = | source-title = | source-date = | source-url = | source-archive-url = | source-archive-date = }} </div> <div style="float:left; width:20%;"> [[File:Eventmath_logo_bitmap_graphic.png|250px]] </div> </div> <div style="width:70%; float:left;"> <!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN 1. Note that "==My heading==" is a level 2 section heading, "===My subheading===" is level 3, etc. Level 1 is reserved for the title. 2. After you fill in a section, please delete the ''italicized instructions.'' 3. Check out the "Advanced" and "Help" menus at the top of the editor if you don't know how to format lists, etc. --> ==Activities== ''This is the main content of the lesson plan. Write as much or as little as needed to convey the main objective, the activities and tasks, and any other essential information. You may organize this content however you'd like.'' ''You can use LaTeX to render mathematical symbols, such as <math>\sum_{k=1}^n k = \frac{1}{2}n(n+1)</math>. See the [https://en.wikiversity.org/wiki/Help:Formula Help:Formula] page for more details.'' ''You may also wish to include helpful materials for an instructor, such as:'' * ''solutions to problems posed by the lesson,'' * ''anticipated points of confusion and suggestions,'' * ''discussion topics that may arise when working with students, or'' * ''any other relevant ideas or materials.'' ==Assignments== ''You're welcome to suggest exercises, activities, assignments, or projects based on the material of this lesson.'' ==Resources== ''(Include these sections at your discretion, keeping in mind that you and others can always edit and add more to these sections later.)'' ===Background=== ''You're welcome to share links to openly-accessible content (e.g. from Khan Academy, YouTube, Wikiversity, etc.) about domain knowledge or math skills students should have. Likewise, you may wish to include contextual information for the instructor.'' ===Explorations=== ''You're welcome to share references for additional learning and exploration, such as links to other articles, videos, spreadsheets, or computer code. When an open-access substitute is unavailable, links to paywalled sites are acceptable in this section.'' ==Feedback== <!--PLEASE DO NOT EDIT THIS SECTION DIRECTLY.--> Have you found this lesson plan helpful? Tell us about it! Just click ''Endorse'' below to open up an editor and type your comments. When you're ready, they'll appear at the bottom of this section to help other educators looking for good lesson plans. (Alternatively, if you see a way to improve this lesson plan, be bold and make an edit! You're also welcome to discuss the lesson plan or provide constructive feedback on its [[{{TALKPAGENAME}}|Discussion page]].) <div style="float:left;"> <inputbox> type=commenttitle hidden=yes page={{FULLPAGENAME}} buttonlabel=Endorse arialabel=Endorse the lesson plan editintro=Eventmath/Lesson plan endorsement instructions preload=Template:Eventmath lesson plan endorsement </inputbox> </div> </div> <div style="width:10%; padding:4em; float:left;"> __TOC__ </div> </div> <!--CATEGORY TAG INSTRUCTIONS Please follow these instructions so that people can find this lesson plan! 1. Select a primary event type from the following list: business, culture, economics, education, finance, government, health, politics, science, technology 2. Select a primary math type from the following list: algebra, arithmetic, calculus, combinatorics, geometry, logic, probability, statistics 3. Type your category tags directly underneath "[[Category: Eventmath lesson plans]]," at the bottom of this editor. For example, if you choose calculus and economics, you would have the following category tags on consecutive lines: [[Category: Eventmath lesson plans]] [[Category: Eventmath calculus]] [[Category: Eventmath economics]] NOTE: Be sure to use sentence casing for the category names (e.g. "Eventmath calculus" not "Eventmath Calculus"). FAQ Q: Can I apply multiple event or math tags? A: Yes! For example, you might want both a probability tag and a statistics tag. Q: Can I make a new category? A: Yes! Just type it in the same format. Example: [[Category: Eventmath number theory]] However, this category will not automatically have its own display box on the Lesson plans page. If you think there should be one, you can add one. If you don't know how, leave a comment on the Lesson plans "Discuss" tab. Alternatively, you might interpret an existing category more broadly. For example, if you have a lesson plan for differential equations, you might decide to include that under the calculus category. It's up to you. --> [[Category: Eventmath lesson plans]] kzsne5umrctdnfrlkv3h1x23lzbcmku 2416273 2416270 2022-08-18T19:43:39Z Davidejehle 2948441 wikitext text/x-wiki <!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --><!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DAF3EF; padding: 1em; overflow:hidden;"> <div style="overflow:hidden;"> <div style="width:50%; margin-right:4em; float:left;"> {{Eventmath lesson plan overview <!--EDIT BELOW THIS LINE ONLY--> <!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--> |assumed-knowledge = <!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--> Basic knowledge of percents and simple calculations of a weighted average |activities = <!--Below, please briefly summarize what students will do during this lesson and what they will learn.--> Based on a contingency table (cross-tabulation) of 2020 Presidential voting choices and voter age categories, calculate the percentage of voters that voted for Biden without knowing the actual number of votes cast. |class-time = <!--Below, please type one of the following, exactly as it appears: 00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes. Your best guess is fine! It can be changed later.--> 00-15 minutes <!--Next, you'll provide all citation data for the source (article/social media post/video). Article example: | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2021-10-28 Post example: | source-website = Twitter | source-title = Remember: Germany is supposed to be a SUCCESS STORY. | source-date = 2020-12-22 | source-url = https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-url = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-date = 2021-02-16 Notes: 1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD, including leading zeros: January is 01, not 1, for example. 2. If you cannot find a date, use n.d. as the date (no date). 3. To ensure that the article/post will remain accessible for educators and students... a. Archive it at https://archive.org/web/ (use the "Save Page Now" form to generate a permanent URL). b. Paste that URL here as the source-archive-url. c. Enter the date you archived the source as the source-archive date. 4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may avoid spreading disinformation by omitting the url of the original site entirely. In that case, use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date. --> | source-website = Twitter | source-title = How old are you? | source-date = 2022-07-05 | source-url = https://twitter.com/umichvoter/status/1544376893121642497 | source-archive-url = https://web.archive.org/web/20220818194056/https://twitter.com/umichvoter/status/1544376893121642497 | source-archive-date = 2022-08-18 }} </div> <div style="float:left; width:20%;"> [[File:Eventmath_logo_bitmap_graphic.png|250px]] </div> </div> <div style="width:70%; float:left;"> <!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN 1. Note that "==My heading==" is a level 2 section heading, "===My subheading===" is level 3, etc. Level 1 is reserved for the title. 2. After you fill in a section, please delete the ''italicized instructions.'' 3. Check out the "Advanced" and "Help" menus at the top of the editor if you don't know how to format lists, etc. --> ==Activities== ''This is the main content of the lesson plan. Write as much or as little as needed to convey the main objective, the activities and tasks, and any other essential information. You may organize this content however you'd like.'' ''You can use LaTeX to render mathematical symbols, such as <math>\sum_{k=1}^n k = \frac{1}{2}n(n+1)</math>. See the [https://en.wikiversity.org/wiki/Help:Formula Help:Formula] page for more details.'' ''You may also wish to include helpful materials for an instructor, such as:'' * ''solutions to problems posed by the lesson,'' * ''anticipated points of confusion and suggestions,'' * ''discussion topics that may arise when working with students, or'' * ''any other relevant ideas or materials.'' ==Assignments== ''You're welcome to suggest exercises, activities, assignments, or projects based on the material of this lesson.'' ==Resources== ''(Include these sections at your discretion, keeping in mind that you and others can always edit and add more to these sections later.)'' ===Background=== ''You're welcome to share links to openly-accessible content (e.g. from Khan Academy, YouTube, Wikiversity, etc.) about domain knowledge or math skills students should have. Likewise, you may wish to include contextual information for the instructor.'' ===Explorations=== ''You're welcome to share references for additional learning and exploration, such as links to other articles, videos, spreadsheets, or computer code. When an open-access substitute is unavailable, links to paywalled sites are acceptable in this section.'' ==Feedback== <!--PLEASE DO NOT EDIT THIS SECTION DIRECTLY.--> Have you found this lesson plan helpful? Tell us about it! Just click ''Endorse'' below to open up an editor and type your comments. When you're ready, they'll appear at the bottom of this section to help other educators looking for good lesson plans. (Alternatively, if you see a way to improve this lesson plan, be bold and make an edit! You're also welcome to discuss the lesson plan or provide constructive feedback on its [[{{TALKPAGENAME}}|Discussion page]].) <div style="float:left;"> <inputbox> type=commenttitle hidden=yes page={{FULLPAGENAME}} buttonlabel=Endorse arialabel=Endorse the lesson plan editintro=Eventmath/Lesson plan endorsement instructions preload=Template:Eventmath lesson plan endorsement </inputbox> </div> </div> <div style="width:10%; padding:4em; float:left;"> __TOC__ </div> </div> <!--CATEGORY TAG INSTRUCTIONS Please follow these instructions so that people can find this lesson plan! 1. Select a primary event type from the following list: business, culture, economics, education, finance, government, health, politics, science, technology 2. Select a primary math type from the following list: algebra, arithmetic, calculus, combinatorics, geometry, logic, probability, statistics 3. Type your category tags directly underneath "[[Category: Eventmath lesson plans]]," at the bottom of this editor. For example, if you choose calculus and economics, you would have the following category tags on consecutive lines: [[Category: Eventmath lesson plans]] [[Category: Eventmath calculus]] [[Category: Eventmath economics]] NOTE: Be sure to use sentence casing for the category names (e.g. "Eventmath calculus" not "Eventmath Calculus"). FAQ Q: Can I apply multiple event or math tags? A: Yes! For example, you might want both a probability tag and a statistics tag. Q: Can I make a new category? A: Yes! Just type it in the same format. Example: [[Category: Eventmath number theory]] However, this category will not automatically have its own display box on the Lesson plans page. If you think there should be one, you can add one. If you don't know how, leave a comment on the Lesson plans "Discuss" tab. Alternatively, you might interpret an existing category more broadly. For example, if you have a lesson plan for differential equations, you might decide to include that under the calculus category. It's up to you. --> [[Category: Eventmath lesson plans]] smsblnoft6b0k6yjwddj04e7xbb5ywo 2416274 2416273 2022-08-18T19:48:16Z Davidejehle 2948441 wikitext text/x-wiki <!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --><!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DAF3EF; padding: 1em; overflow:hidden;"> <div style="overflow:hidden;"> <div style="width:50%; margin-right:4em; float:left;"> {{Eventmath lesson plan overview <!--EDIT BELOW THIS LINE ONLY--> <!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--> |assumed-knowledge = <!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--> Basic knowledge of percents and simple calculations of a weighted average |activities = <!--Below, please briefly summarize what students will do during this lesson and what they will learn.--> Based on a contingency table (cross-tabulation) of 2020 Presidential voting choices and voter age categories, calculate the percentage of voters that voted for Biden without knowing the actual number of votes cast. |class-time = <!--Below, please type one of the following, exactly as it appears: 00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes. Your best guess is fine! It can be changed later.--> 00-15 minutes <!--Next, you'll provide all citation data for the source (article/social media post/video). Article example: | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2021-10-28 Post example: | source-website = Twitter | source-title = Remember: Germany is supposed to be a SUCCESS STORY. | source-date = 2020-12-22 | source-url = https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-url = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-date = 2021-02-16 Notes: 1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD, including leading zeros: January is 01, not 1, for example. 2. If you cannot find a date, use n.d. as the date (no date). 3. To ensure that the article/post will remain accessible for educators and students... a. Archive it at https://archive.org/web/ (use the "Save Page Now" form to generate a permanent URL). b. Paste that URL here as the source-archive-url. c. Enter the date you archived the source as the source-archive date. 4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may avoid spreading disinformation by omitting the url of the original site entirely. In that case, use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date. --> | source-website = Twitter | source-title = How old are you? | source-date = 2022-07-05 | source-url = https://twitter.com/umichvoter/status/1544376893121642497 | source-archive-url = https://web.archive.org/web/20220818194056/https://twitter.com/umichvoter/status/1544376893121642497 | source-archive-date = 2022-08-18 }} </div> <div style="float:left; width:20%;"> [[File:Eventmath_logo_bitmap_graphic.png|250px]] </div> </div> <div style="width:70%; float:left;"> <!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN 1. Note that "==My heading==" is a level 2 section heading, "===My subheading===" is level 3, etc. Level 1 is reserved for the title. 2. After you fill in a section, please delete the ''italicized instructions.'' 3. Check out the "Advanced" and "Help" menus at the top of the editor if you don't know how to format lists, etc. --> ==Activities== ''This is the main content of the lesson plan. Write as much or as little as needed to convey the main objective, the activities and tasks, and any other essential information. You may organize this content however you'd like.'' ''You can use LaTeX to render mathematical symbols, such as <math>\sum_{k=1}^n k = \frac{1}{2}n(n+1)</math>. See the [https://en.wikiversity.org/wiki/Help:Formula Help:Formula] page for more details.'' ''You may also wish to include helpful materials for an instructor, such as:'' * ''solutions to problems posed by the lesson,'' * ''anticipated points of confusion and suggestions,'' * ''discussion topics that may arise when working with students, or'' * ''any other relevant ideas or materials.'' ==Assignments== ''You're welcome to suggest exercises, activities, assignments, or projects based on the material of this lesson.'' ==Resources== ''(Include these sections at your discretion, keeping in mind that you and others can always edit and add more to these sections later.)'' ===Background=== ''You're welcome to share links to openly-accessible content (e.g. from Khan Academy, YouTube, Wikiversity, etc.) about domain knowledge or math skills students should have. Likewise, you may wish to include contextual information for the instructor.'' ===Explorations=== ''You're welcome to share references for additional learning and exploration, such as links to other articles, videos, spreadsheets, or computer code. When an open-access substitute is unavailable, links to paywalled sites are acceptable in this section.'' ==Feedback== <!--PLEASE DO NOT EDIT THIS SECTION DIRECTLY.--> Have you found this lesson plan helpful? Tell us about it! Just click ''Endorse'' below to open up an editor and type your comments. When you're ready, they'll appear at the bottom of this section to help other educators looking for good lesson plans. (Alternatively, if you see a way to improve this lesson plan, be bold and make an edit! You're also welcome to discuss the lesson plan or provide constructive feedback on its [[{{TALKPAGENAME}}|Discussion page]].) <div style="float:left;"> <inputbox> type=commenttitle hidden=yes page={{FULLPAGENAME}} buttonlabel=Endorse arialabel=Endorse the lesson plan editintro=Eventmath/Lesson plan endorsement instructions preload=Template:Eventmath lesson plan endorsement </inputbox> </div> </div> <div style="width:10%; padding:4em; float:left;"> __TOC__ </div> </div> <!--CATEGORY TAG INSTRUCTIONS Please follow these instructions so that people can find this lesson plan! 1. Select a primary event type from the following list: business, culture, economics, education, finance, government, health, politics, science, technology 2. Select a primary math type from the following list: algebra, arithmetic, calculus, combinatorics, geometry, logic, probability, statistics 3. Type your category tags directly underneath "[[Category: Eventmath lesson plans]]," at the bottom of this editor. For example, if you choose calculus and economics, you would have the following category tags on consecutive lines: [[Category: Eventmath lesson plans]] [[Category: Eventmath calculus]] [[Category: Eventmath economics]] NOTE: Be sure to use sentence casing for the category names (e.g. "Eventmath calculus" not "Eventmath Calculus"). FAQ Q: Can I apply multiple event or math tags? A: Yes! For example, you might want both a probability tag and a statistics tag. Q: Can I make a new category? A: Yes! Just type it in the same format. Example: [[Category: Eventmath number theory]] However, this category will not automatically have its own display box on the Lesson plans page. If you think there should be one, you can add one. If you don't know how, leave a comment on the Lesson plans "Discuss" tab. Alternatively, you might interpret an existing category more broadly. For example, if you have a lesson plan for differential equations, you might decide to include that under the calculus category. It's up to you. --> [[Category: Eventmath lesson plans]] [[Category: Eventmath arithmetic]] [[Category: Eventmath politics]] du5vd81kc6m48jjim55m2fkof4i65ta 0 286533 2416272 2022-08-18T19:30:43Z Phillip Andreae 2948490 New resource with "<!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --><!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DA..." wikitext text/x-wiki <!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --><!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DAF3EF; padding: 1em; overflow:hidden;"> <div style="overflow:hidden;"> <div style="width:50%; margin-right:4em; float:left;"> {{Eventmath lesson plan overview <!--EDIT BELOW THIS LINE ONLY--> <!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--> |assumed-knowledge = <!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--> Familiarity with percentages |activities = <!--Below, please briefly summarize what students will do during this lesson and what they will learn.--> ... |class-time = <!--Below, please type one of the following, exactly as it appears: 00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes. Your best guess is fine! It can be changed later.--> 15-30 minutes <!--Next, you'll provide all citation data for the source (article/social media post/video). Article example: | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2021-10-28 Post example: | source-website = Twitter | source-title = Remember: Germany is supposed to be a SUCCESS STORY. | source-date = 2020-12-22 | source-url = https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-url = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-date = 2021-02-16 Notes: 1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD, including leading zeros: January is 01, not 1, for example. 2. If you cannot find a date, use n.d. as the date (no date). 3. To ensure that the article/post will remain accessible for educators and students... a. Archive it at https://archive.org/web/ (use the "Save Page Now" form to generate a permanent URL). b. Paste that URL here as the source-archive-url. c. Enter the date you archived the source as the source-archive date. 4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may avoid spreading disinformation by omitting the url of the original site entirely. In that case, use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date. --> | source-website = | source-title = | source-date = | source-url = | source-archive-url = | source-archive-date = }} </div> <div style="float:left; width:20%;"> [[File:Eventmath_logo_bitmap_graphic.png|250px]] </div> </div> <div style="width:70%; float:left;"> <!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN 1. Note that "==My heading==" is a level 2 section heading, "===My subheading===" is level 3, etc. Level 1 is reserved for the title. 2. After you fill in a section, please delete the ''italicized instructions.'' 3. Check out the "Advanced" and "Help" menus at the top of the editor if you don't know how to format lists, etc. --> ==Activities== ''This is the main content of the lesson plan. Write as much or as little as needed to convey the main objective, the activities and tasks, and any other essential information. You may organize this content however you'd like.'' ''You can use LaTeX to render mathematical symbols, such as <math>\sum_{k=1}^n k = \frac{1}{2}n(n+1)</math>. See the [https://en.wikiversity.org/wiki/Help:Formula Help:Formula] page for more details.'' ''You may also wish to include helpful materials for an instructor, such as:'' * ''solutions to problems posed by the lesson,'' * ''anticipated points of confusion and suggestions,'' * ''discussion topics that may arise when working with students, or'' * ''any other relevant ideas or materials.'' ==Assignments== ''You're welcome to suggest exercises, activities, assignments, or projects based on the material of this lesson.'' ==Resources== ''(Include these sections at your discretion, keeping in mind that you and others can always edit and add more to these sections later.)'' ===Background=== ''You're welcome to share links to openly-accessible content (e.g. from Khan Academy, YouTube, Wikiversity, etc.) about domain knowledge or math skills students should have. Likewise, you may wish to include contextual information for the instructor.'' ===Explorations=== ''You're welcome to share references for additional learning and exploration, such as links to other articles, videos, spreadsheets, or computer code. When an open-access substitute is unavailable, links to paywalled sites are acceptable in this section.'' ==Feedback== <!--PLEASE DO NOT EDIT THIS SECTION DIRECTLY.--> Have you found this lesson plan helpful? Tell us about it! Just click ''Endorse'' below to open up an editor and type your comments. When you're ready, they'll appear at the bottom of this section to help other educators looking for good lesson plans. (Alternatively, if you see a way to improve this lesson plan, be bold and make an edit! You're also welcome to discuss the lesson plan or provide constructive feedback on its [[{{TALKPAGENAME}}|Discussion page]].) <div style="float:left;"> <inputbox> type=commenttitle hidden=yes page={{FULLPAGENAME}} buttonlabel=Endorse arialabel=Endorse the lesson plan editintro=Eventmath/Lesson plan endorsement instructions preload=Template:Eventmath lesson plan endorsement </inputbox> </div> </div> <div style="width:10%; padding:4em; float:left;"> __TOC__ </div> </div> <!--CATEGORY TAG INSTRUCTIONS Please follow these instructions so that people can find this lesson plan! 1. Select a primary event type from the following list: business, culture, economics, education, finance, government, health, politics, science, technology 2. Select a primary math type from the following list: algebra, arithmetic, calculus, combinatorics, geometry, logic, probability, statistics 3. Type your category tags directly underneath "[[Category: Eventmath lesson plans]]," at the bottom of this editor. For example, if you choose calculus and economics, you would have the following category tags on consecutive lines: [[Category: Eventmath lesson plans]] [[Category: Eventmath calculus]] [[Category: Eventmath economics]] NOTE: Be sure to use sentence casing for the category names (e.g. "Eventmath calculus" not "Eventmath Calculus"). FAQ Q: Can I apply multiple event or math tags? A: Yes! For example, you might want both a probability tag and a statistics tag. Q: Can I make a new category? A: Yes! Just type it in the same format. Example: [[Category: Eventmath number theory]] However, this category will not automatically have its own display box on the Lesson plans page. If you think there should be one, you can add one. If you don't know how, leave a comment on the Lesson plans "Discuss" tab. Alternatively, you might interpret an existing category more broadly. For example, if you have a lesson plan for differential equations, you might decide to include that under the calculus category. It's up to you. --> [[Category: Eventmath lesson plans]] d382lew4t9qjr85q4l07bgbb3sqyd1h 2416276 2416272 2022-08-18T19:51:24Z Phillip Andreae 2948490 wikitext text/x-wiki <!--CONTRIBUTOR INSTRUCTIONS 1. This text is a comment: it'll be hidden from the published lesson plan, so it's just for you! If you see unfamiliar code, look for an explanation in a comment like this. 2. Heads up: you'll find a comment at the bottom with simple instructions on adding "category tags," which will help people find this lesson plan. Please follow them :) --><!-- DON'T MESS WITH THIS BIT --> {{Eventmath draft header}} <div style="background-color: #DAF3EF; padding: 1em; overflow:hidden;"> <div style="overflow:hidden;"> <div style="width:50%; margin-right:4em; float:left;"> {{Eventmath lesson plan overview <!--EDIT BELOW THIS LINE ONLY--> <!-- The summary below is displayed as a guide for visitors and contributors. You can publish without it, but it's required for removal from draft status.--> |assumed-knowledge = <!--Below, please briefly describe any mathematical skills or background knowledge students should have before completing this lesson.--> Familiarity with percentages |activities = <!--Below, please briefly summarize what students will do during this lesson and what they will learn.--> The goal is to investigate how Simpson's paradox leads to misleading vaccine efficacy data. Students could also create a graphical display of the data. |class-time = <!--Below, please type one of the following, exactly as it appears: 00-15 minutes, 15-30 minutes, 30-45 minutes, 45-60 minutes, 60-90 minutes, more than 90 minutes. Your best guess is fine! It can be changed later.--> 15-30 minutes <!--Next, you'll provide all citation data for the source (article/social media post/video). Article example: | source-website = Huffington Post | source-title = Voters In Wyoming Have 3.6 Times The Voting Power That I Have. It’s Time To End The Electoral College. | source-date = 2016-11-10 | source-url = https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-url = https://web.archive.org/web/20201210161426/https://www.huffpost.com/entry/its-time-to-end-the-electoral-college_b_12891764 | source-archive-date = 2021-10-28 Post example: | source-website = Twitter | source-title = Remember: Germany is supposed to be a SUCCESS STORY. | source-date = 2020-12-22 | source-url = https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-url = https://web.archive.org/web/20201223064324/https://twitter.com/Neoavatara/status/1341581271991013376 | source-archive-date = 2021-02-16 Notes: 1. To ensure lesson plans can be sorted by date, please enter all dates in the format YYYY-MM-DD, including leading zeros: January is 01, not 1, for example. 2. If you cannot find a date, use n.d. as the date (no date). 3. To ensure that the article/post will remain accessible for educators and students... a. Archive it at https://archive.org/web/ (use the "Save Page Now" form to generate a permanent URL). b. Paste that URL here as the source-archive-url. c. Enter the date you archived the source as the source-archive date. 4. Linking to a website helps the site. If the lesson plan is debunking disinformation, you may avoid spreading disinformation by omitting the url of the original site entirely. In that case, use the URL of the archived version as the value of source-url, and omit values for archive-url and archive-date. --> | source-website = Twitter | source-title = There has been lots of talk about recent data from Israel that seem to suggest a decline in vaccine efficacy against severe disease due to Delta, waning protection, or both. | source-date = 2021-08-17 | source-url = https://twitter.com/CT_Bergstrom/status/1427767356600688646 | source-archive-url = https://web.archive.org/web/20220818191600/https://twitter.com/CT_Bergstrom/status/1427767356600688646 | source-archive-date = 2022-08-18 }} </div> <div style="float:left; width:20%;"> [[File:Eventmath_logo_bitmap_graphic.png|250px]] </div> </div> <div style="width:70%; float:left;"> <!--INSTRUCTIONS FOR THE BODY OF THE LESSON PLAN 1. Note that "==My heading==" is a level 2 section heading, "===My subheading===" is level 3, etc. Level 1 is reserved for the title. 2. After you fill in a section, please delete the ''italicized instructions.'' 3. Check out the "Advanced" and "Help" menus at the top of the editor if you don't know how to format lists, etc. --> ==Activities== ''This is the main content of the lesson plan. Write as much or as little as needed to convey the main objective, the activities and tasks, and any other essential information. You may organize this content however you'd like.'' ''You can use LaTeX to render mathematical symbols, such as <math>\sum_{k=1}^n k = \frac{1}{2}n(n+1)</math>. See the [https://en.wikiversity.org/wiki/Help:Formula Help:Formula] page for more details.'' ''You may also wish to include helpful materials for an instructor, such as:'' * ''solutions to problems posed by the lesson,'' * ''anticipated points of confusion and suggestions,'' * ''discussion topics that may arise when working with students, or'' * ''any other relevant ideas or materials.'' ==Assignments== ''You're welcome to suggest exercises, activities, assignments, or projects based on the material of this lesson.'' ==Resources== ''(Include these sections at your discretion, keeping in mind that you and others can always edit and add more to these sections later.)'' ===Background=== ''You're welcome to share links to openly-accessible content (e.g. from Khan Academy, YouTube, Wikiversity, etc.) about domain knowledge or math skills students should have. Likewise, you may wish to include contextual information for the instructor.'' ===Explorations=== ''You're welcome to share references for additional learning and exploration, such as links to other articles, videos, spreadsheets, or computer code. When an open-access substitute is unavailable, links to paywalled sites are acceptable in this section.'' ==Feedback== <!--PLEASE DO NOT EDIT THIS SECTION DIRECTLY.--> Have you found this lesson plan helpful? Tell us about it! Just click ''Endorse'' below to open up an editor and type your comments. When you're ready, they'll appear at the bottom of this section to help other educators looking for good lesson plans. (Alternatively, if you see a way to improve this lesson plan, be bold and make an edit! You're also welcome to discuss the lesson plan or provide constructive feedback on its [[{{TALKPAGENAME}}|Discussion page]].) <div style="float:left;"> <inputbox> type=commenttitle hidden=yes page={{FULLPAGENAME}} buttonlabel=Endorse arialabel=Endorse the lesson plan editintro=Eventmath/Lesson plan endorsement instructions preload=Template:Eventmath lesson plan endorsement </inputbox> </div> </div> <div style="width:10%; padding:4em; float:left;"> __TOC__ </div> </div> <!--CATEGORY TAG INSTRUCTIONS Please follow these instructions so that people can find this lesson plan! 1. Select a primary event type from the following list: business, culture, economics, education, finance, government, health, politics, science, technology 2. Select a primary math type from the following list: algebra, arithmetic, calculus, combinatorics, geometry, logic, probability, statistics 3. Type your category tags directly underneath "[[Category: Eventmath lesson plans]]," at the bottom of this editor. For example, if you choose calculus and economics, you would have the following category tags on consecutive lines: [[Category: Eventmath lesson plans]] [[Category: Eventmath calculus]] [[Category: Eventmath economics]] NOTE: Be sure to use sentence casing for the category names (e.g. "Eventmath calculus" not "Eventmath Calculus"). FAQ Q: Can I apply multiple event or math tags? A: Yes! For example, you might want both a probability tag and a statistics tag. Q: Can I make a new category? A: Yes! Just type it in the same format. Example: [[Category: Eventmath number theory]] However, this category will not automatically have its own display box on the Lesson plans page. If you think there should be one, you can add one. If you don't know how, leave a comment on the Lesson plans "Discuss" tab. Alternatively, you might interpret an existing category more broadly. For example, if you have a lesson plan for differential equations, you might decide to include that under the calculus category. It's up to you. --> [[Category: Eventmath lesson plans]] k35pn32uricltj2nv7p3j0dv0qnx2cm 6 286534 2416315 2022-08-18T22:02:21Z Young1lim 21186 {{Information |Description=MP3.1F: Mutability - Lambda Calculus (20220817 - 20220816) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=MP3.1F: Mutability - Lambda Calculus (20220817 - 20220816) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 4udpd7us3f26j7zsbhypmtr2rh7yj2o 6 286535 2416319 2022-08-18T22:06:52Z Young1lim 21186 {{Information |Description=MP3.1F: Mutability - Lambda Calculus (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=MP3.1F: Mutability - Lambda Calculus (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} luyyneme3xgfnhx6va39sdsrle38063 6 286536 2416327 2022-08-18T22:32:46Z Young1lim 21186 {{Information |Description=ARM.2ASM: Vectored Interrupt Programming (20220817 - 20220816) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=ARM.2ASM: Vectored Interrupt Programming (20220817 - 20220816) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} pbe91nt69w3nj6vflwc38x9yp3aqw6v 6 286537 2416329 2022-08-18T22:34:08Z Young1lim 21186 {{Information |Description=ARM.2ASM: Vectored Interrupt Programming (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=ARM.2ASM: Vectored Interrupt Programming (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} kbyml7bzey66673ry2aezgihqlrc909 6 286538 2416337 2022-08-18T22:56:31Z Young1lim 21186 {{Information |Description=ARM.4ASM: UART Architecture (20220817 - 20220816) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=ARM.4ASM: UART Architecture (20220817 - 20220816) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 1ux4s1evvci9rkq4tmrylx2hf73or7f 6 286539 2416339 2022-08-18T22:57:32Z Young1lim 21186 {{Information |Description=ARM.4ASM: UART Architecture (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} wikitext text/x-wiki == Summary == {{Information |Description=ARM.4ASM: UART Architecture (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{GFDL}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 7dqh1u6cqwbuoleqwgi9tahkvrno3jp 3 286540 2416343 2022-08-18T23:01:42Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] U3200859!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:01, 18 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} i7n4yzeksyws7aa75fne1ik1z3heyxq 3 286541 2416344 2022-08-18T23:01:48Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Phillip Andreae!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:01, 18 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} tqzyflj2bpdf5odhayhwwej1ff950nj 3 286542 2416345 2022-08-18T23:02:00Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Davidejehle!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:01, 18 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} 1bedjoft2by52lkfb3gbkmiew1pjc54 3 286543 2416347 2022-08-18T23:05:35Z Dave Braunschweig 426084 Welcome wikitext text/x-wiki {{Robelbox|theme=9|title=Welcome!|width=100%}} <div style="{{Robelbox/pad}}"> '''Hello and [[Wikiversity:Welcome|Welcome]] to [[Wikiversity:What is Wikiversity|Wikiversity]] Nsebhastian!''' You can [[Wikiversity:Contact|contact us]] with [[Wikiversity:Questions|questions]] at the [[Wikiversity:Colloquium|colloquium]] or [[User talk:Dave Braunschweig|me personally]] when you need [[Help:Contents|help]]. Please remember to [[Wikiversity:Signature|sign and date]] your finished comments when [[Wikiversity:Who are Wikiversity participants?|participating]] in [[Wikiversity:Talk page|discussions]]. The signature icon [[File:OOjs UI icon signature-ltr.svg]] above the edit window makes it simple. All users are expected to abide by our [[Wikiversity:Privacy policy|Privacy]], [[Wikiversity:Civility|Civility]], and the [[Foundation:Terms of Use|Terms of Use]] policies while at Wikiversity. To [[Wikiversity:Introduction|get started]], you may <!-- The Left column --> <div style="width:50.0%; float:left"> * [[Help:guides|Take a guided tour]] and learn [[Help:Editing|to edit]]. * Visit a (kind of) [[Wikiversity:Random|random project]]. * [[Wikiversity:Browse|Browse]] Wikiversity, or visit a portal corresponding to your educational level: [[Portal: Pre-school Education|pre-school]], [[Portal: Primary Education|primary]], [[Portal:Secondary Education|secondary]], [[Portal:Tertiary Education|tertiary]], [[Portal:Non-formal Education|non-formal education]]. * Find out about [[Wikiversity:Research|research]] activities on Wikiversity. * [[Wikiversity:Introduction explore|Explore]] Wikiversity with the links to your left. </div> <!-- The Right column --> <div style="width:50.0%; float:left"> * Enable VisualEditor under [[Special:Preferences#mw-prefsection-betafeatures|Beta]] settings to make article editing easier. * Read an [[Wikiversity:Wikiversity teachers|introduction for teachers]] and find out [[Help:How to write an educational resource|how to write an educational resource]] for Wikiversity. * Give [[Wikiversity:Feedback|feedback]] about your initial observations. * Discuss Wikiversity issues or ask questions at the [[Wikiversity:Colloquium|colloquium]]. * [[Wikiversity:Chat|Chat]] with other Wikiversitans on [[:freenode:wikiversity|<kbd>#wikiversity</kbd>]]. </div> <br clear="both"/> You do not need to be an educator to edit. You only need to [[Wikiversity:Be bold|be bold]] to contribute and to experiment with the [[wikiversity:sandbox|sandbox]] or [[special:mypage|your userpage]]. See you around Wikiversity! --[[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:05, 18 August 2022 (UTC)</div> <!-- Template:Welcome --> {{Robelbox/close}} kp9paoh3r3jg5hzkty6yixslaqw61rl 2 286544 2416351 2022-08-18T23:15:47Z U3200859 2947964 Started author page wikitext text/x-wiki About me Hi, I am Zoe and I am a student at the University of Canberra. I am studying a double degree of Psychology and Business. rx7ajp59l8rnj3tz2wjc5e7185j4210 2416352 2416351 2022-08-18T23:17:10Z U3200859 2947964 heading wikitext text/x-wiki == About me == Hi, I am Zoe and I am a student at the University of Canberra. I am studying a double degree of Psychology and Business. qvzl1fj4iaoe79gbzfnn5orcmq803xp 2416355 2416352 2022-08-18T23:19:51Z U3200859 2947964 Headings wikitext text/x-wiki == About me == Hi, I am Zoe and I am a student at the University of Canberra. I am studying a double degree of Psychology and Business. This semester I am enrolled in [[Motivation and emotion|Motivation and Emotion]]. === Hobbies === === Book Chapter === The book chapter that I am working on is [[Motivation and emotion/Book/2022/Honesty motivation|honesty motivation]] === Social Contributions === f527uy1ves7ysmzliuob1fbjb5uexth 2416356 2416355 2022-08-18T23:20:19Z U3200859 2947964 wikitext text/x-wiki == About me == Hi, I am Zoe and I am a student at the University of Canberra. I am studying a double degree of Psychology and Business. This semester I am enrolled in [[Motivation and emotion|Motivation and Emotion]]. === Hobbies === == Book Chapter == The book chapter that I am working on is [[Motivation and emotion/Book/2022/Honesty motivation|honesty motivation]] == Social Contributions == dq23h664nqy2ibgmmmxci47avn4xzoy 2416359 2416356 2022-08-18T23:26:48Z U3200859 2947964 wikitext text/x-wiki == About me == Hi, I am Zoe and I am a student at the [https://www.canberra.edu.au/ University of Canberra]. I am studying a double degree of Psychology and Business. This semester I am enrolled in [[Motivation and emotion|Motivation and Emotion]]. === Hobbies === == Book Chapter == The book chapter that I am working on is [[Motivation and emotion/Book/2022/Honesty motivation|honesty motivation]] == Social Contributions == ckeekuprlnkc1t3y7x8ntikqbj747g6 2416360 2416359 2022-08-18T23:28:03Z U3200859 2947964 wikitext text/x-wiki == About me == Hi, I am Zoe and I am a student at the [https://www.canberra.edu.au/ University of Canberra]. I am studying a double degree of [[Psychology]] and [[Business]]. This semester I am enrolled in [[Motivation and emotion|Motivation and Emotion]]. === Hobbies === == Book Chapter == The book chapter that I am working on is [[Motivation and emotion/Book/2022/Honesty motivation|honesty motivation]] == Social Contributions == nddq504r4qsf00m3vtcjxfweju4y0hz 2416361 2416360 2022-08-18T23:29:09Z U3200859 2947964 wikitext text/x-wiki == About me == Hi, I am Zoe and I am a student at the [https://www.canberra.edu.au/ University of Canberra]. I am studying a double degree of [[wikipedia:Psychology|Psychology]] and [[wikipedia:Business|Business]]. This semester I am enrolled in [[Motivation and emotion|Motivation and Emotion]]. === Hobbies === == Book Chapter == The book chapter that I am working on is [[Motivation and emotion/Book/2022/Honesty motivation|honesty motivation]] == Social Contributions == 06fsp2l7np6y4jil76h6zexm6ups8he 6 286545 2416354 2022-08-18T23:19:45Z Young1lim 21186 {{Information |Description=ELF1.1E: Weak and Common Symbols (20220817 - 20220816) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=ELF1.1E: Weak and Common Symbols (20220817 - 20220816) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} 65v8mo4lj1m5amp3ibpgg5m585expqd 6 286546 2416358 2022-08-18T23:20:56Z Young1lim 21186 {{Information |Description=ELF1.1E: Weak and Common Symbols (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=ELF1.1E: Weak and Common Symbols (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-18 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} noofde88lw8dsk50dss9x8p8c623xom 2 286547 2416368 2022-08-19T00:14:46Z Cedevlin9 2947652 added bio details wikitext text/x-wiki About me Hi, I am a student of ''Psychology'' and ''Counselling'' at the '''University of Canberra'''. qrleqeehafyv6tl6hukjd07r7fvj4qt 2416370 2416368 2022-08-19T00:20:26Z Cedevlin9 2947652 headings wikitext text/x-wiki == About me == Hi, I am a student of ''Psychology'' and ''Counselling'' at the '''University of Canberra'''. === Hobbies === == Book chapter I'm working on == == Social contributions == 1zyn0kbedjcb42hclm6njlp7jkkhinp 2416372 2416370 2022-08-19T00:26:41Z Cedevlin9 2947652 /* Hobbies */ wikitext text/x-wiki == About me == Hi, I am a student of ''Psychology'' and ''Counselling'' at the '''University of Canberra'''. === Hobbies === * Yoga * Writing * Camping with my husband in our 1975 T2 Kombi, ''Vernie'' == Book chapter I'm working on == == Social contributions == 4jio27gky3c3ldtcqdo7tuvbzcralhy 2416375 2416372 2022-08-19T00:41:17Z Cedevlin9 2947652 /* About me */ wikitext text/x-wiki == About me == Hi, I am a student of ''[[w:Psychology|Psychology]]'' and ''[[wikipedia:Counseling_psychology|Counselling]]'' at the '''[https://www.canberra.edu.au/ University of Canberra]'''. I am studying [[motivation and emotion]] this semester. === Hobbies === * Yoga * Writing * Camping with my husband in our 1975 T2 Kombi, ''Vernie'' == Book chapter I'm working on == [[Motivation and emotion/Book/2022/Freedom and motivation|Freedom and Motivation]] == Social contributions == ii7kleiv1dom11e0p6mq3ucvw3lnq91 2416378 2416375 2022-08-19T00:53:56Z Cedevlin9 2947652 wikitext text/x-wiki == About me == Hi, I am a student of ''[[w:Psychology|Psychology]]'' and ''[[wikipedia:Counseling_psychology|Counselling]]'' at the '''[https://www.canberra.edu.au/ University of Canberra]'''. I am studying [[motivation and emotion]] this semester. === Hobbies === * Yoga * Writing * Camping with my husband in our 1975 T2 Kombi, ''Vernie'' == Book chapter I'm working on == [[Motivation and emotion/Book/2022/Freedom and motivation|Freedom and Motivation]] == Social contributions == 9xzxpwuzx68ukbnuc5ifmrrd31oqe26 0 286548 2416402 2022-08-19T02:40:04Z U3215976 2947554 New resource with "{{subst:ME/BCS}}" wikitext text/x-wiki {{title|Chapter title:<br>Subtitle?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] thp5r91aziue2l80cis4wxipwwn2uzy 2416403 2416402 2022-08-19T02:42:42Z U3215976 2947554 wikitext text/x-wiki {{title|Social cure - What is the social cure and how can it be applied?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] 1wcjz9shhx28rx8bnzjbo7632ukvn7c 0 286549 2416437 2022-08-19T05:00:05Z U3216256 2942574 New resource with "{{subst:ME/BCS}} ~~~~" wikitext text/x-wiki {{title|Chapter title:<br>Subtitle?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[User:U3216256|U3216256]] ([[User talk:U3216256|discuss]] • [[Special:Contributions/U3216256|contribs]]) 05:00, 19 August 2022 (UTC) ikx50uy002c7e63567i2ld8hpyfh11d 2416439 2416437 2022-08-19T05:01:39Z U3216256 2942574 Added title and subtitle wikitext text/x-wiki {{title|Cognitive dissonance reduction:<br>What strategies do people use to reduce cognitive dissonance and how effective are they?}} {{MECR3|1=https://yourlinkgoeshere.com}} __TOC__ ==Overview== You are underway {{smile}}! This template provides tips for [[Motivation and emotion/Assessment/Topic|topic development]]. Gradually remove these suggestions as you develop the chapter. Also consult the [[Motivation and emotion/Assessment/Chapter|author guidelines]]. At the top of the chapter, the title and sub-title should match the ''exact'' wording and casing as shown in the {{Motivation and emotion/Book}}. The sub-titles all end with a question mark. This Overview section should be concise but consist of several paragraphs which engage the reader, illustrate the problem, and outline how psychological science can help. {{RoundBoxTop|theme=3}} '''Focus questions:''' * What is the first focus question? * What is the second focus question? * What is the third focus question? {{RoundBoxBottom}} {{tip| Suggestions for this section: * What is the problem? Why is it important? * How can specific motivation and/or emotion theories and research help? * Provide an example or case study. * Conclude with Focus questions to guide the chapter. }} ==Main headings== How you are going to structure the chapter? Aim for three to six main headings between the [[#Overview|Overview]] and [[#Conclusion|Conclusion]]. {{tip|Suggestions for this section: * For the [[Motivation and emotion/Assessment/Topic|topic development]], provide at least 3 bullet-points about key content per section. Include key citations. * For the [[Motivation and emotion/Assessment/Chapter|book chapter]], expand the bullet points into paragraphs. * If a section has a lot of content, arrange it into two to five sub-headings such as in the [[#Interactive learning features|interactive learning features section]]. Avoid having sections with only one sub-heading. }} ==Learning features== What brings an online book chapter to life are its interactive learning features. Case studies, feature boxes, figures, links, tables, and quiz questions can be used throughout the chapter. ===Case studies=== Case studies describe real-world examples of concepts in action. Case studies can be real or fictional. A case could be used multiple times during a chapter to illustrate different theories or stages. It is often helpful to present case studies using [[#Feature boxes|feature boxes]]. ===Boxes=== Boxes can be used to highlight content, but don't overuse them. There are many different ways of creating boxes (e.g., see [[Help:Pretty boxes|Pretty boxes]]). Possible uses include: * Focus questions * Case studies or examples * Quiz questions * Take-home messages {{RoundBoxTop|theme=3}} ;Feature box example * Shaded background * Coloured border {{RoundBoxBottom}} ===Figures=== [[File:Monkey-typing.jpg|right|205px|thumb|''Figure 1''. Example image with descriptive caption.]] Use figures to illustrate concepts, add interest, and provide examples. Figures can be used to show photographs, drawings, diagrams, graphs, etcetera. Figures can be embedded throughout the chapter, starting with the Overview section. Figures should be captioned (using a number and a description) in order to explain their relevance to the text. Possible images can be found at [[commons:|Wikimedia Commons]]. Images can also be uploaded if they are licensed for re-use or if you created the image. Each figure should be referred to at least once in the main text (e.g., see Figure 1). ===Links=== Where key words are first used, make them into [[Help:Links|interwiki links]] such as Wikipedia links to articles about famous people (e.g., [[w:Sigmund Freud|Sigmund Freud]] and key concepts (e.g., [[w:Dreams|dreams]]) and links to book chapters about related topics (e.g., would you like to learn about how to overcome [[Motivation and emotion/Book/2020/Writer's block|writer's block]]?). ===Tables=== Tables can be an effective way to organise and summarise information. Tables should be captioned (using APA style) to explain their relevance to the text. Plus each table should be referred to at least once in the main text (e.g., see Table 1 and Table 2). Here are some [[Motivation and emotion/Wikiversity/Tables|example 3 x 3 tables]] which could be adapted. ===Quizzes=== Quizzes are a direct way to engage readers. But don't make quizzes too hard or long. It is better to have one or two review questions per major section than a long quiz at the end. Try to quiz conceptual understanding, rather than trivia. Here are some simple quiz questions which could be adapted. Choose the correct answers and click "Submit": <quiz display=simple> {Quizzes are an interactive learning feature: |type="()"} + True - False {Long quizzes are a good idea: |type="()"} - True + False </quiz> To learn about different types of quiz questions, see [[Help:Quiz|Quiz]]. ==Conclusion== The Conclusion is arguably the most important section. It should be possible for someone to read the [[#Overview|Overview]] and the Conclusion and still get a good idea of the topic. {{tip|Suggestions for this section: * What is the answer to the question in the sub-title (based on psychological theory and research)? * What are the answers to the focus questions? * What are the practical, take-home messages? }} ==See also== Provide up to half-a-dozen [[Help:Contents/Links#Interwiki_links|internal (wiki) links]] to relevant Wikiversity pages (esp. related [[Motivation and emotion/Book|motivation and emotion book chapters]]) and [[w:|Wikipedia articles]]. For example: * [[Motivation and emotion/Book/2016/Anorexia nervosa and extrinsic motivation|Anorexia nervosa and extrinsic motivation]] (Book chapter, 2016) * [[w:David McClelland|David McClelland]] (Wikipedia) * [[Motivation and emotion/Book/2018/Loss aversion|Loss aversion]] (Book chapter, 2018) * [[w:Maslow's hierarchy of needs|Maslow's hierarchy of needs]] (Wikipedia) {{tip|Suggestions for this section: * Present in alphabetical order. * Include the source in parentheses. }} ==References== List the cited references in [[w:APA style|APA style]] (7th ed.) or [[w:Wikipedia:Citing sources|wiki style]]. APA style example: {{Hanging indent|1= Blair, R. J. R. (2004). The roles of orbital frontal cortex in the modulation of antisocial behavior. ''Brain and Cognition'', ''55''(1), 198–208. https://doi.org/10.1016/S0278-2626(03)00276-8 Buckholtz, J. W., & Meyer-Lindenberg, A. (2008). MAOA and the neurogenetic architecture of human aggression. ''Trends in Neurosciences'', ''31''(3), 120–129. https://doi.org/10.1016/j.tins.2007.12.006 Eckardt, M., File, S., Gessa, G., Grant, K., Guerri, C., Hoffman, P., & Tabakoff, B. (1998). Effects of moderate alcohol consumption on the central nervous system. ''Alcoholism, Clinical and Experimental Research'', ''22''(5), 998–1040. https://doi.org/10.1111/j.1530-0277.1998.tb03695.x }} {{tip|Suggestions for this section: * Important aspects for APA style include: ** Wrap the set of references in the hanging indent template. Using "Edit source": <nowiki>{{Hanging indent|1= the full list of references}}</nowiki> ** Author surname, followed by a comma, then author initials separated by full stops and spaces ** Year of publication in parentheses ** Title of work in lower case except first letter and proper names, ending in a full-stop. ** Journal title in italics, volume number in italics, issue number in parentheses, first and last page numbers separated by an en-dash(–), followed by a full-stop. ** Provide the full doi as a URL and working hyperlink * Common mistakes include: ** incorrect capitalisation ** incorrect italicisation ** providing a "retrieved from" date (not part of APA 7th ed. style). ** citing sources that weren't actually read or consulted }} ==External links== Provide up to half-a-dozen [[Help:Contents/Links#External_links|external links]] to relevant resources such as presentations, news articles, and professional sites. For example: * [https://students.unimelb.edu.au/academic-skills/explore-our-resources/essay-writing/six-top-tips-for-writing-a-great-essay Six top tips for writing a great essay] (University of Melbourne) * [http://www.skillsyouneed.com/write/structure.html The importance of structure] (skillsyouneed.com) {{tip|Suggestions for this section: * Only select links to major external resources about the topic * Present in alphabetical order * Include the source in parentheses after the link }} [[Category:{{#titleparts:{{PAGENAME}}|3}}]] [[User:U3216256|U3216256]] ([[User talk:U3216256|discuss]] • [[Special:Contributions/U3216256|contribs]]) 05:00, 19 August 2022 (UTC) ai9h35z3x6r7dhyz74kihilt3dfw4n5 6 286550 2416461 2022-08-19T09:02:45Z Young1lim 21186 {{Information |Description=5MRV.3B: Stationary Random Process Examples (20220817 - 20220816) |Source={{own|Young1lim}} |Date=2022-08-19 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=5MRV.3B: Stationary Random Process Examples (20220817 - 20220816) |Source={{own|Young1lim}} |Date=2022-08-19 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} a5zsy48iuq2lqxfkii36igu7523jit0 6 286551 2416463 2022-08-19T09:05:19Z Young1lim 21186 {{Information |Description=5MRV.3B: Stationary Random Process Examples (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-19 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} wikitext text/x-wiki == Summary == {{Information |Description=5MRV.3B: Stationary Random Process Examples (20220818 - 20220817) |Source={{own|Young1lim}} |Date=2022-08-19 |Author=Young W. Lim |Permission={{cc-by-sa-3.0,2.5,2.0,1.0}} }} == Licensing == {{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}} qt6kdmtop6vzyajjujzz30tddngo01x